p-adic elliptic polylogarithms and arithmetic applications

P -ADIC ELLIPTIC POLYLOGARITHMS AND ARITHMETIC APPLICATIONS

Thesis submitted in partial fulfillment of the requirement for the degree of “DOCTOR OF PHILOSOPHY”

by Noam Solomon

Submitted to the Senate of Ben-Gurion University of the Negev

December 2008 Beer-Sheva

P -ADIC ELLIPTIC POLYLOGARITHMS AND ARITHMETIC APPLICATIONS

Thesis submitted in partial fulfillment of the requirement for the degree of “DOCTOR OF PHILOSOPHY”

by Noam Solomon

Submitted to the Senate of Ben-Gurion University of the Negev

Approved by the advisor Approved by the Dean of the Kreitman School of Advanced Graduate Studies

December 2008 Beer-Sheva

This work was carried out under the supervision of Prof. Amnon Besser

In the Department of Mathematics Faculty of Natural Sciences

BEN-GURION UNIVERSITY DEPARTMENT OF MATHEMATICS The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled “p-adic elliptic polylogarithms

and arithmetic applications”

by Noam Solomon in partial fulfillment of the requirements for the degree of “DOCTOR OF PHILOSOPHY”.

Dated: December 2008

External Examiner:

Research Supervisor: Amnon Besser

Examing Committee:

ii

BEN-GURION UNIVERSITY Date: December 2008 Author:

Noam Solomon

Title:

p-adic elliptic polylogarithms and arithmetic applications

Department: Mathematics Degree: Ph.D.

Convocation: December

Year: 2008

Permission is herewith granted to Ben-Gurion University to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions.

Signature of Author THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, AND NEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAY BE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR’S WRITTEN PERMISSION. THE AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINED FOR THE USE OF ANY COPYRIGHTED MATERIAL APPEARING IN THIS THESIS (OTHER THAN BRIEF EXCERPTS REQUIRING ONLY PROPER ACKNOWLEDGEMENT IN SCHOLARLY WRITING) AND THAT ALL SUCH USE IS CLEARLY ACKNOWLEDGED.

iii

To my grandfathers Pesach Pluda and Avraham Solomon

iv

Table of Contents Table of Contents

v

Abstract

ix

Acknowledgements

xi

Summary of Research 0.1 The elliptic polylogarithm . . . . . . . . . . . . . . . . 0.1.1 The motivic polylogarithm . . . . . . . . . . . . 0.1.2 The absolute class of the elliptic polylogarithm 0.1.3 The de Rham class of the elliptic polylogarithm 0.2 Keywords . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 3 13

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Introduction 0.2.1 Objectives of the thesis . . . . . . . . . . . . . . . . . . . . . . 0.2.2 Importance of the work, scientific background and works published on the subject . . . . . . . . . . . . . . . . . . . . . . .

15 15

1 Preliminaries 1.1 Hodge theory and the logarithmic pro-sheaf . . . . . . 1.1.1 Algebraic variations of mixed Hodge structures 1.1.2 Relatively unipotent variations . . . . . . . . . 1.2 The logarithmic pro-sheaf . . . . . . . . . . . . . . . . 1.2.1 Other incarnations of Log (1) . . . . . . . . . . . 1.2.2 Log (n) . . . . . . . . . . . . . . . . . . . . . . .

20 20 20 25 25 28 32

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15

2 Algebraic realization of the polylogarithms for a single elliptic curve 34 2.1 Explicit description of Log (1) over a single elliptic curve . . . . . . . . 35

v

2.2 2.3 2.4

2.5

2.6

Description of the algebraic de Rham cohomology class corresponding to Log (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of the underlying vector bundle with connection of Log (1) (1) 1 Description of the residue map on HDR (U0 , LogU0 (1)) . . . . . . . . . 2.4.1 A general remark on the residue map in the category V . . . . (1) 1 2.4.2 Explicit computation of the residue map on HDR (U0 , LogU0 (1)) Polylogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 the logarithm sheaf . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 The residue maps . . . . . . . . . . . . . . . . . . . . . . . . . (n) 1 2.5.3 Explicit computation of HDR (U0 , LogU0 (1)) . . . . . . . . . . 2.5.4 The period matrix for Pol(n) . . . . . . . . . . . . . . . . . . . 2.5.5 Computing Pol(n) . . . . . . . . . . . . . . . . . . . . . . . . . Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Elliptic polylogarithms over a general base scheme 3.1 Motivation: the classical polylogarithms over P1 − {0, 1, ∞} . . . . . 3.2 The elliptic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Eisenstein functions and series . . . . . . . . . . . . . . . . . . 3.2.2 Geometric construction of the elliptic logarithmic extension . . 3.2.3 The corresponding multiplier . . . . . . . . . . . . . . . . . . 3.2.4 Algebraization of Log . . . . . . . . . . . . . . . . . . . . . . . 3.3 Quasi-Hodge sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Definition of Quasi-Hodge sheaves . . . . . . . . . . . . . . . . 3.4 The Euler elliptic polylogarithmic extension . . . . . . . . . . . . . . 3.4.1 The elliptic polylogarithm functions . . . . . . . . . . . . . . . 3.4.2 A two variable Jacobi form . . . . . . . . . . . . . . . . . . . . 3.4.3 computation of ∇(Liell ) . . . . . . . . . . . . . . . . . . . . . 3.4.4 Another computation of ∇(Liell ) . . . . . . . . . . . . . . . . 3.5 Algebraization of Liell . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Kronecker theta function . . . . . . . . . . . . . . . . . . . . . 3.5.2 Explicit formula for Lell al . . . . . . . . . . . . . . . . . . . . . . 3.5.3 The algebraization of the Euler polylogarithmic extension Falell 3.6 The elliptic polylogarithm . . . . . . . . . . . . . . . . . . . . . . . . 0 ell 3.6.1 The extension PalV . . . . . . . . . . . . . . . . . . . . . . . . 00 ell ˜ 3.6.2 The extension PalV . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 The construction of the elliptic polylogarithm . . . . . . . . . 3.7 Algebraic description of the elliptic polylogarithms . . . . . . . . . . 3.8 The C ∞ realization of the elliptic polylogarithm . . . . . . . . . . . .

vi

35 37 38 38 39 40 40 42 43 45 46 50 52 52 57 57 58 60 63 66 66 69 69 71 72 76 79 79 82 84 85 85 86 88 94 97

3.8.1 3.8.2 3.8.3 3.8.4 3.8.5 3.8.6 3.8.7

Kronecker’s double serier . . . . . . . . . . R-Hodge structures . . . . . . . . . . . . . . Quasi-Hodge sheaves . . . . . . . . . . . . . The Eisenstein-Kronecker series . . . . . . . The Eisenstein-Kronecker series - continued The g/L correspondence in the C ∞ category The relation between the R- and Q-Hodge elliptic polylogarithm . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . structures . . . . . .

4 The syntomic theory 4.0.8 Functoriality . . . . . . . . . . . . . . . . . . . . . 4.0.9 Comparison between de Rham and rigid complexes 4.0.10 Absolute geometric cohomology . . . . . . . . . . . 4.0.11 Syntomic cohomology . . . . . . . . . . . . . . . . . 4.0.12 Admissible coefficients . . . . . . . . . . . . . . . .

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. . . . 97 . . . . 97 . . . . 98 . . . . 99 . . . . 101 . . . . 103 of the . . . . 108

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111 121 124 127 127 129

5 The Leray spectral sequence for syntomic cohomology 141 5.1 Paranjape’s work and homological algebraic background . . . . . . . 141 5.1.1 First proposition . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.1.2 Second proposition . . . . . . . . . . . . . . . . . . . . . . . . 143 5.1.3 Third proposition . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.1.4 Fourth proposition . . . . . . . . . . . . . . . . . . . . . . . . 147 5.1.5 Fifth proposition . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.1.6 Sixth proposition . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.2 The spectral sequence of a mapping cone . . . . . . . . . . . . . . . . 149 5.2.1 The spectral sequence of a good filtered complex . . . . . . . . 149 5.3 The l’th mapping cone spectral sequence corresponding to a mapping cone of two filtered complexes . . . . . . . . . . . . . . . . . . . . . . 153 5.4 Algebraic differential equations with regular singular points . . . . . . 155 5.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.4.2 Remarks on the definition of regular singular points . . . . . . 156 5.4.3 Relative de Rham cohomology revisited . . . . . . . . . . . . . 157 5.5 Canonical extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.6 Generalized semi-stable morphisms . . . . . . . . . . . . . . . . . . . 161 5.7 The regularity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.7.1 Good compactification . . . . . . . . . . . . . . . . . . . . . . 164 5.7.2 Logarithmic geometry . . . . . . . . . . . . . . . . . . . . . . 170 5.7.3 The logarithmic Riemann Hilbert correspondence (after [50] 0.4) 179 5.7.4 Proof of Theorem 5.7.16 . . . . . . . . . . . . . . . . . . . . . 180 vii

5.8 5.9 6 The 6.1 6.2 6.3 6.4 6.5

Relative geometric cohomology and π−admissible coefficients . . . . . 184 The Leray spectral sequence theorem . . . . . . . . . . . . . . . . . . 191 syntomic elliptic polylogarithm Gysin isomorphism and localization sequences . . . The first logarithm Log (1) . . . . . . . . . . . . . . The nth logarithm Log (n) . . . . . . . . . . . . . . Definition of the elliptic syntomic polylogarithm . . Comparison with the motivic elliptic polylogarithms 6.5.1 The motivic polylogarithm . . . . . . . . . .

Bibliography

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197 197 198 203 204 207 208 219

viii

Abstract Elliptic polylogarithms are certain elements in the K−theory of elliptic curves. In this thesis we develop a theory of syntomic coefficients, which is the p-adic analog of variation of mixed Hodge structures, using which we define syntomic cohomology theory with syntomic coefficients, which generalize [5] and [15]. We then use this theory to describe the syntomic classes of the elliptic polylogarithms, and show that they agree with the realization of the motivic elliptic polylogarithm. In Chapter 1 we give an introduction to the subject. In Chapter 2 we provide a procedure to compute the algebraic de Rham classes of the polylogarithms based on their residues in the case where the base field is specK. In Chapter 3 we do the same for the case of a general base scheme S, using a different approach. This gives explicit formulae for the algebraic de Rham classes of the polylogarithms, generalizing a result of [7] that treated only the case S = spec(K). In Chapter 4 we develop the syntomic theory. We define the p-adic analog of variation of mixed Hodge structures, which are called syntomic coefficients, and then define a cohomological theory, called syntomic theory with coefficients, following [5] and [15]. In Chapter 5 we state and prove the Leray spectral sequence theorem for syntomic cohomology with coefficients. In Chapter 6 we define the syntomic elliptic polylogarithms as elements of a certain syntomic cohomology group with coefficients. We then prove that these elements are ix

x

the realizations of the elliptic motivic polylogarithms.

Acknowledgements I would first like to thank Professor Amnon Besser, my supervisor, for his many suggestions and constant support during this research. His mathematical vision and intuition were and still are an inspiration to me. It was a privilege for me to go on this mathematical journey with him. I would also like to thank him for his friendship. Second, I think this is a good opportunity to thank Professor Beno Arbel for his help and guidance during my undergraduate studies. I would also like to thank Professor Amnon Yekutieli and Doctor Barak Weiss for interesting discussions over the years. Professor Kenichi Bannai expressed his interest in this work and supplied us with the preprints of some of his recent works, which gave us a better perspective on some of the results in this thesis. He was always extremely kind and helpful, and for this I would like to express my utmost gratitude. I would like to thank Professor Joseph Bernstein for valuable discussions. I would like to thank Professor Vladimir Berkovich for useful advice. I had the pleasure of meeting Professor P. Deligne. Our conversation was very helpful. I would like to thank Professor N. Katz, Professor F. Kato, Professor J. Steenbrink, Professor E. Grosse-Klönne, and Professor T. Fujisawa for their advice. I also want to thank Professor Yefim Dinitz for our joint work, for his friendship and support, and for believing in me in the early stages of my studies. Above all, I am grateful to my parents for their patience and love. Without them this work would never have come into existence (literally); to my mother Ma’ayana, for loving me at times when I’m least lovable, for her constant caring through my life, and for the tears I made her cry; to my father Arie, the most generous person I know, for being the rock we all lean on, and for teaching me what giving is all about.

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Finally, I wish to thank the following: Ya’ara, Hadas, and Anat (for their love); Hagar and Ronen (for their friendship); Ori (Parzan) for his help on the last minute; My aunt Chaviva for her constant support and invaluable help with my grandparents; my grandmother Tzipora for everything she has given me with all her heart and love (I always think of you); my grandmother Rachel for her love, caring, and generosity and for the smell of fine cuisine that fills my childhood memories; I would also like to mention my grandfathers to whom this dissertation is dedicated; Pesach Pluda, who influenced me in so many ways. I will always cherish the memories I have from him, and remember his determination and strong spirit, together with the faith he had in me; Avraham Solomon, for the love that was always spread in his home; my beloved baby sister Michal, who I love with all my heart, for being; and my baby twin brother Shay for the strong bond we have had since day one, and for teaching me a valuable lesson in life: winning is a state of mind. Beer-Sheva, Israel December 15, 2008

Noam Solomon

Summary of Research This thesis consists of six chapters. The first chapter introduces preliminary material that the reader should be familiar with upon reading the thesis; the notion of (algebraic) variation of mixed Hodge structures and the notion of logarithmic pro-sheaf. These notions are defined over the complex numbers, and are considered classical. In the thesis, amongst other things, we develop p-adic analogs of these notions in very general settings.

The other chapters of the thesis systematically develop the theory of p-adic elliptic polylogarithms. These polylogarithms have motivic origin.

0.1

The elliptic polylogarithm

The notion of Polylogarithmic extensions on families of Abelian varieties (or more generally on mixed Shimura varieties) was introduced by Jorg Wildeshaus [81], relying on the important work of [9], which is the elliptic analog to the construction of [8]. The best exposition for this subject is found in [56], which we explain below.

0.1.1

The motivic polylogarithm

We follow Kings’s definition of the motivic polylogarithm, where the abelian variety is an elliptic curve.

1

2

Definition 0.1.1. For a regular scheme X we denote by i HM (X, j) := Grγj K2j−i (X) ⊗ Q, 0 HiM (X, j) := Grjγ Ki−2j (X) ⊗ Q,

the motivic cohomology and homology of X. Here K• and K•0 are Quillen’s K-groups. For their properties see [74]. The setup is as follows: E is an elliptic curve over a base S. Denote by e : S → E its zero section, and by π the projection from E to S. We define U = (E − e(S)) ×S (E − e(S)) = {(p1 , p), πp1 = πp, p1 , p 6∈ e(S)} . Now we define V ⊂ U to be the complement of the diagonal section ∆ : (E − e(S)) → U , so that V = {(p1 , p) ∈ U, p1 6= p} . We consider these as schemes over E − e(S) via the second projection. For every scheme W/T we define its n-th fiber power W n := W ×T . . . ×T W n-times. We can write U n as follows: U n = {(p1 , . . . , pn , p), πpi = πp, pi , p 6∈ e(S)}. Next, for every I ⊂ {1, . . . , n} we define V I ⊂ U n by V I = {(p1 , . . . , pn , p) ∈ U n , pi = p iff i ∈ / I}. The following holds (1) •−n • HM (V n , ∗)signn ∼ (E \ e(S), ∗ − n)(1) = HM

where the sign operation is defined in 6.6.5, and the sign eigenspace is taken with respect to it. Since all of the above is valid over an arbitrary base, we can replace E by the elliptic curve E ×S E considered as an elliptic curve over E (base change), and obtain:

3

Definition 0.1.2. The n0 th motivic polylogarithm corresponding to E/S is the ele(1) n+2 ment of the motivic cohomology group HM (E × V n , n + 1)signn mapping under the isomorphism (1) n+2 2 HM (E × V n , n + 1)signn ∼ (E × (E \ e(S)), 1)(1) = HM

to the class of the diagonal. One of the main goals in our research was to compute the p-adic regulator of these elements. To this end, we first compute the de Rham Chern classes of these elements.

0.1.2

The absolute class of the elliptic polylogarithm

Following [9] sections 6.1.11–6.1.12, Kings showed ([56] corollary 2.3.3) that the regulator to the absolute cohomology maps the nth motivic polylogarithm to the image of the small polylogarithm PolE , defined as the class in Ext1V(E−e(S)) (πE∗ H∨ , LogE (1)) ∼ = HomS (H∨ ,

Y

Symk H∨ )

k>0

corresponding to the map in

Q

∨

k>0

k

∨

HomS (H , Sym H ), which is the identity for

k = 1, and zero for k 6= 1, where H is the first cohomology group of E over S, LogE is the logarithm pro-sheaf, and V(E − e(S)) is the category of algebraic variation of mixed Hodge structures (cf. definition 1.1.1). This work motivated our p-adic analog.

0.1.3

The de Rham class of the elliptic polylogarithm

Using the work of Kings [56], we observed that the algebraic de Rham realization of the elliptic polylogarithm corresponds to a map from H∨ to R1 πE−e(S) ∗ Log, sending h ∈ H∨ to the algebraic differential form with values in LogE\e(S) (1) whose residue at e(S) is h. Chapter 2 is devoted to the description of these algebraic classes in the case S = spec(K), where K is any field of characteristic 0. Let V be the pullback of the first relative homology group of E/S to E (cf. Definition 1.2.5. The first logarithmic sheaf Log (1) is defined by the characterization of Lemma 1.2.12. This characterization

4

1 1 implies that given some basis ω1 , ω2 to HDR (E), the class in HDR (E, V) corresponding

to Log (1) is ω1 ⊗ ω1∨ + ω2 ⊗ ω2∨ . Therefore Log (1) is a vector bundle with integrable connection. It admits also a Hodge and weight filtration (cf. Equations 2.3.3, 2.3.4) and therefore Log (1) is an element of V(E). We define the logarithmic pro-sheaf Log := lim Symn Log (1) ∈ V(E), where the ←− n connection is given by ∇Log = ∇S • + ω1 ⊗ ω1∨ + ω2 ⊗ ω2∨ .

(0.1.3)

(n)

1 (E \ e(S), LogE\e(S) (1)) . We give an explicit basis for HDR

We are taking here ω1 =

dx , ω2 y

=

xdx . y (n)

1 (E \ e(S), LogE\e(S) (1)) is given Theorem 0.1.4. With our notations a basis to HDR by:

{

xdx j k ⊗ ei0 ⊗ ω1∨ ⊗ ω2∨ , i + j + k = n, y

0 ≤ i, j, k}

[ dx n−i { ⊗ ei0 ⊗ ω2∨ , i = 0, . . . , n}. y

Definition 0.1.5. The small polylogarithm Pol (we omit the subscript for E) is defined as the class in Ext1E\e(S) (V, LogE\e(S) (1)), corresponding under the isomorphism 6.5.10

1 Ext1E\e(S) (V, LogE\e(S) (1)) ∼ (E \ e(S), LogE\e(S) (1)) ∼ = HomS (H∨ , HDR =

HomS (H∨ ,

Q

k ∨ k>0 Sym H ) =

Q

k>0

HomS (H∨ , Symk H∨ ),

Q to the map in k>0 HomS (H∨ , Symk H∨ ) which is the identity for k = 1, and zero for (n) k 6= 1. Poln are defined as the projection of Pol to Ext1E\e(S) (V, LogE\e(S) (1)). It follows from the definition that finding Pol(n) is equivalent to finding two differ(n)

(n)

(n)

1 ential forms ζ1 , ζ2 ∈ HDR (E \ e(S), LogE\e(S) (1)), whose modified residue satisfies \ \ (n) (n) (n) (n) Res (ζ ) = ω ∨ , Res (ζ ) = ω ∨ . ∞

1

1

∞

2

2

5

(n)

1 Recalling the explicit basis to HDR (E \ e(S), LogE\e(S) (1)) presented in Theorem (n)

2.5.13, we write ζi X

as

xdx j ai,n e0 j,k,l y

{j,k,l:j+k+l=n}

We have ai,n j,k,l =

⊗

(ω1∨ )k

⊗

(ω2∨ )l

+

n X j=0

bi,n j

dx j e0 ⊗ (ω2∨ )n−j . y

(0.1.6)

1 i,k+l a . j! 0,k,l

Let us fix throughout this section i = 1 (the case

i = 2 is treated similarly).

We may view the series {ai,n j,k,l } as a series in two

variables {ak,l }. We investigate this series via its two variable generating function P P (x, y) := k,l≥0 ak,l · xk · y l . We similarly may regard {bi,n j } as a series in one variP able {bn } and investigate its generating function Q(x) = n≥0 bn · xn . 1 Taking the infinite version of Theorem 0.1.4, gives an explicit basis to HDR (E \ (n)

e(S), LogE\e(S) (1)). We take the projective limit over n of the ζ1 tion 0.1.6

and rewrite Equa-

∞ X xdx ∨ k dx ∨ l ak,l (ω1 ) ⊗ (ω2 ) + bn (ω2∨ )n . y y n=0 k,l≥0

X

(0.1.7)

The main result of Chapter 2 is the following equation in x, y for P (x, y), Q(x, y)

Rest=0 ((P (ω1 , ω2 ) ·

dx xdx + Q(ω2 ) · ) · exp(F (t) · ω1 + G(t) · ω2 )) = ω1 . y y

(0.1.8)

We also showed that this equation has a unique solution. In Chapter 3 we treat the general case of a general base scheme S. Following the approach of [9] we found explicit formulae for these algebraic de Rham classes, generalizing a result of [7] that treated the case S = spec(K). There is a complex analytic description of the elliptic polylogarithms found by [9]. For an algebraic description of the elliptic polylogarithms, the following function, introduced by Kronecker [58], turns out to be fundamental. P 1 1 m n −m −n mn F (ξ, η; τ ) := (2πi) 1 − 1−z − 1−w − ∞ w )q , m,n=1 (z w − z (0.1.9) =(τ ) > =(ξ) > 0, =(τ ) > =(η) > 0,

6

where q := exp(2πiτ ).

We recall the Weierstrass σ-function σ(z; τ ) = z

Y

1−

γ∈Γτ \0

z z2 z exp + 2 . γ γ 2γ

We have the following theta function θ(z; τ ), whose relation to the Weierstrass σ-function is e∗ θ(z; τ ) = exp − 2 z 2 σ(z; τ ), 2

(0.1.10)

where e∗2 together with this definition appears in Subsection 3.5.1. We define the Kronecker theta function Θ(z, w; τ ) to be the function Θ(z, w; τ ) :=

θ(z + w; τ ) . θ(z; τ )θ(w; τ )

(0.1.11)

This function is a reduced theta function (cf. [6] definition 1.8) associated to the Poincaré line bundle of C/Γτ . We have Θ(z, w; τ ) = exp(zw/A)F (z, w; τ ),

(0.1.12)

where F (z, w; τ ) is defined in Equation 3.4.8. Definition 0.1.13 ([6] definition 1.3). Define the function Ξ(z, w; τ ) := exp(−F1 (z; τ )w)Θ(z, w; τ ).

(0.1.14)

The main result of Chapter 3 is the explicit algebraic description of the elliptic polylogarithms, which uses the “algebraic” function Ξ(z, w; τ ). This is the content of Section 3.7. Next, we investigate the relation between the C ∞ realization, expressed via the Eisentein-Kronecker series and the Q Hodge realization as described above. This is described in detail in Section 3.8.

7

In Chapter 4 we develop the syntomic theory. We Define the p-adic analog of variation of mixed Hodge structures, which are called syntomic coefficients, and then define a cohomological theory, called syntomic theory with coefficients, following [5], [15]. Our main goal is to find the syntomic classes of the motivic elliptic polylogarithms. To this end we first need to define the category where these elements “live”; our approach is an amalgamation of the methods developed in [15] and [5]. In this Chapter, we assume that K is a finite extension of Qp with ring of integers OK and residue field k. We let K0 be the maximal unramified extension of Qp in K and W its ring of integers. σ is the lifting to K0 of the Frobenius automorphism on k. We also assume given a scheme X, smooth and of finite type over OK .

The philosophy behind the syntomic mechanism creates a sophisticated way to patch together two classical “worlds”.

The first of these two “worlds” is the algebraic world, which is generally referred to as the de Rham world. In a sense, one might say that this world is the classical one. A de Rham datum for X is a compactification of XK . Definition 0.1.15. Let Y be a de Rham datum for XK . We define CdR (Y ) to be the category of vector bundles with integrable connection on Y , having logarithmic singularity along D := Y \ XK . We define SdR (XK ) := limY CdR (Y ), where the limit −→ is over all de Rham data. For an element MdR ∈ CdR (Y ), we say that Y is a de Rham datum for MdR . The second “world” is the rigid analytic world, which took more time to be discovered. We refer to [11] for a thorough reading on the subject. Definition 0.1.16. We define Srig (Xk ) to be the category of F -isocrystsals overconvergent over Xk /K0 , σ (cf. [11], Definitions 2.3.6 and 2.3.7).

8

In order to glue two objects, an intersection face along which the objects are glued is required. In our case, this intersection is a third “world” which, in a sense, underlies both worlds described above.

Definition 0.1.17. We define Svec (X) to be the category of isocrystsals over X/K. The patching is done via a general homological algebraic construction, which we called “GLUE” (cf. definition 4.0.21), and is given here without details. The definition for the syntomic world is given by: Definition 0.1.18 (definition 4.0.26). We define the category of syntomic coefficients on X, Ssyn (X) to be GLUE(SdR (XK ), Svec (Xk ), Srig (Xk ), Fan , FK ). For M := (MdR , Mrig , ϕ) ∈ Ssyn (X), let Y 0 be a de Rham datum for MdR . For a • de Rham datum i : Y ,→ Y 0 , let D = Y \ XK , and define DRdR (M )Y by • DRdR (M )Y = i∗ MdR ⊗ Ω•Y (logD).

(0.1.19)

This is a filtered complex with filtration given on each term by q F m DRdR (M )Y = i∗ F m−q MdR ⊗ ΩqY (logD).

(0.1.20)

• DRrig (M ) = Mrig ⊗ Ω•]X k [P ,

(0.1.21)

We define X

where the complex Ω•]X

k [P X

is independent up to canonical isomorphism of the

choice of rigid datum and similarly for the vector datum The cohomology of these complexes gives the corresponding de Rham and rigid cohomologies. This construction is functorial in X as we showed in Subsection 4.0.8. There is also a comparison morphism between de Rham and rigid complexes.

θk : F k RΓdR (XK /K, M ) → RΓrig (X/K0 , M ) ⊗W OK ∼ = RΓrig (X/K, M ), (0.1.22)

9

which is functorial in X, M . There is also an action of the Frobenius ϕM on RΓrig (X/K0 , M ). We then define the notion of absolute geometric cohomology, which is in a sense a p-adic analog of the singular cohomology Definition 0.1.23. Let M be a syntomic datum in Ssyn (X). We define the ith geometric cohomology of X with coefficients in M , denoted HAi (X, M ) = (MdR , Mrig , ϕ) to be the following element of Ssyn (K): i (XK /K, M ) with Filtration defined in 4.0.33. 1. MdR = HdR i (Xk /K0 , M ) with Frobenius induced by ϕM . 2. Mrig = Hrig i i (Xk /K0 , M ) ⊗W K induced by θ. 3. ϕ : HdR (XK /K, M ) → Hrig

It is also functorial and satisfies the expected properties. We then arrive at one of the highlights of this Chapter, namely, the definition of syntomic cohomology with coefficients, generalizing [14], [5]. Definition 0.1.24. The syntomic complex of X with coefficient M twisted by n is defined to be ∼ ϕM RΓsyn (X, M ; n) := Cone(1 − n )[−1] ×RΓrig (Xk /K,M ) F n (RΓdR (XK /K, M )). p i (X, M ; n). The i’th cohomology of RΓsyn (X, M ; n) will be denoted Hsyn

When n is omitted, we assume it is set to 0. We have functoriality with respect to morphisms of schemes over OK . Again, all the expected properties which are proved in [14] and [5] are generalized. For many purposes, we need to work with certain classes of coefficients which are called admissible (cf. Definition 5.8.12). The following is a fundamental theorem in our theory. Theorem 0.1.25 (Theorem 4.0.70). Let M be an admissible syntomic coefficient over X. Then there exist canonical and functorial isomorphisms, both denoted by Ξ, that is: 1 (X, M ), Ξ : Ext1Ssyn (X) (K(0), M ) → Hsyn

10

1 Ξ : Ext1Srig (X) (K(0), M ) → Hrig (X, M ),

such that the obvious diagram connecting these maps (via the forgetful functors) commute. In Chapter 5 we state and prove the Leray spectral sequence theorem for syntomic cohomology with coefficients. For this, we proved a new theorem in homological algebra, namely Theorem 0.1.26. Let (A• , F ), (B • , F ) be good filtered complexes (cf. Definition 5.1.7), together with a filtered morphism ϕ : A• → B • . Let D• := M C(ϕ) be their mapping cone. That is, Dn = An ⊕ B n−1 and dD (a, b) = (−da, db − ϕ(a)). Given l ∈ N+ , there exists a filtration F on D• such that the spectral sequence associated to the good filtered complex (D• , F ), Erp,q (D• , F ) ⇒ H p+q (D), satisfy

p,q p,q Elp,q (D• , F ) ∼ = M C(ϕ : El (A, F ) → El (B, F )).

Following [27], we reintroduce the notion of regularity. That is, we work with syntomic coefficients whose de Rham part has regular singular points along the divisors. ¿From this point on, it is assumed that K ⊂ C is a number field, so that complex analytic methods are applicable here. We intend to relax this condition in the future. There is then the notion of canonical extension. Definition 0.1.27. Let (M, ∇) be a regular algebraic differential equation on X with logarithmic singularities along D, as in Definition 5.4.3. For each branch Di of D, we denote by M (Di ) the locally free sheaf ODi ⊗OX M on Di . Composing ∇ with the map “residue along Di ” Ω1X (log D) ⊗OX M

“residue along Di00 ⊗1

−→

ODi ⊗OX M = M (Di ).

(0.1.28)

We obtain an ODi -linear endomorphism Li of M (Di ). As Di is proper, the characteristic polynomial of Li , Pi (x) = det(xI − Li ; M (Di )) lies in K[x]. We call Pi the indicial polynomial of (M , ∇) around Di , and its roots are called exponents of (M , ∇) around Di . Lemma 0.1.29. The numbers exp(2πi), an exponent, are the eigenvlues of the local monodromy transformation “turning once around Di ” of the space of local holomorphic horizontal sections of (M , ∇)|X\D .

11

Corollary 0.1.30. The exponents that depend on M , are determined modulo Z by (M , ∇)|X\D . Definition 0.1.31. Let QR(X) denote the full subcategory of algebraic differential equations, whose objects are quasi-unipotent and regular, where quasi-unipotent means that the exponents of the monodromy over points of X \ X are rational. This condition is independent of the compactification by Corollary 5.5.4. Proposition 0.1.32. Let (M, ∇) ∈ QR(X). There exists a unique algebraic differential equation (M , ∇) over X verifying 1. (M , ∇)|X = (M, ∇). 2. The residue Resi (Γ) of the connection along Di , (1 ≤ i ≤ n) has eigenvalues λ with 0 ≤ λ < 1. This canonical extension admits a filtration extending the filtration on (M, ∇). Moreover, the functor M → M is functorial for horizontal maps, is exact, and is compatible with Hom, tensor products and exterior powers. Definition 0.1.33. We call the extension (M , ∇) established by Proposition 5.5.6 the canonical extension of (M, ∇). QR (X) denote the full subcategory of Ssyn (X) whose objects Definition 0.1.34. Let Ssyn consist of all syntomic coefficients M whose de Rham part (MdR , ∇dR , F • ) is the canonical extension of (MdR |X , ∇dR |X , F • |X ).

¿From now on, we restrict attention to the category of quasi-unipotent regular QR syntomic coefficients, namely, to Ssyn (X).

For the proof of the Leray spectral sequence, we assume that our smooth morphism π : X → S admits a good compactification. This is a technical condition, which means that π : X → S comes from a log smooth morphism π : Y → T of proper schemes. We define Ω1Y /T (log D) to be the subsheaf of j∗ Ω1X/S having logarithmic singularities along D. Then Ω1Y /T (log D) is locally free. We define the locally free OY -modules Ω•Y /T (log D)

:=

p ^

Ω1Y /T (log D).

12

By our assumptions, the bundle ΩiY /T (log D) is a subbundle of j∗ ΩiX/S , and a quotient of ΩiY (log D). Thus, it is induced with a differential operator, and Ω•Y /T (log D) becomes a complex of locally free OY -modules. The following theorem is one of our important theorems. Theorem 0.1.35. With the above notations, if (M, ∇) ∈ QR(X), q q (X/S, (M, ∇))), δq ). then (F p (HdR (Y /T, (M , ∇))), δ q ) is the canonical extension of (F p (HdR The next ingredient needed for the proof of the Leray spectral sequence theorem is to show that the higher direct images have the structure of a syntomic coefficient. For this we introduced the notion of relative geometric cohomology and π−admissible coefficients. Many results generalize from the global to the relative case. Ideally, the syntomic cohomology would be independent of the choice of the particular de Rham datum for X. Unfortunately, this is not always the case. However, it turns out that when the de Rham coefficient has regular singular points with non-integral exponents and local unipotent monodromy this is true. This allows us to define the syntomic cohomology for these “nice” coefficients. We have Theorem 0.1.36. Let M = (MdR , Mrig , Mvec ) ∈ Ssyn (X) and assume MdR has regular singular points. Assume MdR is defined over some de Rham datum Y . Then the i syntomic cohomology Hsyn (X, M ) may be computed by replacing RΓdR (XK /K, M ) • in Definition 4.0.56 with RΓ(Y, DRdR (M )Y ). The Leray spectral sequence theorem is the following: π−ad,QR Theorem 0.1.37. Let M ∈ Ssyn (X). Then, there is a Leray spectral sequence for syntomic cohomology q p p+q E2p,q = Hsyn (S, HA (X/S, M )) ⇒ Hsyn (X, M ),

(0.1.38)

q where the connection on the complex Ω•S,syn ⊗ HA (X/S, M )) is induced by the GaussManin connection (cf. [52], [77], [33]).

13

Using the Leray spectral sequence theorem, and the machinery developed in the previous Chapters, in Chapter 6 we define the syntomic elliptic polylogarithm, as an element of a certain syntomic cohomology group with coefficients. To be precise: Definition 0.1.39. We define the syntomic elliptic polylogarithm to be the system (n) 1 of elements polE ∈ Hsyn (E \ e(S), H∨ ⊗ Log (n) (1)) such that δ(lim pol(n) )E = id ∈ End(H). ←− n

It remains to establish the fact that the elliptic polylogarithm defined in 6.4.10 coincides with the syntomic realization of the motivic elliptic polylogarithm defined by definition 6.5.8. To this end, following the method of [56] corollary 2.3.3, we need to prove: Theorem 0.1.40. There is an isomorphism n+2 Ext1Ssyn (E\e(S)) (H, Log (n) (1))(1,1) ∼ (E ×S V n , n + 1)(1,1) = Hsyn sgnn ,

And residue maps, which appear in Corollary 6.5.5, that give isomorphisms Ext1Ssyn (E\e(S)) (H, Log (n) (1))(1,1) → Ext1Ssyn (E\e(S)) (H, Log (n) (1))(1,1) induced by Log (n) (1) → Log (n−1) (1). We then deduce what we wanted, namely Corollary 0.1.41 (compare [56] corollary 2.3.3). The syntomic regulator (1,1)

(1,1)

n+2 n+2 rsyn : HM (E × V n , n + 1)signn → Hsyn (E × V n , n + 1)signn (n)

(n)

maps PolM,E (cf. definition 6.5.8) to polE (cf. definition 6.4.10).

0.2

Keywords

Elliptic polylogarithms. Elliptic curves. Hodge theory.

14

Variations of Hodge structures. Syntomic regulator. Motivic cohomology. Syntomic cohomology. Leray spectral sequence.

Introduction The p-adic elliptic polylogarithms are the subject of this dissertation. In this Introduction we will describe their origin and explain their importance.

0.2.1

Objectives of the thesis

To develop a theory of p-adic elliptic polylogarithms, which are p-adic functions of Coleman type on an elliptic curve, analogous, on the one hand to the theory of elliptic polylogarithms as developed by [9], [34], [81] and others, and on the other hand to theory of p-adic polylogarithms as developed by Coleman. One of the main objectives is to establish a relation between elements in certain K-theories of elliptic curves representing the motivic elliptic polylogarithms and elements in certain syntomic cohomologies of elliptic curves.

0.2.2

Importance of the work, scientific background and works published on the subject

For an algebraic variety X over a field K and any non-negative integer, there corresponds a “motive”, H n (X), giving a universal description for the n’th cohomology of X. More generally, there exist motives that are parts of the motives of the algebraic varieties. A precise definition of the category of mixed motives MM, that is, of the

15

16

motives corresponding to non-projective varieties, is still sought, in spite of impressive developments over the past few years. For many practical purposes this does not pose a problem since we have realizations of the motives, that is, their cohomology, in any give cohomology theory, like singular cohomology HB , which gives us a realization which is a mixed Hodge structure, or the étale cohomology, which gives us a representation of the absolute Galois group of K. In particular, one can correspond to a motive M his L function L(M, s), which is a complex function of the complex variable s. In addition, it is possible to define motivic cohomology Ext1MM (Q, M ). This description of extensions in the undefined category of mixed motives, is not precise, and another description using algebraic K-theory or algebraic cycles is required, but is very convenient for heuristic explanations, as we do hereinafter. The Beilinson conjectures [10] are central conjectures in arithmetic geometry. They predict the special value L(M, m) for an integer number m up to a non-zero rational number, using a map, called the regulator map, from motivic cohomology to absolute. The regulator map can be described as a map Ext1MM (Q, M ) → Ext1M HS (Q, HB (M )) that is induced by the functor HB . In general, this conjecture is far from our reach. In fact, except for the theorem of Borel [21], which proves the conjecture in dimension 0, we have no full result to reinforce these conjectures fully. The main problem lies in our limited understanding of motivic cohomology. Therefore, the research in this field deals with finding partial results: certain elements in motivic cohomology are built, their regulator is computed and the result is compared to the conjectured value of the corresponding L function. The results that are relevant to us in this direction are those dealing with polylogarithms. The classical polylogarithms are multi-valued complex functions whose

17

Taylor expansion at 0 is given by Lk =

∞ X zn n=1

nk

.

˜ • (K) (the In [26] de Jeu defined for a number field K, complexes of vector spaces M (n) Bloch complex) and an injection ˜ • (K)) ,→ Ext1 (Q, K(n)) H 1 (M (n) MM where n stands for torsion. de Jeu also proved that the regulator map can be described on known generators ˜ 1 (K) using Ln . This result is known as the weak version of the Zagier {x}n of M (n) conjecture, where the strong version is that the injection defined by de Jeu is an isomorphism, which is known for n = 2, 3. This conjecture allows, using the theorem of Borel, to express the Dedekind Zeta function of number fields at positive integers using values of polylogarithms. The elliptic polylogarithm was invented by Bloch [20] for the purpose of generalizing the polylogarithm, and the regulator computation using it, to elliptic curves over number fields. The work done on this subject [9], [34], [80], [56], [63] brings us to a similar understanding to that over number fields, except for the fact that this theory is based on unproven conjectures in K theory. Under these assumptions, the elliptic Bloch complexes B • (E, n) were constructed, which conjecturally compute the motivic cohomology Ext1MM (Q, Symn−2 (H 1 (E))(n − 1)) and the regulator on the cohomology of these complexes is computed using the elliptic polylogarithm. To be more precise, one takes the real part of the elliptic polylogarithm, called the Eisenstein-Kronecker series (cf. subsections 3.8.4, 3.8.4). This series is transcendental, that is, it uses a non-algebraic series. It is worthwhile mentioning that the work of [81] uses a very deep theory of variations of mixed Hodge structures.

18

The regulator into absolute cohomology has a p-adic analog – the syntomic regulator. This regulator can be written similarly to the regulator to absolute cohomology by replacing the category of Hodge structures with that of filtered Frobneius modules and replacing the singular cohomology with the rigid cohomology. (This will be thoroughly discussed in the body of the thesis). Therefore, it seems reasonable to write a p-adic analog of the Beilinson conjectures. However, one should keep in mind that the situation here is more complicated since p-adic L functions are much more difficult for definition and computation. These conjectures were never written explicitly, but appear implicitly in [68]. However, the validity of these conjectures is known to us in several cases: cyclotomic fields [35], abelian extension of Q [57], totally real fields (for the value at 1) [25], and elliptic curves with complex multiplication (in a weaker version and for the value at 2) [15, 24]. It also seems that the conjectures may be proven, again in a weaker version, and only for the value 2 for modular curves, relying on the work of Kato [73]. There are tools for the computation of syntomic regulator [14, 15, 17, 18]. These tools rely on the p-adic integration theory of Coleman [16, 23, 24]. Applications of these tools should give p-adic analogs to classical results. In particular, the result of [17] is an a complete p-adic analog of [26] in the sense that the syntomic regulator is computed on the same elements using the p-adic polylogarithm instead of the classical polylogarithm. Besser and de Jeu also computed an algorithm to compute Coleman’s p-adic polylogarithms [19]. These works motivate the p-adic elliptic analogical questions. In this thesis we develop the theory of p-adic elliptic polylogarithms, which generalize [4]. For this, we developed a theory of syntomic coefficients, which is in a sense the p-adic analog of the notion of variation of mixed Hodge structures, and then defined a syntomic cohomology theory with syntomic coefficients in the most general settings, generalizing [14, 5].

19

One of the main goals of our project was to prepare the grounds for the formulation of the p-adic elliptic analog of the Zagier conjectures. The soil is now ready for this.

Chapter 1 Preliminaries 1.1

Hodge theory and the logarithmic pro-sheaf

1.1.1

Algebraic variations of mixed Hodge structures

The classical origin of Hodge theory lies in the discussion of periods of integrals of the first kind on algebraic curves over the complex numbers. The appropriate language to describe this uses the notion of cohomology. Hodge’s own work on the theory of Harmonic integrals gave the celebrated Hodge Decomposition Theorem: for a compact Kähler manifold X, its cohomology groups decompose as follows: H n (X, C) =

M

H p,q ,

p+q=n

with H

p,q

(X) = H q,p . This work has also led to the famous Hodge conjecture, which

asserts that for a projective complex algebraic manifold X, every rational class in H p,p is represented by a codimension p algebraic cycle with Q-coefficients. The modern era of Hodge theory began in the late 1960s with the work of Griffiths, generalizing previous works to higher dimensional varieties and higher codimensional cycles. This gave the notion of variation of Hodge structure and classifying spaces for Hodge structures; the intermediate Jacobians and Abel-Jacobi mappings. Shortly after Griffiths’ work appeared, Deligne’s work on mixed Hodge theory introduced a set of new 20

21

ideas and techniques into the subject – the notion of a mixed Hodge structure, which is, loosely speaking, an iterated extension of Hodge structures of increasing weight. The main point with these mixed structures is that the cohomology groups of any complex algebraic variety carry mixed Hodge structures. This idea was influenced by Grothendieck’s philosophy of motives. Grothendieck thought that there should exist “mixed motives”, associated to the cohomology groups of general algebraic varieties. Such motives should be successive extensions of pure motives. For a very nice exposition we refer the reader to [59]. The usual polylogarithms are considered by now to be classical, and were studied extensively. The classical polylogarithmic system admits a very natural variation of mixed Hodge structures on it (cf. e.g. [36] section 7). The elliptic polylogarithmic system was defined in the pioneering work of [9] as the elliptic analog of the classical system. It has since been studied by many people ([81], [56], [61], [62], [4] and others). The elliptic polylogarithmic system also admits a variation of mixed Hodge structures on it (cf. [81]). We find it important to explain these notions, as these are the foundations of the theory. Later, when we construct the syntomic analog, the coefficients are no longer Hodge structures, but rather an analog thereof (cf. Definition 4.4.0.26). Definition 1.1.1. Let X be a scheme smooth and locally of finite type over C. An algebraic variation of mixed Q-Hodge structures V on X is a triple (V, W• , F • ), where 1. V is a Q-local system on X. 2. W• is an ascending filtration by Q-local subsystems. 3. F • is a descending filtration on V ⊗ OX such that • These data induce a mixed Hodge structure on each fiber Vx , x ∈ X. • The connection ∇ induced by the local system V on V ⊗ OX by setting ∇(v ⊗ f ) = v ⊗ df for local sections v, f of V and OX , respectively, satisfies the Griffiths’ transversality condition ∇F i ⊂ F i−1 .

22

The category of algebraic variations of mixed Hodge structures over X is denoted by V(X), or V when it is clear what is the underlying scheme. In the special case where X = spec C, one obtains that V(C) is the category of algebraic mixed Q-Hodge structures over C. Let (V, W• , F • ), (U, W• , F • ) ∈ V(X). We define V ⊗U := (V ⊗Q U, W• , F • ) ∈ V(X) by setting 1. The local system is V ⊗Q U . 2. Wm (V ×Q U ) =

X

Wi (V ) ⊗Q Wj (U );

i+j=m

3. As (V ⊗Q U ) ⊗ OX is canonically isomorphic to (V ⊗ OX ) ⊗OX (U ⊗ OX ) we define X

F p ((V ⊗Q U ) ⊗ OX ) =

F i (V ⊗ OX ) ⊗OX Wj (U ⊗ OX ).

i+j=p

We define Hom(U, V ) := (HomQ (U, V ), W• , F • ) ∈ V(X) by setting 1. The local system is HomQ (U, V ). 2. Wm (HomQ (U, V )) = {ϕ : U → V | ϕ(Wj (U )) ⊂ Wm+j (V ) for all j}; 3. As (V ⊗Q U ) ⊗ OX is canonically isomorphic to (V ⊗ OX ) ⊗OX (U ⊗ OX ) we define F p (HomOX (U ⊗ OX , V ⊗ OX )) = {ϕ : U ⊗ OX → V ⊗ OX | ϕ(Fr (U ⊗ OX )) ⊂ Fp+r (V ⊗ OX ) for all r}.

23

• The Tate-Hodge objects (as mixed Hodge structures): For n ∈ Z let Q(n) be the (pure) Hodge structure (Q, F • ) of weight −2n. That is  Q(n), m ≤ −n F m (Q(n)) = 0, m > −n. This generalizes to the category V(X) by taking the trivial variation. For A ∈ V, we let A(n) := A ⊗ Q(n), and call it the nth Hodge-Tate twist of A. What is the weight filtration on Q(n)? Recall that the condition is that GriW V should be a pure object of weight i. This implies that W−2n−1 = 0, W−2n = Q(n). It is well known that V(X) is a Tannakian category. • Mixed Hodge structure on the first co- and homology groups of an elliptic curve: Let E/K be a projective algebraic curve. Then H 1 (E) is a pure Hodge structure of weight 1. Indeed, for a scheme X smooth of finite type over C, there is the p+q Hodge to de Rham spectral sequence E1p,q = H q (X, ΩpX ) ⇒ HdR (X). It induces

a decomposition 1 HDR (E) = H 1 (E, OE ) + H 0 (E, Ω1E ).

It is common to let H p,q (X) denote H q (X, ΩpX ). If X is proper, the Hodge to de Rham spectral sequence degenerates at E1 . This is the Hodge degeneration theorem. There is thus a canonical isomorphism p+q p,q E1p,q = H q (X, ΩpX ) ∼ = F p /F p+1 (HdR (X)). = E∞

Since E is proper, we deduce that 1 1 F k (HdR (E)) = HdR (E), k ≤ 0,

1 F 1 (HdR (E)) = H 0 (E, Ω1E ),

1 F k (HdR (E)) = 0, k > 1.

1 The Hodge filtration on HdR (E) is the one induced by the Hodge to de Rham

spectral sequence. There is a similar decomposition for homology. Algebraically,

24

this can be thought of as the dual to the cohomology, or by taking the first 1 Hodge-Tate twist of HdR (E), that is H1 (E) ∼ = H 1 (E)(1) = H 1 (E) ⊗ Q(1).

Notice that (H 1,0 ⊕ H 0,1 ) ⊗ (F −1,1 Q(1)) = H0,−1 C ⊕ H−1,0 C . We see that H1 (E) is a pure object of weight−1. Since H1 (E) ∼ = H 1 (E) ⊗ C(1), we deduce that the filtration on homology is given by F k H1 (E) := H 1 (E), k ≤ −1,

F 0 H1 (E) := H 0 (Ω1 )∨ ,

F k H1 (E) = 0, k ≥ 1.

Now, we compute the weight filtration. Recall that the condition is that GriW V should be a pure object of weight i. Hence, we define Wk H 1 (E) = 0, for k ≤ 0,

Wk H 1 (E) = H 1 (E) for k ≥ 1.

Since H1 (E) ∼ = H 1 (E) ⊗ Q(1), we deduce: Wk H1 (E) = 0, for k ≤ −2,

Wk H1 (E) = H1 (E), for k ≥ −1.

• Variation of mixed Hodge structures of the first relative co- and homology groups of a family of elliptic curves: Let π : E → S be a family of elliptic curves over a base scheme S of characteristic zero. R1 π∗ Q ∈ V(S). The local system is R1 π∗ Q, which is clearly a local system over S whose fiber at s ∈ S is H 1 (Es , Q). The Hodge and weight filtration are defined by taking fiberwise the mixed Hodge structure defined in the previous item and varying it as follows F k (R1 π∗ Q) = R1 π∗ Q, k ≤ 0,

F 1 (R1 π∗ Q) = R0 π∗ Ω1E/S ,

F k (R1 π∗ Q) = 0, k > 1.

For more details, we refer to [53], algebraic solutions, remark 1.4.1.8.1. Similarly, Wk (R1 π∗ Q) = 0, for k ≤ 0,

Wk (R1 π∗ Q) = R1 π∗ Q for k ≥ 1.

25

Let H = R1 π∗ Q and let H∨ = R1 π∗ Q(1) be the algebraic variation corresponding to the first relative homology (cf. Equation 1.2.7 below for more details). We obtain similarly, F k (H∨ ) := H∨ , k ≤ −1,

F 0 H1 (E) := (R0 π∗ Ω1E/S )∨ ,

Wk (H∨ ) = 0, for k ≤ −2,

1.1.2

F k H1 (E) = 0, k ≥ 1.

Wk (H∨ ) = H∨ , for k ≥ −1.

Relatively unipotent variations

Definition 1.1.2. Let π : X → Y be a morphism of smooth and separated Q-schemes of finite type. π − U V(X) is defined as the full subcategory of unipotent objects of V(X), i.e., those objects whose graded parts are pullbacks of objects in V(Y ).

1.2

The logarithmic pro-sheaf

We aim at a description of the logarithmic pro-sheaf as an element of πE − U V(E), where πE : E → S is the given family of elliptic curves. For this we need the following theorem Theorem 1.2.1. Let π : X → Y be a morphism of smooth and separated Q-schemes of finite type, which identifies X as the complement in a smooth, proper Y -scheme of an N C-divisor relative to Y , all of whose irreducible components are smooth over Y . Let i ∈ X(Y ). The functor i∗ : π − U V(X) → V(Y ) is pro-representable in the following sense: 1. There is an object Geni ∈ pro − π − U V(X), the generic pro-π-unipotent sheaf with base point i on X, which has a weight filtration satisfying Geni /W−n Geni ∈ π − U V(X) for all n. This implies that the direct system (R0 π∗ Hom(Geni /W−n Geni , V))n∈N of smooth sheaves on Y becomes constant for any V ∈ π−U V(X). This constant value is denoted by R0 π∗ Hom(Geni , V).

26

2. There is a morphism of variations on Y 1 ∈ HomV(Y ) (Q(0), i∗ Geni ). 3. The natural transformation of functors from π − U V(X) to V(Y ) ev : R0 π∗ Hom(Geni , V) → i∗ , ϕ 7→ (i∗ ϕ)(1) is an isomorphism. Proof. This is [81] Rem. d) after theorem 3.6 or [82] theorem 1.2.1. By applying the functor HomV (Y )(F (0), •) to the result in (c), one obtains: Corollary 1.2.2. The natural transformation of functors Homπ−U V(X) (Geni , •) → HomV (Y )(F (0), i∗ •), ϕ 7→ (i∗ ϕ)(1) is an isomorphism. Now, let S be a smooth separated Q-scheme of finite type, πE : E → S an elliptic curve with zero section e : S → E, U0 := E − e(S), j : U0 ,→ E, πU0 := πE ◦ j : U0 → S . We may form the generic pro-πE unipotent sheaf with base point 0 on E. Definition 1.2.3. Let Log := Gene ∈ pro − πE − U V(E). It is called the elliptic logarithmic pro-sheaf (or system). Remark 1.2.4. Our definition of Log coincides with that of [9] 1.2.8. There, the logarithmic pro-sheaf is denoted by G. Definition 1.2.5. Let V := π ∗ (H∨ ) ∈ V be defined as follows: the local system is πE∗ (H∨ ). The weight filtration W is induced via the above calculation: W−2 = 0, W−1 = V. The vector bundle is VO := πE∗ (H∨ ) ⊗C OE with the trivial connection; and the filtration is the induced by the above calculation: F −1 VO = VO , F 0 V = πE∗ H 0 (Ω1 ).

27

We would like to compute Rq πU0 ∗ Log(1)|U0 . Theorem 1.2.6.

1. Rq πU0 ∗ Log(1)|U0 = 0 for q 6= 0.

2. There is a canonical isomorphism ∼

R0 πU0 ∗ Log(1)|U0 → W−1 (e∗ Log). 3. The sheaf e∗ Log is split e∗ Log = e∗ Gr•W Log. 4. For every k ≥ 0, there is an isomorphism ∼

GrkW Log → Symk V. Consequently, there is an isomorphism ∼

R0 πU0 ∗ Log(1)|U0 →

Y

Symk H∨ .

k≥1

Proof. This is [82] Theorem 1.2.7. We work with a canonical isomorphism ∼

κ : GrkW Log → Symk V. This is defined in [81] page 263 and [82] after theorem 1.2.7. We recall that by Corollary 1.2.2, there is a canonical projection : Log → Q(0). Furthermore, there is a canonical isomorphism ∼

W Log → V. γ : Gr−1

For this we refer to [81] page 263. There is a cup product ∪πE

2 ^ ∼ : R1 πE∗ Q → Q(−1).

28

Both R1 πE∗ and ∪πE are compatible with bas change. Let π1 : E1 → S, π2 : E2 → S be elliptic curves over S. Any S-isogeny Ψ : E1 → E2 induces an isomorphism ∼

Ψ∗ : R1 π1∗ Q → R1 π2∗ Q, and the formula deg Ψ · ∪π1 = ∪π2 ◦

2 ^

Ψ∗ :

2 ^

∼

R1 πE∗ Q → Q(−1)

holds. This pairing is also called the Poincaré pairing. It induces an isomorphism ∼

α : R1 πE∗ Q(1) → (R1 πE∗ K(1))∨ (1),

(1.2.7)

by sending v to α(v) = ∪πE (v, •). We notice that both Gr•W Log and

Q

k≥0

Symk V carry a canonical multiplicative

structure: for Gr•W , this is a formal consequence of [81] corollary 3.4.ii). We define ∼

κ : GrkW Log → Symk V

(1.2.8)

to be the unique isomorphism compatible with , γ, α and the multiplicative structure of both sides. We define Log (n) := Log/W−n−1 (Log). Then, Log (1) is an extension in V(E)

0 → V → Log (1) → Q(0) → 0. That is to say Log (1) ∈ Ext1V(E) (1, V).

1.2.1

Other incarnations of Log (1)

For any elliptic curve π : E → S there is the Abel-Jabobi morphism [ ] : E(S) → Ext1V(S) (Q(0), H∨ )

(1.2.9)

29

. For a nice description of this morphism, see [81] page 272. We can form the base change pr1 : E ×S E → E and look at the extension [Λ] ∈ Ext1V(E) (Q(0), V). Proposition 1.2.10. The class of Log (1) in Ext1V(E) (Q(0), V) equals [Λ]. Proof. This is [81] proposition III.2.5. This characterization leads to another characterization which is used by [56] (cf. definition 1.1.2, and by Bannai in his work on the p-adic elliptic polylogarithms (e.g., [5] definition 3.1). The Leray spectral sequence for the category V induces the short exact sequence π∗

E 0 → Ext1V(S) (Q(0), H∨ ) → Ext1V(E) (Q(0), V) → HomV(S) (Q, H ⊗ H∨ ) → 0 (1.2.11)

that splits as e is a section of πE . The splitting of πE∗ is given by e∗ . Lemma 1.2.12. Log (1) is the class of Ext1V(E) (Q(0), V), which is mapped to zero under e∗ and to the standard map in HomV(S) (Q, H⊗H∨ ) (this is the map corresponding to the identity via the identification of HomV(S) (Q, H ⊗ H∨ ) with HomV(S) (H∨ , H∨ )). Proof. This follows from the properties of the Abel-Jacobi morphism. There is a geometric description of Log (1) , which appears inside the proof of proposition 2.1.3 [56]. We explain it with more details here. We need the base change theorem in cohomology Theorem 1.2.13. Given a cartesian diagram q

Y −−−→ X     gy fy p

T −−−→ S

30

where X,S,T,Y locally compact spaces, then for any abelian sheaf F on X, one has a canonical morphism p∗ Rk f∗ F → Rk g∗ q ∗ F that is an isomorphism if f is proper. Proof. It appears in [44]. First make the following notations: define the schemes U¯ := (E − e(S)) ×S E and V¯ := U¯ − ∆(E). Let p : E × E → E is the second projection and will also denote the second projection of U, U¯ , V, V¯ . Proposition 1.2.14. We have R1 p∗ QV¯ (1) ∼ = Log (1) . Proof. We recall first the notion of cohomological purity: Let U ,→ X be an open embedding and let Z = X \ U , of codimension d. Then we have the following long exact localization sequence: H i (X) → H i (U ) → H i+1 (X, Z) ∼ = H i+1−2d (Z)(−d) → · · ·

(1.2.15)

(cf. [45] page 82(section k)). Purity is the property that H i (X, Z) ∼ = H i−2d (Z)(−d), implying that H i (X) = H i (U ) for i < 2d − 1, and that the above long exact sequence takes the form 0 → H 2d−1 (X) → H 2d−1 (U ) → H 2d (X, Z) ∼ = H 0 (Z)(−d) → H 2d (X) → H 2d (U ) → H 2d+1 (X, Z) ∼ = H 1 (Z)(−d) → · · ·

(1.2.16)

Purity naturally generalizes to the relative case, where one takes d to be the relative codimension of Z. We first apply purity to the inclusion of V¯ in U¯ over E \e(S). Here, the relative codimension is maximal, namely 1, and purity means that Ri p∗ QU¯ (1) = Ri p∗ QV¯ (1) for i < 1, and the above long exact sequence becomes: 0 → R1 p∗ QU¯ (1) → R1 p∗ QV¯ (1) → Q → 0

(1.2.17)

The reason that the sequence is exact in Q, is that the image of R1 p∗ QV¯ (1) → Q is the kernel of Q → R2 p∗ QU¯ (1). We want to show therefore, that this kernel is Q. This is equivalent to show that the image of Q → R2 p∗ QU¯ (1) is 0. This image is the kernel of the morphism R2 p∗ QU¯ (1) → R2 p∗ QV¯ (1), which is zero as R2 p∗ QU¯ (1) = 0. The proof of this is given below.

31

We now apply purity in the following situation: the closed set (denoted by Z above) corresponds to the embedding e × id : E → E ×S E, that implies that in the above notation X = E ×S E, U = U¯ . Purity therefore implies Ri p∗ QE⊗S E (1) = Ri p∗ QU¯ (1) for i < 1, and we obtain the short exact sequence (it is not short exact): 0 → R1 p∗ QE⊗S E (1) → R1 p∗ QU¯ (1) → Q

(1.2.18)

Since E ×S E is proper, the residue theorem (in each fiber) implies that there is no residue (it is not possible to have residue only at one point, since by the residue theorem the sum of residues is zero), hence we in fact have an isomorphism R1 p∗ QU¯ (1) ∼ = R1 p∗ QE×S E (1). Applying the base change theorem in cohomology (Theorem 1.2.13 above) corresponding to the cartesian square p

E ×S E −−−→ E     py πE y

(1.2.19)

π

E E −−− → S implies it is equal to πE∗ R1 πE∗ QE (1) ∼ = πE∗ HE , where the last isomorphism is Poincaré duality. We have therefore shown that R1 p∗ QV (1) sits in the same short exact se(1) quence as LogE . We turn to prove that R2 p∗ QU¯ (1) = 0. Since R1 p∗ QU¯ (1) ∼ = R1 p∗ QE×S E (1), we deduce that the image of R1 p∗ QU¯ (1) → Q is 0. But this implies that the morphism Q → R2 p∗ QE⊗S E (1) is injective. By base change and Poincaré duality we deduce that R2 p∗ QE⊗S E (1) = πE∗ R0 πE∗ QE = Q. Thus Q → R2 p∗ QE⊗S E (1) is also subjective, implying that the morphism R2 p∗ QE⊗S E (1) → R2 p∗ QU¯ (1) is the zero morphism. This implies that the morphism R2 p∗ QU¯ (1) → R1 p∗ Qe(s)×S E is injective. However, R1 p∗ Qe(s)×S E = 0 since fiberwise this is the first cohomology of a one point space which is 0, implying R2 p∗ QU¯ (1) = 0 as claimed. (1) We know that R1 p∗ QV (1) sits in the same short exact sequence as LogE . It remains to show that it satisfies the conditions of Lemma 1.2.12. The first step is to show that the Sequence 6.2.13 is split after applying e∗ . We use base change in cohomoloy with respect to the cartesian square

id×e

E ×S E ←−−− E     py πE y E

e

←−−− S

restricted to V¯ , and we obtain a base change morphism: α : e∗ R1 p∗ QV¯ (1) → R1 πE∗ QU0 (1).

(1.2.20)

32

[Note that in this case the map p is not proper, so the induced map need not be an Isomorphism.] Notice that by the above argument (Poincaré duality and the residue theorem) we have R1 πE∗ QU0 (1) ∼ = HE . Notice that the horizontal maps of the Cartesian square 6.2.15 commute with composition with the horizonal maps of this Cartesian square, implying that we obtain a commutative diagram p

id×e

E −−−→ E ×S E −−−→ E       py πE y πE y S −−−→ e

E

π

E −−− → S

all of whose resulting squares are Cartesian. Base change for the right square provides a morphism β : πE∗ R1 πE∗ QU0 (1) → R1 p∗ QV¯ (1). (1.2.21) The composition of the two inner squares gives the identity square. By functoriality of base change we obtain that id = α ◦ e∗ (β). Thus, α provides a splitting of the exact sequence as claimed. We now turn to prove that R1 p∗ QV¯ (1) ∈ Ext1E (Q, πE∗ HE ) ∨ is mapped to the standard map in HomS (Q, HE ⊗HE ). In [45], there is an explanation 2 ¯ that this class is the image of the class of the diagonal [∆] ∈ H∆ (U , Q(1)) under the composition of maps: 2 ¯ H∆ (U , Q(1)) → H 2 (U¯ , Q(1)) ∼ = Ext2U¯ (Q, Q(1)) → Ext1A (Q, R1 p∗ QU¯ (1))

where the second map is the edge morphism of the spectral sequence (cf. [45] proof (1) of Lemma 9.2). By Proposition 1.2.10 the image of LogE ∈ Ext1E (Q, πE∗ HE ) equals the image of [∆] under the above map (cf. [81], page 273 proposition V.2.5). The claim is therefore proved. Proposition 1.2.22. Log 1 is isomorphic to the path space variation {H0 (P0,x E; C)}x∈E(C) (cf. [69] section 3 and proposition 4.20). Proof. This follows from [81] theorem I.3.3. See also loc. cit. proposition V.2.5 and [9] lemma 4.3.3.

1.2.2

Log (n)

The morphism : Log → Q(0)

33

induces a morphism n : Log (n) → Q(0). This is the projection of Log (n) = Log/W−n−1 Log on 1.2.8

Log (0) = Log/W−1 = Sym0 V = Q(0). Theorem 1.2.23.

1. There is a unique isomorphism ∼

(n)

ϕ0 : Log (n) → Symn Log(1) such that the diagram

n −−− →

Log (n)  (n)  ϕ0 y

Q(0) x o 

Symn 1

Symn Log(1) −−−−→ Symn Q(0) commutes, where the right vertical map is given by multiplication. 2. There is a unique isomorphism ∼

(n) ϕ(n) → Symn Log(1) n : Log

such that the diagram Symn V   ||y

κ−1

−−−→

Log (n)  o y

Symn κ−1

Symn V −−−−−→ Symn Log (1) commutes, where the right vertical map is given by multiplication. 3. We have the relation (n)

ϕ0 = n! · ϕ(n) n . Proof. This is [81] Theorem III.2.6.

Chapter 2 Algebraic realization of the polylogarithms for a single elliptic curve The work of this chapter is of a combinatorial nature. The results here are restricted to the case where S = specK, but it should not be difficult to generalize them to the case of a general base scheme S, following the same approach. The description of the algebraic de Rham classes of the elliptic polylogarithms is given in concrete terms, but still these results do not shed a geometrical light on the subject. In the next chapter we follow a different method, which results in a deeper geometric understanding of the elliptic polylogarithms, in the context of the general base case. We finally decided to include this chapter in the thesis because we feel the ideas and combinatorial methods used here may serve to study other related questions, and because everything done here is in very concrete terms. Throughout, we assume that K is a finite extension of Qp with ring of integers OK and residue field k. We let K0 be the maximal unramified extension of Qp in K and W its ring of integers. σ is the lifting to K0 of the Frobenius automorphism on k.

34

35

2.1

Explicit description of Log (1) over a single elliptic curve

Let E/K be an elliptic curve defined over the field K. Assume it is given by the affine equation E : y 2 = x3 + ax + b and the projective equation Y 2 Z = X 3 + aXZ 2 + bZ 3 . We regard E as a curve over the point spec(K), with the trivial projection πE : E → spec(K). We work with V(E), from Definition 1.1.1.1, where local systems are replaced by vector bundles with integrable connections.

2.2

Description of the algebraic de Rham cohomology class corresponding to Log (1)

Let V be the analog of Definition 1.1.2.5. Let Ψ denote the map induced by the 1 Leray spectral sequence for de-Rham cohomology Ψ : Ext1V (1, V) → F 0 HDR (E, V), 1 where the filtration on HDR (E, V) is the hodge filtration. By an analog of [45] Lemma 1 1 (E) ⊗ V. The first (E, V) = HDR 9.2, Ψ is an isomorphism. We also recall that HDR

logarithmic pro-sheaf Log (1) is defined by the characterization of Lemma 1.1.2.12 (cf. also [56] definition 1.1.2). This characterization implies that given some basis ω1 , ω2 1 1 to HDR (E), the class in HDR (E, V) corresponding to Log (1) is ω1 ⊗ ω1∨ + ω2 ⊗ ω2∨ .

We recall a few basic facts about the computation of the algebraic de Rham cohomology of any scheme of finite type X over K: By definition, it is the hypercohomology of the complex Ω•X . If U = (Ui ) is an affine covering of X, then this cohomology may be computed as the Cech hypercohomology of Ω•X with respect to U. This is, by definition the total cohomology of the double complex C p (U, ΩqX ) of Cech p-cochains with coefficients in ΩqX , with total differential equal to δ + (−1)p d, where d is the differential of Ω•X and δ is the Cech differential (cf. [22]).

36

In our setting, X = E is an elliptic curve, and we take U = (U0 , U1 ), where U0 = ((x, y, z) ∈ E : z 6= 0) = E − ∞, U1 = ((x, y, z) ∈ E : y 6= 0) in projective coordinates. Throughout this chapter (only), the “zero section” is denoted by the symbol “∞”. Let j denote the inclusion of U0 ,→ E. For any F ∈ V, We denote FU0 := j ∗ F . Since E is a smooth projective curve (dimension = 1), we have Ω≥2 E = 0. This implies that the first cohomology group of the corresponding double complex can be written as: {(α0 ∈ Ω1E (U0 ), α1 ∈ Ω1E (U1 ), f ∈ OE (U0 ∩U1 ))} {(df1 , df2 , f1 −f2 |U0 ∩U1 )|f1 , f2 ∈ OE (Ui )} (2.2.1) Next, we describe a basis for the cohomology in this form. We first recall that { dx , xdx } is a basis for the algebraic de Rham cohomology of E − ∞. We then y y notice that the invariant differential

dx y

is holomorphic on E, and that

xdx y

− d( 2y ) is x

holomorphic on U1 . These facts, together with the fact that the algebraic de Rham cohomology of E is two dimensional, implies that n dx dx xdx xdx 2y 2y o ω1 := , , 0 , ω2 := , − d( ), y y y y x x

(2.2.2)

is a basis for the de Rham cohomology of E. We deduce that the de Rham cohomology class of Log (1) is: dx dx xdx xdx 2y 2y , , 0 ⊗ ω1∨ + , − d( ), ⊗ ω2∨ = y y y y x x dx dx xdx xdx 2y 2y ⊗ ω1∨ , ⊗ ω1∨ , 0 + ⊗ ω2∨ , − d( ) ⊗ ω2∨ , ⊗ ω2∨ = y y y y x x dx xdx dx xdx 2y 2y ⊗ ω1∨ + ⊗ ω2∨ , ⊗ ω1∨ + − d( ) ⊗ ω2∨ , ⊗ ω2∨ y y y y x x

37

2.3

Description of the underlying vector bundle with connection of Log (1)

In the previous section, we found the one-form corresponding to Log (1) : xdx xdx 2y 2y dx dx 1 , , 0 ⊗ ω1∨ + , − d( ), ⊗ ω2∨ ∈ HDR (E, V). y y y y x x 1 (E, V). Let us describe the inverse to the isomorphism Ψ : Ext1 (1, V) ∼ = F 0 HDR

This description is true whenever VO is quasi-coherent; we describe Log (1) as a one extension in V. First, we have the inclusion in V, which is: V ,→ Log (1) . Then we notice that Log (1) |U0 , Log (1) |U1 are trivial vector bundles. Indeed, since VO is quasi-coherent, and Ui are affine, this is asserted by proposition 5.6 (Hartshorne). (1)

(1)

We therefore have the sections e0 , e1 on LogU0 , LogU1 , respectively, which are the preimages of the constant section 1 on U0 , U1 , respectively. Log (1) |U0 , Log (1) |U1 are glued together by the rule e0 |U01 − e1 |U01 =

2y x

⊗ ω2∨ . This rule defines the underlying

vector bundle of Log (1) . We next define the connection on it. Again, the inclusion V ,→ Log (1) implies that the global sections ω1∨ , ω2∨ ∈ Log (1) are flat. It remains to define ∇e0 =

dx xdx dx xdx 2y ⊗ ω1∨ + ⊗ ω2∨ , ∇e1 = ⊗ ω1∨ + − d( ) ⊗ ω2∨ . y y y y x

(2.3.1)

This defines Log (1) as a vector bundle with integrable connection. It remains to describe the filtration F on Log (1) . F is determined by The filtration on V and the short exact sequence defining Log (1) . Since this exact sequence is in V, we have short exact sequences on the level of filtrations. On the F −1 level, we have: 0 → F 1 V → F 1 Log (1) → F 1 1 → 0.

(2.3.2)

Since F −1 1 = 1 and F −1 VO = VO , the five lemma implies that F −1 Log (1) = Log (1) (the equality holds since the inclusion map is also an isomorphism).

38

On the F 0 level, we have: 0 → F 0 V → F 0 Log (1) → F 0 1 → 0.

(2.3.3)

The description of the zeroth filtration is: F 0 1 = 1 and F 0 VO = H 1 (Ω1E ) Similarly F k = Log (1) = Log (1) for k < −1 and F k = 0 for k > 0. Next we describe the weight filtration. On the W−1 level we have: 0 → W−1 V → W−1 Log (1) → W−1 1 → 0.

(2.3.4)

Since W−1 V = V and W−1 V = 0, we deduce that W−1 Log (1) = V. Similarly, Wk Log(1) = Log (1) for k > −1 and F k = 0 for k < −1. It is readily verified that Gr−1 Log (1) = V is a pure Hodge structure of weight -1 and Gr0 Log (1) = 1 is the trivial Hodge structure (of weight 0). This describes Log (1) as an element of V.

2.4

(1)

1 Description of the residue map on HDR (U0, LogU0 (1))

In this section we compute the first residue map, Res(1) ∞ (cf. chapter 3, Section 2.5.2).

2.4.1

A general remark on the residue map in the category V

1 For every W ∈ V there exists a residue map: ResW ∞ : HDR (U0 , WU0 (1)) → W |∞ . The

picture is the following: let D denote the formal completion of E at ∞. We have a flat isomorphism (W, ∇) ∼ = (W |∞ ⊗ OD , d).

(2.4.1)

This isomorphism induces an isomorphism Ω1E (D − ∞, W ) ∼ = W |∞ ⊗ Ω1D−∞ . Under 1 this isomorphism ResW ∞ is given by id⊗Res∞ : W |∞⊗ΩD−∞ → W |∞, where Res∞ is

39

the residue map at ∞. In order for ResW ∞ to be well-defined, one should show that it is independent of the representative for the cohomology class, or put otherwise, that it is zero on elements which are co-boundaries. Indeed, let n = ∇w ∈ B 1 (U0 , WU0 (1)). P By Isomorphism 2.4.1 we may write w = i wi ⊗ αi where αi ∈ OD−∞ and wi are flat P P sections of W|D−∞ . Thus n = i wi ⊗dαi , and it is clear that id×Res∞ ( i wi ⊗dαi ) = 0. The twist of the weights is a consequence of the filtration on the cohomology, and we don’t elaborate on it here. Notice that a description of the first isomorphism above is equivalent to finding a local flat basis of sections for W “near” ∞. We use this remark to compute the residue (1)

1 map on HDR (U0 , LogU0 (1)) in explicit terms.

(1)

1 Explicit computation of the residue map on HDR (U0 , LogU0 (1))

2.4.2

(1)

Let α = η1 ⊗ ω1∨ + η2 ⊗ ω2∨ + η3 ⊗ e0 ∈ Ω1DR (U0 , LogU0 (1)). By the previous section, we want to write this form as a combination of elements of a local flat basis of sections for Log (1) near ∞. We first use change of coordinates near infinity to get: η1 ⊗ω1∨ +η2 ⊗ω2∨ −η3 · 2y ⊗ω2∨ +η3 ⊗e1 ∈ Ω1DR (U1 , Log (1) (1)). Since ω1∨ , ω2∨ x ∨ 2y xdx ∨ are flat, we need to modify only the part of e1 . Since ∇e1 = dx ⊗ω + −d( ) ⊗ω2 , 1 y y x we would like to find a section g of V near infinity such that ∇(e1 − g) = 0. Assume g(∞) = 0. Let t = xy , then t is a uniformizer at ∞. We may compute the Laurent expansion of g as a function of t. t−2 = x=

1 t2

− at2 − bt4 + . . . ,

y=

x t

=

1 t3

dx = − t23

− at − bt3 + . . . ,

1 y

x3 +ax+b x2

a x

+ xb2 . This implies that − 2at − 4bt3 + . . . dt. =x+

= t3 + at7 + bt9 + . . ..

We may continue the expansion of x a bit more and get x=

1 − at2 − bt4 − a2 t6 − 3abt8 + . . . 2 t

(2.4.2)

40

We deduce: dx = −2 − 4at4 − 6bt6 + . . . dt (2.4.3) y xdx 2 = − 2 − 2at2 − 4bt4 − 6a2 t6 + . . . dt (2.4.4) y t 7 3 5 2 7 5 Hence, g = −2t − 4at5 − 6bt7 + . . . ⊗ ω1∨ + − 2at3 − 4bt5 − 6a7 t + . . . ⊗ ω2∨ R R xdx We summarize the above calculation: Let F (t) = dx , G(t) = , the correy y sponding integrals in a neighborhood of infinity. Then e0 → (e1 − g) + F (t) ⊗ ω1∨ + G(t) ⊗ ω2∨ . For simplicity we denote e01 := e1 − g. Thus, the original differential form may be written near ∞ as: η1 + F (t) · η3 ⊗ ω1∨ + η2 + G(t) · η3 ⊗ ω2∨ + η3 ⊗ e01

(2.4.5)

And then: ∨ Res(1) ∞ (α) = Rest=0 η1 + F (t) · η3 ⊗ ω1 + Rest=0 η2 + G(t) · η3 ⊗ ω2∨ + Rest=0 (η3 ) ⊗ e01 Remark 2.4.6. Under the Isomorphism 2.5.6, we further map e01 → 1. We will use this identification in the future.

2.5

Polylogs

2.5.1

the logarithm sheaf

Definition 2.5.1.

• Let Log (n) := Symn Log (1) .

We have the short exact sequences for Log (n) : 0 → Symn V → Log (n) → Log (n−1) → 0 induced by the short exact sequence 1.2.9 of chapter 2, defining Log (1) .

(2.5.2)

41

• Define the logarithm Log := proj lim Log (n) ∈ V, where the transition maps appear in Equation 2.5.2. • Let p denote the projection in the short exact sequence 1.2.9. Then, the projection it induces on the short exact sequence 2.5.2, denoted by p(n) : Log (n) → P Log (n−1) is defined by p(n) (l1 ⊗ · · · ⊗ ln ) = ni=1 p(li )l1 ⊗ · · · ⊗ lî ⊗ · · · ⊗ ln , where li ∈ Log (1) , i = 1, . . . , n. The connection on Log n |U0 We recall that ∇e0 =

dx y

⊗ ω1∨ +

xdx y

⊗ ω2∨ , ∇ω1∨ = 0, ∇ω2∨ = 0.

By Definition 2.5.1, a basis for Log (n) ⊗OU0 (U0 ) is given by {ek0 (ω1∨ )l (ω2∨ )m |k +l +m = n}. We know that Log n is characterized by describing the connection on this basis. We have: ∇ek0 (ω1∨ )l (ω2∨ )m = k

dx k−1 ∨ l+1 ∨ m xdx k−1 ∨ l ∨ m+1 e0 (ω1 ) (ω2 ) + k e (ω1 ) (ω2 ) . y y 0

(2.5.3)

In the sequel, we will use the following ∇Log = ∇S • +

dx xdx ⊗ ω1∨ + ⊗ ω2∨ . y y

(2.5.4)

Further properties of Log Theorem 2.5.5 ([9] Lemma 1.2.7). πE ∗ Log (n) ∼ = Symn (H∨ ). R1 πE ∗ Log (n) ∼ = Symn+1 (H∨ )(−1). R2 πE ∗ Log (n) ∼ = Q(−1). We recall again the zero section, ∞ : spec(K) → E, which is defined by ∞(spec(K)) = P∞ = ∞ (note that we use the ambiguous notation ∞ for both the zero section, and the point at ∞ of E). Again, by definition (1.1.2, [56]) we have ∞∗ Log (1)

42

is trivial, which means that it is isomorphic to H∨ ⊕ Q. Taking the projective limit of the symmetric product, we deduce that ∞∗ Log = ∞∗ (proj lim Log (n) ) ∼ = Q proj lim ∞∗ (Symn Log (1) ) ∼ = k≥0 Symk H∨ . ˜ := p(n) ◦ p(n−1) · · · p(1) denote the projection from Log (n) to Q. Then denote Let p(n) ˜ the projection from Log to Q. p(∞) := proj lim p(n) Define I := ker(p(∞) ); I is called the augmentation ideal of Log. For any sheaf F on E we write F[∞] := ∞∗ F. Hence, Log[∞] ∼ =

Y

Symk H∨ , and I[∞] ∼ =

Y

Symk H∨ .

(2.5.6)

k>0

k≥0

Theorem 2.5.7. Log (n) = Log/I n+1 , and hence: Log ∼ = proj lim Log/I n .

2.5.2

The residue maps

There are residue maps: (n)

1 (n) Res(n) [∞] := Log[∞]/I n+1 [∞] ∞ : H (U0 , LogU0 (1)) → Log

(2.5.8)

These residue maps commute with the corresponding maps on the cohomology groups induced by the projections Log (n) → Log (n−1) . (n)

We Let p∗

(n)

(n−1)

denote the maps H • (U0 , LogU0 (1)) → H • (U0 , LogU0

(1)), induced by

the projection Log (n) → Log (n−1) , for n ≥ 1. Theorem 2.5.9 (cf. [9] Lemma 1.3.2). There exists a unique isomorphism of additive groups: (n)

H 1 (U0 , LogU0 (1)) → I[∞]/I n+2 [∞] (n+1)

such that the composition with the projection p∗ gives Res(n+1) . ∞

(n+1)

: H 1 (U0 , LogU0

(n)

(1)) → H 1 (U0 , LogU0 (1))

43

\ (n) We denote this isomorphism by Res ∞ . (n)

Corollary 2.5.10. dim H 1 (U0 , LogU0 (1)) =

Pn+1 k=1

|Symk H1 | =

Pn+1 k=1

k+1 =

n2 +5n+4 2

Theorem 2.5.11 (cf. [9] corollary 1.3.3). H 0 (U0 , LogU0 (1)) = 0,

2.5.3

Res∞ : H 1 (U0 , LogU0 (1)) ∼ = I[∞]

(2.5.12)

(n)

1 (U0 , LogU0 (1)) Explicit computation of HDR (n)

1 Our goal in this section is to find a basis for HDR (U0 , LogU0 (1)). (n)

1 (U0 , LogU0 (1)) is given by: Theorem 2.5.13. With our notations, a basis to HDR

{

xdx j k ⊗ ei0 ⊗ ω1∨ ⊗ ω2∨ , i + j + k = n, y

0 ≤ i, j, k}

[ dx n−i { ⊗ ei0 ⊗ ω2∨ , i = 0, . . . , n} y

Proof. The short exact sequence (2.5.2) defining Log (n) gives a long exact sequence in cohomology: (n−1)

0 . . . → HDR (U0 , LogU0

(1)) (2.5.14)

1 →HDR (U0 , Symn V(1)) →

(n)

1 HDR (U0 , LogU0 (1))

(n−1)

1 → HDR (U0 , LogU0

(1))

2 →HDR (U0 , Symn V(1))

E is an elliptic curve, implying Ω≥2 E = 0. Together with the fact that U0 is 2 2 2 affine, we have HDR (U0 ) = 0, implying that HDR (U0 , Symn V(1)) ∼ (U0 ) ⊗ = HDR (n)

(n−1)

1 1 Symn H∨ (1) = 0. Hence, HDR (U0 , LogU0 (1)) → HDR (U0 , LogU0

(1)) is a surjection.

(This fact is important beyond the scope of this proof.) By Corollary 2.5.10 there is a (n−1)

1 basis for HDR (U0 , LogU0

n2 +3n elements. We exploit the affinity of U0 2 (n−1) 0 HDR (U0 , LogU0 (1)) ∼ = Symn−1 H∨ (1), since it is isomor-

(1)) containing

once more, to deduce that

(n) phic to R0 (πE ◦ j)∗ LogU0 (1) ∼ = πE ∗ j∗ j ∗ (Log (n) )(1) ∼ = πE ∗ (Log (n) )(1) ∼ = Symn−1 H∨ (1)

44

by Theorem 2.5.5.

(It is also described as the flat sections of Log (n−1) (1) over (n−1)

0 U0 .) The mapping δ : HDR (U0 , LogU0

1 (1)) → HDR (U0 , Symn V(1)) is the bound-

ary map in cohomology. We explain how to compute δ; given some element s ∈ (n−1)

0 HDR (U0 , LogU0

(1)) = Symn−1 H∨ (1), we pull it to Log (n) (1) and get e0 ⊗ s (it

−1

clearly lies in p(n) (s)). Then we apply ∇ to it, to obtain

dx y

⊗ s ⊗ ω1∨ + xdx ⊗ s ⊗ ω2∨ ∈ y

1 HDR (U0 , Symn V(1)). The resulting element is δ(s).

We have the short exact sequence: (n)

(n−1)

1 1 1 0 → HDR (U0 , Symn V(1))/Im(δ) → HDR (U0 , LogU0 (1)) → HDR (U0 , LogU0

(1)) → 0

1 1 (U0 ) ⊗ Symn H∨ (1), we deduce that it contains (U0 , Symn V(1)) ∼ Since HDR = HDR dx ⊗ω1∨ i ⊗ω2∨ n−i , xdx ⊗ω1∨ i ⊗ω2∨ n−i , i = 0, . . . , n. y y (n−1) 0 HDR (U0 , LogU0 (1)), these elements are mapped injectively to

a basis with 2(n+1) elements, namely Modulo the image of (n)

1 HDR (U0 , LogU0 (1)). There are therefore 2(n + 1) − n = n + 2 independent elements in (n−1)

(n)

1 1 (U0 , LogU0 (U0 , LogU0 (1)), which are mapped to zero in HDR HDR (n−1)

1 (U0 , LogU0 gether with some preimage of a basis of HDR

(1)). Hence, to-

(1)), we get n2 + 3n + 2n + 4

(n)

1 independent elements in HDR (U0 , LogU0 (1)). By Corollary 2.5.10, this gives a basis (n)

1 for HDR (U0 , LogU0 (1)). (n)

1 (U0 , LogU0 (1)) with relaLet us use this calculation to write a spanning set for HDR

tions on it. The spanning set is n,

dx y

⊗ ei0 ⊗ ω1∨ j ⊗ ω2∨ k , xdx ⊗ ei0 ⊗ ω1∨ j ⊗ ω2∨ k , i + j + k = y

0 ≤ i, j, k. These are taken modulo the relations generated by:

dx y

⊗ ei0 ⊗ ω1∨ j+1 ⊗

ω2∨ k + xdx ⊗ ei0 ⊗ ω1∨ j ⊗ ω2∨ k+1 , i + j + k = n − 1, 0 ≤ i, j, k. Notice that if we sum over y P P 2 i=0,. . . ,n, the degree of e0 , we get ni=0 2(n−i+1)−(n−i) = ni=0 n−i+2 = n +5n+4 , 2 independent elements, which is exactly what we wanted. The basis stated in the theorem immediately follows. Definition 2.5.15. The small polylogarithm Pol (we omit the subscript for E) is defined as the class in Ext1U0 (V, LogU0 (1)), corresponding under the isomorphism 5 Q 1 Ext1U0 (V, LogU0 (1)) ∼ (U0 , LogU0 (1)) ∼ = HomS (H∨ , HDR = HomS (H∨ , k>0 Symk H∨ ) =

45

Q

k>0

HomS (H∨ , Symk H∨ ), to the map in

Q

k>0

HomS (H∨ , Symk H∨ ) which is the

identity for k = 1, and zero for k 6= 1. Poln are defined as the projection of Pol (n)

to Ext1U0 (V, LogU0 (1)). Similarly, by Theorem 2.5.9, Poln corresponds to the map in Qn+1 ∨ k ∨ k>0 HomS (H , Sym H ), which is the identity for k = 1 and zero for k 6= 1. In particular, Pol0 corresponds to the identity map in HomS (H∨ , H∨ ). We note that under that correspondence, this is exactly the differential form computed for Log (1) . Remark 2.5.16. It follows from the definition that finding Pol(n) is equivalent to (n) (n) (n) \ (n) (n) (ζ ) = finding two differential forms ζ , ζ ∈ H 1 (U , Log (1)), such that Res 1

\ (n) (n) ω1∨ , Res ∞ (ζ2 )

=

ω2∨ ,

2

DR

0

∞

U0

1

under the Isomorphism 2.5.6. (n)

Remark 2.5.17. In Theorem 2.5.13, we proved that p∗

is a surjection, for n ≥ 1.

(n)

Notice that p∗ (Pol(n) ) = Pol(n−1) , implying also that Pol = proj limPol(n) .

2.5.4

The period matrix for Pol(n)

Write the short exact sequence for Pol(n) : 0 → Log (n) (1) → Pol(n) → πE∗ (H1 ) → 0 Hence, a basis for Pol(n) is given by {2πi ek0 (ω1∨ )l (ω2∨ )m |k+l+m = n}

(2.5.18) S

{ω˜1 ∨ , ω˜2 ∨ }.

The connection is defined on {2πi ek0 (ω1∨ )l (ω2∨ )m |k + l + m = n} as before. We define ∇ω˜1 ∨ =

(n) 1 en−1 ζ1 , ∇ω˜2 ∨ n−1 0

=

(n) 1 en−1 ζ2 . n−1 0

46

2.5.5

Computing Pol(n)

Pol(0) By Section 2.4 of chapter 2, we know that given α = η1 ⊗ ω1∨ + η2 ⊗ ω2∨ + η3 ⊗ e0 ∈ Ω1DR (U0 , Log (1) (1)), its residue is given by: ∨ Res(1) ∞ (α) = Res∞ η1 + F (t) · η3 ⊗ ω1 + Res∞ η2 + G(t) · η3 ⊗ ω2∨ + Res∞ (η3 ) ⊗ e01 We notice that since ηi ∈ Ω1DR (U0 ), its residue must be zero. This implies that ∨ ∨ Res(1) ∞ (α) = Res∞ F (t) · η3 ⊗ ω1 + Res∞ G(t) · η3 ⊗ ω2 (1) We see that choosing ζ˜1 =

xdx 4y

⊗ e0 ,

(1) ζ˜2 =

dx 4y

˜ (1) ⊗ e0 , gives: Res(1) ∞ (ζ1 ) =

(1) (1) (1) ∨ ˜ (1) = ζ˜1 ⊗ ω1 + ζ˜2 ⊗ ω2 ∈ Ω1DR (U0 , Log (1) (1)) ⊗ H. ω1∨ , Res(1) ∞ (ζ2 ) = ω2 . Define ζ 1 By Definition 2.5.15 and Theorem 2.5.9 its projection to HDR (U0 , Log (0) (1)) ⊗ H ∼ = 1 HomS (H∨ , HDR (U0 , Log (2) (1))), namely the element

xdx 4y

⊗ ω1 +

dx 4y

⊗ ω2 , gives Pol(0) .

(1) Notice that under the Isomorphism 2.5.9, Pol(0) is mapped to Res(1) ), which is ∞ (ζ

the class of Log (1) as computed in section (1.0.1). Pol(1) and Res(2) ∞ By the exact sequence (2.5.2) we write our differential form as: α2 = η0 ⊗ s2 + η1 ⊗ e0 ⊗ ω1∨ + η2 ⊗ e0 ⊗ ω2∨ + η3 ⊗ e0 ⊗ e0 ∈ Ω1DR (U0 , Log (2) (1)) with s2 ∈ Symn V. One then follows the logics described in Section 2.4.1 to write the sections near infinity as follows: 1. η1 ⊗ e0 ⊗ ω1∨ → F (t) · η1 ⊗ ω1∨ ⊗ ω1∨ + G(t) · η1 ⊗ ω1∨ ⊗ ω2∨ + η1 ⊗ e01 ⊗ ω1∨ . 2. η2 ⊗ e0 ⊗ ω2∨ → F (t) · η2 ⊗ ω1∨ ⊗ ω2∨ + G(t) · η2 ⊗ ω2∨ ⊗ ω2∨ + η2 ⊗ e01 ⊗ ω2∨ .

47

3. η3 ⊗ e0 ⊗ e0 → F (t)2 · η3 ⊗ ω1∨ ⊗ ω1∨ + G(t)2 · η3 ⊗ ω2∨ ⊗ ω2∨ + 2F (t)G(t) · η3 ⊗ ω1∨ ⊗ ω2∨ + 2F (t) · η3 ⊗ e01 ⊗ ω1∨ + 2G(t) · η3 ⊗ e01 ⊗ ω2∨ + η3 ⊗ e01 ⊗ e01 And one should take the residue at t = 0 of the resulting differential form. We again notice that since ηi ∈ Ω1DR (U0 ), its residue must be zero. From this it is easy to write the residue of α2 , and notice that it is independent of s2 . We are interested in Pol(1) . (2) By the same reasoning as above, we write: ζ˜1 = dx 8y

(2)

xdx 8y

⊗ e0 ⊗ e0 ,

(2) ζ˜2 =

(2)

(1) ˜ ∨ ∨ ˜ ⊗ e0 ⊗ e0 , gives: Res(2) ∞ (ζ1 ) = ω1 , Res∞ (ζ2 ) = ω2 , under the Isomorphism

(2) 2.5.6 (namely, in Formulae 2.5.5, we should map e01 → 1). Define ζ (2) = ζ˜1 ⊗ (2) ω1 + ζ˜2 ⊗ ω2 ∈ Ω1DR (U0 , Log (2) (1)) ⊗ H ∼ = HomS (H∨ , Ω1DR (U0 , Log (2) (1))). By (1) Definition 2.5.15 and Theorem 2.5.9 its projection to H 1 (U0 , Log (1)) ⊗ H ∼ = DR

HomS (H

∨

1 (U0 , Log (2) (1))), , HDR

namely

xdx 4y

⊗ e0 ⊗ ω1 +

dx 4y

U0

⊗ e0 ⊗ ω2 gives Pol(1) .

Pol(n) in general We use Theorem 2.5.7 and Remark 2.5.17 of this chapter to deduce that in the previous section we actually found Pol modulo I 2 . To make this statement precise, (n)

we claim that for every n ≥ 1, we have: ζ1

=

xdx 4(n!)y

(n)

⊗en0 +· · · , ζ2

=

dx 4(n!)y

⊗en0 +· · · .

We prove it by induction on n. Recall that in Theorem 2.5.3 we found a basis for (n)

(n)

1 HDR (U0 , LogU0 (1)). This implies that ζi , i = 1, 2 is a linear combination of elements (n)

(n)

(n−1)

of this basis. Moreover, by Remark 2.5.17, p∗ (ζi ) = ζi

, i = 1, 2.

(n)

Hence, by definition of p∗ and the inductive hypothesis the claim follows. (n)

Let s(n) : LogU0 |U0 → Log (n+1) |U0 be the natural right inverse to p(n) on U0 sending ej0 ⊗ (ω1∨ )k ⊗ (ω2∨ )l with j, k, l : j + k + l = n to (n)

OU0 linearly. Then s∗ induces a right inverse

1 j+1 e ⊗ (ω1∨ )k j+1 0 (n) to p∗ |U0 in the

⊗ (ω2∨ )l , and extended cohomology restricted

to U0 . (n)

We now describe an iterative procedure which finds ζi (n)

(n−1)

based on ζi

, i = 1, 2.

1 Recalling the explicit basis to HDR (U0 , LogU0 (1)) presented in Theorem 2.5.13 we

48

(n)

write ζi

as xdx j e0 ai,n j,k,l

X

y

{j,k,l:j+k+l=n}

⊗

(ω1∨ )k

⊗

(ω2∨ )l

(n)

1 j

· ai,n−1 j−1,k,l ,

bi,n j

j=0 (n)

(n−1)

Based on the identity p∗ (ζi ) = ζi have: ai,n j,k,l =

+

n X

bi,n = j

1 j

dx j e ⊗ (ω2∨ )n−j . y 0

(2.5.19)

, i = 1, 2, we deduce that for j > 0 we

· bi,n−1 j−1 . So it remains to find the coefficients

i,n ai,n 0,t,n−t , b0 , t = 0, . . . , n, i = 1, 2. To find them, we recall that by Remark 2.5.16, (n) \ under the Isomorphism 2.5.6 we have: Res = ωi∨ , i = 1, 2. We there∞ (n)ζi (n) (n) (n) \ \ fore compute Res to ∞ (n)ζi . Since Res∞ (n) = Res∞ (n) ◦ s∗ , we first lift ζi (n)

1 (U0 , Log (n+1) (1)) via s∗ . HDR

For that we follow a procedure similar to the procedure described in Section 2.5.5, to (n)

(n)

each term of s∗ (ζi ). We first apply the rule ej0 → (e01 + F (t)ω1∨ + G(t)ω2∨ )j , to the (n)

lift via s∗ of ej0 ⊗ (ω1∨ )k ⊗ (ω2∨ )l times

1 xdx j+1 y

xdx y

⊗ ej+1 ⊗ (ω1∨ )k ⊗ (ω2∨ )l → 0

1 j+1

or

dx , y

Pj+1 i=0

and we get:

j+1 i

i j+1−i F G dG(ω1∨ )i+k ⊗ (ω2∨ )j+1−i+l +

terms with total-deg(ω1∨ , ω2∨ ) < n + 1. (2.5.20)

1 dx j+1 y

⊗ ej+1 ⊗ (ω2∨ )n−j → 0

1 j+1

Pj+1 i=0

j+1 i

i j+1−i F G dF (ω1∨ )i ⊗ (ω2∨ )n+1−i +

terms with total-deg(ω1∨ , ω2∨ ) < n + 1. (2.5.21) Then we take the residue of these expressions, and substitute e01 → 1. The equation (n) \ Res = ωi∨ , i = 1, 2 induces a polynomial identity in K[ω1∨ , ω2∨ ]. Restricting ∞ (n)ζi

to the homogenous part of degree n + 1 would give us a set of n + 2 linear equations i,n in the n + 2 variables ai,n 0,s,n−s , b0 , s = 0, . . . , n, i = 1, 2, (all the other variables are

49

considered to be known at this step) as follows: the coefficient of (ω1∨ )r ⊗(ω2∨ )n+1−r , r = 0, . . . , n + 1 in this system is  1, if (ω ∨ )r ⊗ (ω ∨ )n+1−r = ω ∨ i 2 1 0, otherwise and is denoted by δ i (r). Hence, by the residue computation above, we get the following system of linear equations: 1 j+1 Rest=0 F r−k Gn+1−r−l dG+ (2.5.22) j+1 r−k {j,k,l|0≤r−k≤j+1,j+k+l=n} n X j+1 i,n 1 bj Rest=0 F r Gj+1−r dF = δ i (r), r = 0, . . . , n + 1 j+1 r X

ai,n j,k,l

j=max(r−1,0)

We show that this system is already in diagonal form. Fix 0 ≤ r ≤ n + 1; we examine the number of actual occurrences (that is, with ∨ n+1−r ∨ r . non-zero coefficients) of the variables ai,n 0,s,n−s in the equation for (ω1 ) ⊗ (ω2 )

There are two options: s = r, or s = r − 1. We notice that the contribution would be i,n ai,n 0,r,n−r Rest=0 GdG in the first case, and a0,r−1,n−r+1 Rest=0 F dG in the latter. Since

GdG = 21 d(G2 ) its residue and hence the first term vanishes, whereas Rest=0 F dG = −4 6= 0. Hence, ∨ r ∨ n+1−r the coefficient of ai,n 0,s,n−s in the equation for (ω1 ) ⊗ (ω2 )

(2.5.23) is non-zero if and only if s = r − 1. We now turn to check at which Equations 2.5.22 bi,n 0 appears. We see that there are two options: r = 1 or r = 0. for r = 1 its coefficient is Rest=0 F dF = 0, and for r = 0 its coefficient is Rest=0 GdF 6= 0, implying the coefficient of bi,n 0 is non-zero only in the equation for ω2n+1 . In this equation there are no appearances of any a0,s,n−s , since by Equation 2.5.23 it would imply that s = −1 which is impossible. Hence, in each equation of 2.5.22, there is exactly one non-zero coefficient, implying it is in

50

diagonal form as claimed. It is therefore a trivial task to write down every variable based on ai,n−1 j,k,l ,

bji,n−1 computed in the previous step(s), and explicit computation

of Rest=0 (F l d(Gm )), Rest=0 (d(F l )Gm ), 0 ≤ l, m : l + m ≤ n + 2. Summarizing, finding Pol is an iterative process as claimed.

2.6

Generating functions

1 i,k+l We suggest the following approach: Recall that ai,n j,k,l = j! a0,k,l . Let us fix throughout

this section i = 1 (the case i = 2 is treated similarly). We may view the series {ai,n j,k,l } as a series in two variables {ak,l }. We investigate this series via its two variable genP erating function P (x, y) := k,l≥0 ak,l · xk · y l . We may similarly regard {bi,n j } as a P series in one variable {bn } and investigate its generating function Q(x) = n≥0 bn ·xn . 1 (U0 , LogU0 (1)). Taking the infinite version of Theorem 2.5.13, gives an explicit basis to HDR (n)

We take the projective limit over n of the ζ1

and rewrite Equation 2.5.19

∞ X xdx ∨ k dx ∨ l ak,l (ω1 ) ⊗ (ω2 ) + bn (ω2∨ )n . y y n=0 k,l≥0

X

(2.6.1)

Taking the projective limit of the right hand side of (summing over all possible j’s) Equations 2.5.20, 2.5.21, gives:

∞ X j=0

j+1 X 1 j+1 F i Gj+1−i dG(ω1∨ )i+k ⊗ (ω2∨ )j+1−i+l . i (j + 1)! i=0

(2.6.2)

j+1 X 1 j+1 F i Gj+1−i dF (ω1∨ )i ⊗ (ω2∨ )j+1−i+n . (j + 1)! i=0 i

(2.6.3)

∞ X j=0

51

Therefore the residue of ζ1 is given by the residue at t = 0 of: P

k,l≥0 ak,l

P∞

1 j=0 (j+1)!

Pj+1 i=0

j+1 i

i j+1−i F G dG(ω1∨ )i+k ⊗ (ω2∨ )j+1−i+l + (2.6.4)

P∞

n=0 bn

P∞

1 j=0 (j+1)!

Pj+1 i=0

j+1 i

F i Gj+1−i dF (ω1∨ )i ⊗ (ω2∨ )j+1−i+n

The condition is that this should be equal to ω1∨ (ω2∨ if i = 2). Substituting x for ω1∨ And y for ω2∨ , using standard calculations we obtain: P P P F i Gh i h ( k,l≥0 ak,l xk y l ) · ( ∞ j=1 0≤h,i|h+i=j Rest=0 ( i! h! dG) · x y )+ (2.6.5) P∞ P P F i Gh n i h ( ∞ j=1 0≤i,h|h+i=j Rest=0 ( i! h! dF ) · x y ) = y n=0 bn y )( Notice that we may allow the summation of j to begin from 0, as Rest=0 dF, dG = 0, there is no change. We therefore obtain the following equation in x, y: P (x, y)·Rest=0 exp(F (t)·x) exp(G(t)·y)dG+Q(y)·Rest=0 exp(F (t)·x) exp(G(t)·y)dF = x. (2.6.6) This equation can also be written as: Rest=0 ((P (ω1 , ω2 ) ·

xdx dx + Q(ω2 ) · ) · exp(F (t) · ω1 + G(t) · ω2 )) = ω1 . y y

(2.6.7)

Chapter 3 Elliptic polylogarithms over a general base scheme 3.1

Motivation: the classical polylogarithms over P1 − {0, 1, ∞}

We start with a review of the classical theory of the polylogarithms. For a very detailed exposition of the subject we refer the reader to [81] Chapter IV. This section is concluded by an analog of Equation 2.6.7. The polylogarithmic system is obtained by extending the logarithmic pro-sheaf by adding an element p satisfying ∇(p) =

dz 1−z

· 1 (cf. [3] definition 6.5(ii)).

We make the last statement precise. What does 1 stand for? In the elliptic case we have ∇1 =

dx y

⊗ ω1∨ +

xdx y

⊗ ω2∨ (cf. Equation 2.3.1), whereas over P1 − {0, 1, ∞} we

have: ∇1 = − dz ⊗ e. z We start with Log (1) over C∗ = P1 − {0, ∞}. We recall from [81] Theorem I.3.3 (1) that for each z ∈ C∗ , we have an isomorphism of fibers: Logz ∼ = H1 (C∗ , {1, z}; Q),

where the latter group is the first relative homology of C∗ with respect to {1, z}. Let e represent the homology class corresponding to the closed loop around 0 in a clockwise direction, and let γ represent the relative homology class such that γ(z) corresponds

52

53

to the path from 1 to z. We first notice that e is a global section of Log (1) , whereas γ has a non-trivial monodromy. What does this mean? Log (1) is a local system over C∗ . So, one must examine how an element of π1 (C∗ ) =< e > acts on it, and it suffices to check the action of the generator e on it. The notation of e is ambiguous. In addition to denoting an actual loop e, it also represents its class inside π1 (C∗ ) and its class inside Log (1) . We denote the action of π1 (C∗ ) on Log (1) by “·”. We also notice that locally e, γ is a basis for Log (1) (this follows by the representation of Log (1) as the relative homology H1 (C∗ , {1, z}; Q)). It is easy to verify that e ◦ γ ∼ e + γ. Let 1 := γ − e log(z). Then 1 is a global section of Log (1) ⊗Q OC∗ . Since e, γ are elements of the Q local system, we have: ∇(e) = ∇(γ) = 0. This implies that ∇(1) = − dz e. We denote: Log (n) = Symn Log (1) . Over OC∗ it is spanned by ek 1l z with k + l = n. We have: ∇ek 1l = −l dz ek+1 1l−1 . We let ek,n = z

ek 1l l!

we obtain:

∇(ek,n ) = −ek+1,n dz . We define a map p : Log (1) → Q by sending e to 0 and γ to 1 z (it therefore also sends 1 to 1). It induces a map pn∗ : Log (n+1) → Log (n) as follows: pn∗ (a1 ⊗ a2 ⊗ · · · an+1 ) = Pn+1 î ⊗ · · · ⊗ · · · an+1 . This implies that pn∗ ek,n+1 = ek,n , k ≤ n, and i=1 p(ai )a1 ⊗ · · · ⊗ a pn∗ en+1,n+1 = 0. Hence, by defining Log to be limLog (n) we have a canonical basis e0 , e1 , . . . , where ←− pn∗ Q k k,n e = lime . The Hodge filtration is defined by F j Log = −j≥k≥0 OC∗ ek , W2n Log = ←− pn∗ Q W2n+1 Log = k≥−n OC∗ ek . It is easy to verify that Log/W−2(n+1) ∼ = Log (n) . In the language of [8] page 98, we may write any element of the polylogarithmic pro-sheaf as P i v0 (z)p + ∞ i=0 vi+1 (z)e , or in the coordinate description as (v0 ; v1 , v2 , . . .). One may think of p, e0 , e1 , . . . as the usual coordinates in C∞ . There is an isomorphism of C vector bundles Y k∈N0

OP1 (C)−{0,∞} → LogC

54

given by ek → ek . This provides also the Q-structure on LogQ via the mapping Q k k k∈N0 OP1 (C)−{0,∞} → LogC given by e → e exp(log ze) where the exponent is taken formally. It also provides a way to go from the global sections of the complexified logarithmic pro-sheaf to its Q-local structure. Remark 3.1.1. This is done on pages 215-219 [81], but our ek means ek there (and l k

his ek means our lim γ l!e = ek exp(−e log(z))), and he takes γ to be the path from z ←− l to 1. We now define the polylogarithmic extension. We use coordinate description. Define: 

 0 0 0    1 0   E0 :=  1       O





0 0 0

       ,      

O

1 0



0 0 0

 1 0 0    0 0   E1 :=  0       O

O

0 0

             

0 0 P i Any element of the logarithmic (!) sheaf l may be written as l = ∞ i=0 vi (z)e , 1 0

or in the coordinate description as l = (v1 , v2 , . . .). In this description, we know that: ∇Log (v) = dl −

dz E0[1,∞]×[1,∞] (l). z

Let Li be the Euler polylogarithmic section

of LogOC∗ . It is defined for |z| < 1 by the formula Li(z) :=

∞ X

Lk+1 (z)ek

(3.1.2)

k=0

(cf. [9] page 163), where Lk :=

X zn n≥0

nk

,

k ≥ 1.

Then Li extends to C∗ − {1} as a multi-valued holomorphic section, and Li := Li mod LogQ is a well-defined single valued section of LogO /LogQ on C∗ −{1}. we obtain

55

∇Log (Li) = −d log(1 − z) · e0 . We now extend the logarithmic system, to obtain the polylogarithmic system (Pol, ∇Pol ) as follows. (i) As an OP1 (C)−{0,1,∞} -module it is OP1 (C)−{0,1,∞} · p ⊕ Log. (ii) The connection is defined by ∇Pol (p) =

dz 1−z

· e0 , ∇Pol |Log = ∇Log .

we obtain that for any element v = (v0 ; v1 , v2 , . . .) ∈ Pol, we have ∇Pol (v) = dv − (

dz dz E0 + E1 )v. z z−1

By the computation above, we deduce that ∇Pol (p − Li) = 0,

(3.1.3)

thus providing a horizontal section. Thus, we showed how to trivialize p, starting from the polylogarithmic pro-sheaf. It is easy to verify by this computation that there is a horizonal isomorphism Y OP1 (C)−{0,1,∞} × OP1 (C)−{0,1,∞} → Pol, k∈N0

given by the matrix 



1

 −L  1  L(z) =  −L2   ·  ·

1 log(z) (log(z))2 2!

·

 O    . 1   log(z) 1   · ·

Therefore, its inverse 

1



  Λ  1 O  1      Λ(z) = Λ2 − log(z) 1    2 (− log(z))  · − log(z) 1 2!   · · · ·

56

provides the isomorphism in the opposite direction. Since it is also horizontal, we deduce a way to express p and e0 above (which correspond to the column vectors (1, 0, 0, . . .) and (0, 1, 0, 0, . . .)), in terms of the trivial infinite dimensional complex vector bundle over P1 −{0, 1, ∞}; we express an element of our bundle (v0 (z), v1 (z), v2 (z), . . .) P i i i as ∞ i=0 vi (z)e , where we may think of e as a formal notation for (ade0 ) (e1 ) (cf. [8] page 98). We notice that   0   1     0  e0 = exp(−log z · e) = Λ(z)   . 0     ·   · Let

Λ(z, e) := 1 +

∞ X

Λn (z)en−1

n=1

where Λn (z) :=

1 (n−1)!

Rz 0

  1   0     0  = Λ(z)   , 0   ·   ·

t n−1 d log(1−t) ( log ) are the Debye polylogarithms cf. [9]. 2πi 2πi

Using the formula log dΛk−1 (3.1.4) 2πi dz we deduce that dz Λ(z, 2πie) = 1−z exp(log z · e), thus p(z) = Λ(z, 2πie). Therefore, (k − 1)dΛk =

we found a way to write the polylogarithmic extension in terms of the trivial bundle. The polylogarithmic extension may be described in terms of residues via the equation analog to Equation 2.2.6.7 Resz=1 dz p(z) = 1. We omit the proof thereof, but this is a straightforward computation.

(3.1.5)

57

3.2 3.2.1

The elliptic case Eisenstein functions and series

We recall notations from [79] Chapters III & IV; consider a lattice in C generated by 1 and τ with =(τ ) > 0. Let ξ be a coordinate on C and z := exp(2πiξ), q := exp(2πiτ ). P P The symbol 0 denotes the summation over nonzero elements of a lattice and 0e denotes some variant of regularization of the divergent sums, known as Eisenstein summation. The precise details of this summation are of no concern to us but we only need some of their (well-known) properties. We set ek (τ ) =

0 X

−k

(nτ + m)

0 X and Ek (ξ, τ ) = (ξ + nτ + m)−k .

e

(3.2.1)

e

The ek are modular functions for k ≥ 4, and Ek are elliptic functions for k ≥ 2. The ek are called Eisenstein series. The Ek are called the Eisenstein functions. One can express the Weierstrass elliptic function ℘, its derivative and modular forms as ℘ = E2 − e2 , ℘0 = −2E3 , g2 = 60e4 , g3 = 140e6 . It is known that the ek are rational polynomials in e4 and e6 for k ≥ 8, while the Ek are rational polynomials in E2 − e2 , E3 , e4 and e6 for k ≥ 4. Moreover, the rings Q[ek ] and Q[Ek , ek ] are stable under the derivation (2πi)∂τ . The basic theta function is defined by the Jacobi product formula 1

1

1

θJ (ξ, τ ) = −iq 8 (z 2 − z − 2 )

∞ Y (1 − zq j )(1 − z −1 q j )(1 − q j ).

(3.2.2)

j=0

We have E1 = ∂ξ θJ . We notice that E1 (ξ + 1, τ ) = E1 (ξ, τ ) and E1 (ξ + τ, τ ) = E1 (ξ, τ ) − 2πi.

(3.2.3)

58

3.2.2

Geometric construction of the elliptic logarithmic extension

We use the description of an elliptic curve over the complex numbers as a complex torus, defined by C/Γτ , Im(τ ) > 0, where Γτ :=< 1, τ > is the lattice spanned by 1 and τ (when τ varies or is clear from the context we omit the corresponding subscript). By applying the map z → exp(2πiz), using the notation q = exp(2πiτ ) we may regard this complex torus as C ∗ /q Z . We obtain a family of elliptic curves E/D, where D := {q ∈ C∗ : 0 < |q| < 1}, whose fiber over q ∈ D is given by Eq = C ∗ /q Z . Let (q)n denote the automorphism of E defined by the rule (z, q) → (q n z, q). Notice that U0 = E \ 0(D) corresponds to V := C∗ × D \ {(q n , q), q ∈ D, n ∈ Z}. There is a non-degenerate skew-symmetric intersection pairing < ∗, ∗ >: Γ ∧ Γ → Z(1). This is explained with more details in subsection 3.8.4 below. To describe the logarithmic extension, we first have to describe the Hodge structure H1 (E; Q): for any q ∈ D, We let eq denote the simple loop around 1 in Eq = C∗ /q Z , which is the image under the mapping z → exp(2πiz) of the straight line from 0 to 1 on the complex plane, and fq denotes the section of F0 H1 (E; Q) such that < eq , fq >= 2πi. This implies in particular that f is a global section of the complexification of H1 (E; Q). That is, f has trivial monodromy when q varies on a simple loop around 0. One needs to be careful here. Since we are working with a family of elliptic curves, one needs to check what happens not only for the action of the fundamental group of each elliptic curve, but also to check a loop in the direction of the family. We next notice that e = eq and f = fq form a basis to H1 (E; Q)⊗OE . To describe the Q-lattice H1 (E; Q), we let f˜ = f˜q denote the simple path from 1 to q in Eq ∼ = C∗ /q Z , which is the image under the mapping z → exp(2πiz) of the straight line from 0 to

59

τ on the complex plane. We have f˜ = f +

1 e log q 2πi

= f + eτ . Indeed, when q varies

on a simple loop around 0, the monodromy of f˜ is f˜ → f˜ + e. This implies that f˜ − eτ is some multiple of f . On the other hand < e, f˜ − eτ >=< e, f˜ >, because < ∗, ∗ > is skew-symmetric. Since the image of < e, ∗ >: Γ → Z(1) is Z(1), and the pairing is non-degenerate, we deduce that f˜ = f + eτ . Next, Log (1) is the Hodge structure over E whose fiber over (z, q) ∈ E is H1 (Eq , {1, z}; Q)z,q . For any q ∈ D, we have eq and fq as above, but also the section γz corresponding to a simple path from 1 to z. The monodromy of γ when z varies on the loop e is γ + e and on the loop f˜ is γ + f˜. Let 1 denote the element γ −

1 e log z 2πi

= γ − eξ. Then

1 is a global section of the pullback of Log (1) to C∗ × D, since the monodromies in both the direction of e and the direction of a loop around q = 0 are trivial. Recall the short exact sequence of variations of mixed Hodge structures 1.1.2.9 p

0 → V → Log (1) → Q(0) → 0. The projection p naturally induces a map of variations of mixed Hodge structures pn : Log (n+1) → Log (n) satisfying pn∗ (a1 ⊗ a2 ⊗ · · · an+1 ) =

Pn+1 i=1

p(ai )a1 ⊗ · · · ⊗ aî ⊗ · · · ⊗ · · · an+1 . This

implies that pn∗ ek f l 1n+1−k−l = (n + 1 − k − l)ek f l 1n−k−l , k + l ≤ n, and pn∗ ek f l = 0, for k + l = n + 1. Hence, by defining the pro-sheaf Log to be limLog (n) we have a canonical basis ek f l ←− pn∗

k l

where e f :=

1 lim ek f l 1n+1−k−l . ←− (n+1−k−l)! pn∗

Elements of this basis have no monodromy

when z varies on a loop around 1. If we let vk,l := ek f l γ ∞ := limek f l γ n+1−k−l we ←− k l

pn∗ k l

have: e f = vk,l exp(−eξ). Let us compute the monodromy of e f when z varies on

60

the path f˜. For this we notice that the monodromy of 1 when z varies on the path f˜ is given by 1 = γ − eξ → γ + f˜ − eξ − eτ = γ − eξ + f = 1 + f. That is 1 → 1 + f. This implies that the monodromy of ek f l when z varies on the path f˜ is given by ek f l → ek f l exp(f ).

3.2.3

The corresponding multiplier

Multipliers In this section we explain the notion of multiplier. There is a correspondence between vector bundles over Eq = C ∗ /q Z and q-difference equations on trivial bundles on C∗ . This work is classical, and we refer to the paper of [2] for details. We describe this correspondence in the nilpotent case. For a scheme X, Let N Bdl(X) denote the category of nilpotent vector bundles. The notion of nilpotence refers to unipotent transition matrices. Let V be a C-vector space and m ∈ GL(V ). It is well known that the equation v(qz) = mv(z)

(3.2.4)

gives rise to a vector bundle over E, whose sections over the open subvariety U are given by the analytic solutions to Equation 3.2.4 over π −1 U . We will frequently denote the resulting vector bundle by V. The set of pairs (V, m) where V, m are as above and m is unipotent, form a Tannakian category in a very natural way. We denote this category by V M U (E). For a proof, we refer to [64], proposition 2.1.3. It also shows the following

61

Proposition 3.2.5. The functor G : V M U (E) → N Bdl(E) defined by G((V, m)) := V is an equivalence of categories. The non-trivial fact is that constant multipliers describe all vector bundles over E. A priori, a multiplier may depend on z. That is, we might want to look at equations of the form v(qz) = m(z)v(z), where m : C∗ → GL(V ) satisfies certain conditions. This proposition tells us that nilpotent vector bundles are described using constant multipliers via G. We record here a useful fact, [64] proposition 2.1.5: Proposition 3.2.6. Let m ∈ GL(V ) be unipotent. The global holomorphic sections of the bundle with multiplier m are the constants fixed by m. The logarithmic pro-sheaf in this point of view Let D = {q ∈ C∗ : 0 < |q| < 1} ⊂ C∗ be the coordinate punctured disk. We let ˜ E˜ = C∗ × D. For n ∈ Z, we denote by (q)n the automorphism (z, q) → (q n z, q) of E. ˜ We let E = (q)Z \E. ˜ Then pE : E → D is the standard This defines a Z-action on E. family of elliptic curves over D. We notice that for q ∈ D the curve Eq is equipped with a canonical embedding 2πiZ = H1 (C∗ , Z) ,→ H1 (Eq , Z) such that H1 (Eq , Z)/2πiZ has no torsion and E/D is the universal family of elliptic curves equipped with such data. We let π : E˜ → E be the canonical projection. Fiberwise, we have πq : C∗ → C∗ /q Z , and we also let

Log   y E

62

be the vector bundle corresponding to the local system described above. We will describe an isomorphism of pro-systems (it is an isomorphism for all finite levels, Q n ∗ compatible with morphisms in the directed sets) Θ : V := ∞ n=0 Sym H → π Log. I recall here that π ∗ Log is the sheaf associated to the presheaf: U → limV ⊇π(U ) Log(V ). −→ One must be careful, and remember that the sheaf is not the same as the presheaf. Remark 3.2.7. For each fiber q ∈ D we know that πq∗ Logq is a trivial bundle as a vector bundle over C∗ . Thus, in order to find a trivialization over π ∗ Log, one needs to handle the monodromy as q varies on a loop around 1. We may view sections of π ∗ Log(W ) as functions from x ∈ W to π ∗ Logx = Logπ(x) . We recall the section γz = the straight line from 1 to z on C∗ . We therefore define on 1 e log(z), which we previously called 1. Using this section we C∗ the section θz = γz + 2πi

obtain a basis of global sections to π ∗ Log given by fk,l (z) = ek f l θz∞ . This basis defines a trivialization of π ∗ Log, given as a function from W to π ∗ Log as Θ(ek f l ) = fk,l . These are global section of π ∗ Log. The point is that (γ + eξ)(ξ + τ ) = (γ + eξ)(ξ) + f , hence the pushforwards π∗ fk,l are multivalued over E , but as a section of E˜ they are globally well defined single valued sections. We are now ready to describe the multiplier. Again, the multiplier is defined as the projective limit of all the multipliers in the pro-system. We will not bother with the formality of writing everything for the finite levels and taking the projective limit, but it should be clear that this underlies everything done below. We notice that a section of Log may be pulled back to π ∗ Log (namely π ∗ s(x) = s(π(x))).

Since

π ◦ q = π, the multiplier corresponding to the logarithmic pro-sheaf is given as the composition, ∗

q Θ id Θ m−1 : V := q ∗ V −− −→ q ∗ π ∗ Log −−−→ π ∗ Log −−−→ V . −1

We first notice that the mapping m−1 is given on an element v ∈ V as v → q ∗ (vθ(z)) = vθ(qz) = vθ(z) exp(−f ) → v exp(f )

63

implying that m is a multiplication by exp(−f ). Let us explain this more carefully. The idea is that fibers of the vector bundle over z and qz are identified via a linear transformation. Say T : Vqz → Vz . This is exactly m−1 = exp(f ). If you have a section on the quotient space, it means that you have a section v(z), such that T (v(qz)) = v(z). This implies that v(qz) = exp(−f )v(z).

We also notice that

Θ may be extended to a mapping Θ : V := V ⊗ O → π ∗ LogO , and this induces a mapping M which still makes sense. If Θ(v(z)) is a section of Log, we deduce that q ∗ Θ(v(z)) = Θ(v(z)). This would imply that M (v(z)) = v(z). Explicitly v(z) → q ∗ θ(v(z)) = v(qz)θ(qz) = v(qz)θ(z) exp(f ) → v(qz) exp(f ). this implies that v(qz) = exp(−f )v(z). This shows that our definition for m is the right one.

3.2.4

Algebraization of Log

In this section we describe how to transform the logarithmic pro-sheaf, which is given by the vector space V with multiplier exp(−f ) into its algebraization. Caution: the term algebraic is somewhat ambiguous. In this context, we only mean that the multiplier is trivial, that is, one works with sections of the covering space, as opposed to the usual context of algebraicity for elliptic curves, when algebraic usually means over the universal elliptic curve. Next, we notice that the logarithmic pro-sheaf is not trivial as shown above. However, over U0 = E − 0(S) it is. This means that the logarithmic pro-sheaf over U0 is isomorphic to the vector bundle given by V with trivial multiplier. Thus, we are in the following situation; we are given a C vector bundle P over E given by the vector space Cn and the multiplier m. P over U0 is trivial, that is, it is isomorphic to Cn with multiplier 1. We also assume that π ∗ P|π−1 (E) is trivial. Or even more generally, we are given complex vector spaces W, W 0 giving rise to complex vector bundles W, W 0 on V := C∗ × D \ {(q n , q), n ∈ Z} via multipliers m, m0 respectively. There is also a group action of G =< q > on V , given by (q) ◦ (z, q) = (qz, q). We are looking

64

for a morphism between W, W 0 given by g : V → Hom(W, W 0 ), making the following diagram commutative m(z,q)

Wz,q −−−→ Wqz,q     g(z,q)y g(qz,q)y 0 0 −−0−−→ Wqz,q Wz,q m (z,q)

In our situation we condition that g be an isomorphism. The Gauge transformation We notice that U0 = E − 0(S) corresponds to V := C∗ × D \ {(q n , q), n ∈ Z}. Over U0 there is a trivialization of the logartihmic sheaf. By equation 3.2.3, we deduce that (ξ,τ )f (ξ,τ )f ) is a section of Log over U0 (notice that it means Θ(ek f l exp( E1 2πi )) ek f l exp( E1 2πi

and the monodromy is given by multiplication by exp(f ) times the monodromy of (ξ,τ )f (ξ,τ )f exp( E1 2πi ), which is exp(−f ), thus the monodromy of Θ(ek f l exp( E1 2πi )) is trivial.

Another way is to use the terminology of multipliers, and it is clear that the above section satisfies v(qz) = exp(−f )v(z). Definition 3.2.8. The Gauge transformation is the transformation corresponding to (ξ,τ )f multiplication by exp( −E12πi ).

The Gauge transformation sends this section to ek f l inside the algebraic version of Log. The algebraic logarithmic pro-sheaf The Gauge transformation is nothing more than a (holomorphic) change of variables on each fiber. We therefore define the algebraic logarithmic pro-sheaf to be the logarithmic pro-sheaf under the change of variables corresponding to the Gauge transformation. That is, Definition 3.2.9. Let Logal be the Gauge transformation of Log|U0 .

65

The corresponding vector bundle is trivial (the multiplier is 1). More explicitly, each v ∈ V represents a flat section of Log|U0 . The connection on LogU0 is given by ∇alg = g(z, q)∇g −1 (z, q). Proposition 3.2.10. ∇alg = ∇S • H − e ⊗ ω −

f f ⊗ (η + e2 ω) + E3 dτ. 2πi (2πi)2

Proof. The connection of the analytic log sheaf (i.e., with multiplier exp(f )) is defined as follows: on the homology we define ∇H e = 0, ∇H f = −edτ . Then there is a natural connection induced on S • H, and the connection ∇ = ∇Log := ∇S • H − edξ. This connection induces a connection on the algebraic logarithm sheaf, namely ∇alg = g(z, q)∇g −1 (z, q). ∇alg a = g(z, q)∇(g −1 (z, q)a) = g(z, q)∇S • H (g −1 (z, q))a + −E1 E2 )f −E2 f a 1e dξ + (E3(2πi) dτ − E2πi dτ + ∇a. Hence ∇alg = ∇S • H − 2 2πi −E1 E2 )f E2 f f 1 dξ + (E3(2πi) dτ = ∇S • H − e(dξ + 2πi E1 dτ ) − (E2 − e2 + e2 ) 2πi (dξ + 2 2πi

g(z, q)g −1 (z, q)∇a = 1e edξ − E2πi dτ −

1 E dτ ) 2πi 1

f + E3 (2πi) 2 dτ = ∇S • H − e ⊗ ω −

f 2πi

f ⊗ (η + e2 ω) + E3 (2πi) Above, 2 dτ .

(ξ,τ )f I used the fact that ∇S • H (g −1 (z, q)) = g −1 (z, q)∇S • H ( E1 2πi ), and d(E1 (ξ, τ )) =

−E2 (ξ, τ )dξ +

1 (E3 2πi

− E1 E2 )dτ.

Let us compare this result with known results in the case of a single elliptic curve. In the previous Chapter, we showed (Equation 2.5.4) ∇Log = ∇S • +

dx xdx ⊗ ω1∨ + ⊗ ω2∨ . y y

(3.2.11)

This also appears in [7] corollary 1.16. We notice that in loc. cit. page 19 section 1.5, one might change the basis of differential forms to be ω = dx/y, ω ∗ = −xdx/y−e2 dx/y and the connection becomes: ∇Log = ∇S • + ω ⊗ ω ∨ + ω ∗ ⊗ ω ∗∨ . where ω ∨ , ω ∗∨ are the dual basis for ω, ω ∗ .

(3.2.12)

66

This means that the translation between the algebraization of the analytic logarithm sheaf and the sheaf, as we and [7] found, is given by e → −ω ∨ , f → 2πiω ∗∨ . We recall that the elliptic logarithmic pro-sheaf was defined in Chapter 1 Section 1.2. We used the comparison established by Proposition 1.2.22 in an essential way.

3.3

Quasi-Hodge sheaves

Our goal is to describe the Q-Hodge algebraic realization of the elliptic polylogarithm. Using the Schottky uniformization, we enlarge the category of Hodge sheaves on our elliptic curve to the one of Quasi-Hodge sheaves, which is a clever way to record data on a covering space of the elliptic curve and is introduced in [9] section 4. Needless to say, there is nothing original in this section.

The idea is to represent the polylogarithmic pro-system as a Baer sum of two pro-systems, the first is a direct analog of the classical polylogarithmic pro-system on P1 (C) − {0, 1, ∞}, by taking a η-regularized q-averaging on it and the second is a Koszul correction, which is designed to trivialize the non-trivial q action on the first extension.

3.3.1

Definition of Quasi-Hodge sheaves

Definition 3.3.1. Let S be a connected complex analytic manifold. A quasi Q-Hodge sheaf V on S is a variation of mixed Q-Hodge structures: that is, a Q-local system on S together with a mixed Q-Hodge structure on each fiber Vs , s ∈ S, such that W• is an ascending filtration by Q-local subsystems, F • is a holomorphic descending filtration satisfying Griffiths’ transversality condition ∂F i ⊂ F i−1 . Denote by QH (S)Q the category of quasi Q-Hodge sheaves. This is a Tannakian Q-category. Let π : S˜ → S be a local homeomorphism such that S˜ is nonempty and

67

connected. A quasi Q-Hodge sheaf F on S relative to S˜ is a collection (FS˜ , FS i , ϕi ) ∼ ˜ Q , FS i ∈ QH (S)Q (i ∈ Z), and ϕi : π ∗ FS i → where FS˜ ∈ QH (S) griW (FS˜ ) are

isomorphisms. Clearly FS i is a variation of pure Hodge structures of weight i on S, for each i ∈ Z. Such F ’s form a category QHS˜ (S) which is a Tannakian category in an obvious manner. One has QHS (S) = QH (S)Q . If π 0 : S˜0 → S is another local homeomorphism and f : S˜0 → S˜ is an S-morphism then we have an exact tensor functor f ∗ : QHS˜ (S) → QHS˜0 (S). We have [9] Lemma 4.1.1. The term quasi-Hodge sheaves is given in [9], and we follow their terminology. Lemma 3.3.2. f ∗ is a fully faithful embedding. In particular, π ∗ : QH (S)Q → QHS˜ (S) is a fully faithful embedding. Corollary 3.3.3. We may consider QHS˜ (S) as an enlargement of the category QH (S)Q (and of H (S)Q if S is algebraic). The following is [9] Lemma 4.1.2. Lemma 3.3.4. Let F ∈ QHS˜ (S) be a quasi-Hodge sheaf such that F = W−2 F and F 0 F = 0. Then the Q-vector space Ext1QHS˜ (S) (Q(0)S , F ) is canonically isomorphic −1 to the space of global sections g of FOS˜ /FSQ FS˜ ⊗ Ω1S˜ . ˜ such that ∂g ∈ F

Proof. First, let us explain the following: given a sheaf F and a subsheaf H, what is the quotient sheaf G := F/H? Given an open set U , an element g ∈ G(U ) is described via a covering U =

S

i

Ui

by open sets Ui , and sections gi ∈ F(Ui ), such that on intersections Ui ∩ Uj we have gi − gj ∈ H(Ui ∩ Uj ). We may say that gi is a local lifting of g on Ui . In the proof of Lemma 4.1.2 [9], they let g˜ represent any local lifting. The end result is an extension Pg , and it is described by its restriction to each Ui , together with a system −1 of gluing morphisms. Given a section g ∈ FOS˜ /FSQ FS˜ ⊗ Ω1S˜ (that ˜ with ∂g ∈ F

is, we assume given the compatible system gi as above), we construct an extension Pg ∈ Ext1QHS˜ (S) (Q(0)S , F ).

68

1. Each “local” extension Pi is defined as follows: we define 1i to be a flat, rational section of Pg˜, mapping to 1Q ∈ QUi . We then define F 0 (Pg˜) := OUi · (gi + 1i ). 2. We next define F 1 (Pi ) := 0, F 0 (Pi ) := OUi · (gi + 1i ), F i (Pi ) := F i (F ) + F 0 (Pi ), for i < 0. 3. Over each Ui we obtain extensions Pi , and on them sections gi + 1i . Now, since we know that h := gj − gi ∈ FSQ ˜ (Ui ∩ Uj ) there is a canonical isomorphism ϕi,j : Pi → Pj on the intersection, defined by ϕi,j |FUi ∩Uj = id, ϕi,j (1i ) = 1j + h from which follows that it sends gi + 1i to gj + 1j . Notice that 1j + h is also a flat rational section of Pj |Ui which maps to 1Q in QUi ∩Uj . Altogether, we deduce that these extensions glue together to form an extension Pg over S, and that there is a global section “g + 1” belonging to F 0 (Pg ). 4. We obtain F 1 (Pg ) := 0, F 0 (Pg ) := OUi · (g + 1), F i (Pg ) := F i (F ) + F 0 (Pg ), for i < 0. 5. Let us clarify some subtle points. On a set Ui there is a section 1i ∈ PUi Q . When one continues this section to Uj , we regard 1i as the section 1j + hi,j on Uj and one can then continue this section to Uk and obtain the section 1k + hi,j + hj,k = 1k + hi,k etc. This implies that there is no global section “100 ˜ on Pg (S). 6. The condition ∂g ∈ F −1 FS˜ ⊗Ω1S˜ implies that Pg satisfies Griffiths’ transversality condition. 7. We also remark that even though one may choose different coverings and liftings, by taking a refinement of both coverings, one obtains a canonical isomorphism between the two extensions.

69

For the other direction of the lemma, we refer the reader to the proof of Lemma 4.1.2 in [9].

3.4

The Euler elliptic polylogarithmic extension

The polylogarithmic extension is built upon a simpler extension of Log by the trivial bundle.

Using the description of an elliptic curve over the complex numbers as a

complex torus, we described the logarithmic pro-sheaf over E similarly to describing the logarithmic pro-sheaf over P1 − {0, 1, ∞}. We notice that the morphism π : V → U0 is a local homeomorphism, and V is non-empty and connected. Since U0 is algebraic, and Log|U0 is an algebraic Hodge sheaf, it follows by Corollary 3.3.3 that LogU0 ∈ QHV (U0 ). Moreover, we have W −2 Log(1) = Log, F 0 Log(1) = 0.

3.4.1

The elliptic polylogarithm functions

This section is contained in section 4.4 in [9]. We define a q-averaged version of the classical polylogarithm functions. Let Li, the Euler polylogarithmic section of Log(1)0 over C∗ , be defined for |z| < 1 by the formula Li(z) :=

∞ X

Lk+1 (z)(

k=0

e k ) 2πi

(this was defined in equation 3.1.2 but there is some normalization because the connection is also normalized). Then Li extends to C∗ \ {1} as a multi-valued holomorphic section, and Li := Li mod Log(1)0Q is a well-defined single valued section of Log(1)0O /Log(1)0Q on C∗ \ {1}. One has ∇Li = −d log(1 − z) · 1,

[−1]∗C∗ Li = Li + log0 .

(3.4.1)

70

These arguments establish the fact that Liell is a well-defined (single valued) section of Log(1)O /Log(1)Q over V . For more details, see [9] subsection 4.4. We let [−1] : E → E be the involution induced by z → z −1 on C∗ , and for m ≥ 0 we let: logm (z, q) :=

m!(log z)k+1−m+l (log q)m−l e k l f. l!(m − l)!(k + 1 − m + l)! 2πi k,l≥0k+1≥m−l≥0 X

(3.4.2)

In the formula we choose the branches of log that vanish at 1. To see this, one observes that dz dq + mf m−1 . (3.4.3) z q ˜ It is also easy to see is a well-defined section of Log(1)O /Log(1)Q over E. ∇Log (logm (z, q)) = f m

logm that

(q)n∗ logm =

X na a≥0

a!

logm+a ,

[−1]∗ logm = (−1)m+1 logm .

(3.4.4)

Moreover, for any n, almost all logm s lie in W−n Log(1). Therefore, for any seP quence {am }m≥0 with am ∈ Q, the series m≥0 am logm converges to a section of ˜ We also let Bn denote the Bernouli numbers. Log(1)O /Log(1)Q over E. Then, the function Liell :=

X X X X Ba+1 (q)n∗ Li = (q)n∗ Li + [−1]∗ (q)n∗ Li − loga (a + 1)! n≥0 n≥1 a≥0 n∈Z

(3.4.5)

is a well-defined (single valued) section of Log(1)O /Log(1)Q over V . Since LogU0 ∈ QHV (U0 ) and W −2 Log(1) = Log, F 0 Log(1) = 0 Lemma 3.3.4 Liell defines an extension: 0 → LogV → F ell → 1V → 0

(3.4.6)

of Quasi-Q-Hodge sheaves on V (it is also referred to as Quasi-Q-Hodge sheaves on U0 relative to V ). This is almost the elliptic polylogarithmic extension as we know it, only that this F ell is defined over V and does not descend to U0 (it is not invariant under the q-action).

71

Thus, some modification is needed, which is the content of section 4.5 of [9]. However, as one can easily observe, F ell is the obvious analog of the classical polylogarithmic extension. It actually encodes all the important data. Notice that Liell is the regularized sum of the Li which gives the classical polylogarithmic extension. Following the previous subsection, we notice that there is a global section p ∈ F 0 (F ell ), such that locally (over each Ui ) we have: pUi = 1 + gi where gi is a local lift of Liell over Ui . To compare with the classical situation, we also had a global section p such that ∇(p − Li) = 0 (cf. equation 3.1.3). Here we have the same situation: a global section p and a multi-valued section Liell (which may be thought of as a single-valued section of LogO /LogQ ). By choosing the same functions gi on the same covering, we see that the restriction of p − Liell to Ui is 1, which is a rational flat section of F ell . We also know that this extension is characterized by ∇(p) = ∇an (Liell ). We will compute ∇an (Liell ) in Lemma 3.4.26. We record here the notation p := 1 + Liell .

3.4.2

(3.4.7)

A two variable Jacobi form

The following function was introduced by Kronecker [58]. It appears in [79], II.2, equation (4), and then in [83], and [61] section (3), [64] section 2.2. F (ξ, η; τ ) := (2πi) 1 −

1 1−z

−

1 1−w

−

P∞

m,n=1 (z

m

wn − z −m w−n )q mn , (3.4.8)

=(τ ) > =(ξ) > 0, =(τ ) > =(η) > 0, where q := exp(2πiτ ). By [83], F (ξ, η; τ ) can be continued to a meromorphic function with poles at divisors ξ = m + nτ, η = m0 + n0 τ , where m, m0 , n, n0 ∈ Z and F (ξ, η; τ ) =

θJ0 (0; τ )θJ (ξ + η; τ ) . θJ (ξ; τ )θJ (η; τ )

(3.4.9)

72

There are also transformation formulae for F F (ξ + 1, η; τ ) = F (ξ, η; τ ).

(3.4.10)

F (ξ + τ, η; τ ) = exp(−2πiη)F (ξ, η; τ ).

(3.4.11)

F (ξ, η; τ ) can also be expressed as (cf. [61] equation (6)) ∞ X (−η)k (2πi) exp − (Ek − ek ) . F (ξ, η; τ ) = η k k=1

(3.4.12)

It also satisfies the “mixed heat” equation: 2πi

3.4.3

∂F (ξ, η; τ ) ∂ 2 F (ξ, η; τ ) = . ∂τ ∂ξ∂η

(3.4.13)

computation of ∇(Liell )

The Debye polylogarithms Λk (z) describe the period matrix for the classical polylogarithms (cf. [9] section 4.8, and are defined by Z z log t k−1 d log(1 − t) 1 . Λk (z) := (k − 1)! 0 2πi 2πi

(3.4.14)

The Debye polylogarithms are related to the usual polylogarithms Lk (z) via X (log z)k−i Lk (z) = (−2πi)i Λi (z), (k − i)! k≥i≥1 −k

Λk (z) = (−2πi)

X (− log z)k−i Li (z). (k − i)! k≥i≥1

(3.4.15)

(3.4.16)

Using a similar regularization as for the Debye polylogarithms, defines the elliptic Debye polylogarithms: P P 1 Λm,n (ξ, τ ) = m! { j≥0 j m Λn (q j z) + (−1)m+n+1 j≥1 j m Λn (q j z −1 )+ (3.4.17) P (−1)m+n n≥k≥0

(−ξ)n−k τ k (n−k)!k!

ζ(−m − k) + (−1)m+n+1 Bn!n ζ(−m)

73

where ξ =

1 2πi

1 2πi

log z, τ =

matrix for F ell is:

1

log q. On page 169 in [9] it is mentioned that a period ! 0

Λm,n t1n−1 tm L 2 where L is the period matrix for the logarithm sheaf over V . L is the matrix of P

the linear transformation exp(−ξt1 − τ t1 ∂t2 ) acting on the space C[[t1 , t2 ]]. Definition 3.4.18. Let X

Λ(ξ, τ ; X, Y ) :=

Λm,n (−Y )n−1 X m .

(3.4.19)

m≥0,n≥1 ξ) + (exp Y1−1)X . We Definition 3.4.20. Let Λ(ξ, τ, X, Y ) := Λ(ξ, τ, X, Y ) + (−Yexp(−Y )(−Y τ +X)

also denote S(ξ, τ, X, Y ) := Λ(ξ, τ, X, Y ) − Λ(ξ, τ, X, Y ) =

exp(−Y ξ) (Y )(−Y τ +X)

+

1 . (1−exp Y )X

P + Corollary 3.4.21 (cf. [61] page 272). Λ(ξ, τ, X, Y ) = ∞ j=0 exp(jX)Λ (ξ+jτ ; −Y )+ P∞ exp(−Y ξ) 1 − + exp 1Y −1 exp 1X−1 . j=1 exp(−jX)Λ (−ξ + jτ ; Y ) + −Y exp(−Y τ +X)−1 By applying a formal X-derivative to Λ we deduce: Lemma 3.4.22 (lemma/definition). Λx (ξ, τ, X, Y ) := P∞

j=0

∂ Λ(ξ, τ, X, Y ∂X

)=

j exp(jX)Λ+ (ξ + jτ ; −Y ) −

ξ) exp(−Y τ +X) + exp(−Y − Y (exp(−Y τ +X)−1)2

P∞

j=1

j exp(−jX)Λ− (−ξ + jτ ; Y )

1 . (−Y τ +X)2

We state proposition 3.1 [61] Proposition 3.4.23. 2.

∂ Λ(ξ, τ, X, Y ∂τ

)=

1. 1 2πi

∂ Λ(ξ, τ, X, Y ∂ξ

)=

1 2πi

τ +X exp(−Y ξ)F (ξ, −Y2πi , τ ).

∂ τ +X exp(−Y ξ) ∂X F (ξ, −Y2πi , τ ).

The following is an easy corollary of Definition 3.8.20:

74

∂ S(ξ, τ, X, Y ∂ξ exp(−Y ξ) . (−Y τ +X)2

Corollary 3.4.24. ∂ S(ξ, τ, X, Y ∂ξ

)=

ξ) ) = − exp(−Y , −Y τ +X

We omit the proof of the following proposition Proposition 3.4.25. Liell (ξ, τ, X, Y ) = −2πiΛ(ξ, τ, X + Y τ, Y )eY ξ . X ∂ X Lemma 3.4.26. ∇Liell (ξ, τ, X, Y ) = −(F (ξ, 2πi , τ )dξ+ ∂X F (ξ, 2πi , τ )dτ )+2πi 1 dτ . X2

Lemma 3.4.27. ∇S • = d −

∂ ∂X

1 dξ− X

· Y dτ .

Proof. This follows by ∇H (Y ) = 0, ∇H (X) = −Y dτ and Leibniz rule for the symmetric powers of H. proof of Lemma 3.4.26. By Proposition 3.4.25 we have Liell (ξ, τ, X, Y ) = −2πiΛ(ξ, τ, X + Y τ, Y ) exp Y ξ. By definition and Lemma 3.4.27 ∇ = ∇S • − Y dξ = d −

∂ · Y dτ − Y dξ. ∂X

We deduce that ∇Liell (ξ, τ, X, Y ) = −2πi∇ Λ(ξ, τ, X + Y τ, Y ) exp (Y ξ) .

75

We have ∇(Λ(ξ, τ, X + Y τ, Y ) exp Y ξ) =

d Λ(ξ, τ, X + Y τ, Y ) exp (Y ξ) − Λx (ξ, τ, X + Y τ, Y ) exp(Y ξ)Y dτ

−Λ(ξ, τ, X + Y τ, Y ) exp Y ξY dξ =

dΛ(ξ, τ, X + Y τ, Y ) exp (Y ξ) + Λ(ξ, τ, X + Y τ, Y ) exp (Y ξ)Y dξ

−Λ(ξ, τ, X + Y τ, Y ) exp (Y ξ)Y dξ

−Λx (ξ, τ, X + Y τ, Y ) exp(Y ξ)Y dτ =

(dΛ(ξ, τ, X + Y τ, Y )) exp (Y ξ) − Λx (ξ, τ, X + Y τ, Y )Y dτ. We have dΛ(ξ, τ, X + Y τ, Y ) =

∂ Λ(ξ, τ, X ∂ξ

+ Y τ, Y )dξ +

∂ Λ(ξ, τ, X ∂τ

+ Y τ, Y )dτ + Λx (ξ, τ, X + Y τ, Y )Y dτ.

We have ∂ ∂ ∂ Λ(ξ, τ, X + Y τ, Y ) = Λ(ξ, τ, X + Y τ, Y ) + S(ξ, τ, X + Y τ, Y ). ∂ξ ∂ξ ∂ξ Similarly, we have ∂ ∂ ∂ Λ(ξ, τ, X + Y τ, Y ) = Λ(ξ, τ, X + Y τ, Y ) + S(ξ, τ, X + Y τ, Y ). ∂τ ∂τ ∂τ

76

By Corollary 3.4.24 we deduce: dΛ(ξ, τ, X + Y τ, Y ) =

1 X exp(−Y ξ) 2πi (F (ξ, 2πi , τ )dξ +

ξ) dξ + − exp(−Y X

exp(−Y ξ) dτ X2

∂ X F (ξ, 2πi , τ )dτ ) ∂X

+ Λx (ξ, τ, X + Y τ, Y )Y dτ.

Thus, X ∇Liell (ξ, τ, X, Y ) = −(F (ξ, 2πi , τ )dξ +

3.4.4

∂ X F (ξ, 2πi , τ )dτ ) ∂X

+ (2πi)

1 dξ X

−

1 dτ X2

.

Another computation of ∇(Liell )

In this section ∇ stands for the ∇an . We recall the definition of Liell from Equation 3.4.5 Liell :=

X

(q)n∗ Li + [−1]∗X

n≥0 n∗

We first notice that (q)

X X Ba+1 loga . (q)n∗ Li − (a + 1)! n≥1 a≥0

and [−1]∗X commute with ∇. This follows either by direct

calculation (using Equation 3.4.4) or by using the fact that these actions are endomorphisms of Log as a variation of Hodge structures, implying that they commute with the connection ∇. This implies that: X X X Ba+1 ∇Liell := (q)n∗ ∇Li + [−1]∗X (q)n∗ ∇Li − ∇ loga . (a + 1)! n≥0 n≥1 a≥0 By Formulae 3.4.3 and 3.4.1 we obtain ∇loga = f a dz/z + af a−1 dq/q

and

∇Li = −dlog(1 − z) · 1.

Hence ∇Liell = − −

P

P

n≥0

dlog(1 − q n z)exp(nf ) −

Ba+1 a a≥0 (a+1)! f dz/z

−

P

n≥0

Ba+1 a−1 dq/q. a≥1 (a+1)! af

P

dlog(1 − q n z −1 )exp(−nf )

77

Let L1 := −

P

L3 := −

P

n≥0

dlog(1 − q n z)exp(nf ), L2 := −

Ba+1 a a≥0 (a+1)! f dz/z

P

n≥0

dlog(1 − q n z −1 )exp(−nf ),

Ba+1 a−1 dq/q. a≥1 (a+1)! af

P

−

Setting w := exp(f ), we obtain d(q n z) n n≥0 1−q n z w

L1 =

P

( dz + z

P dq ∂ qn z n )( n≥0 1−q n z w ). q ∂X

We have P

qn z n n≥0 1−q n z w

P

m≥1,n≥0

=

q n dz+nq n−1 zdq n w 1−q n z

P

=

n≥0

P

n≥0

q nm z m wn =

qnz

P

m≥0

P

m≥1,n≥1

=

qn z n dz n≥0 1−q n z (w z

P

q nm z m wn =

q nm z m wn +

P

n,m≥0

1 1−z

+ nwn dqq ) =

q n(m+1) z m+1 wn =

−1 .

Similarly,

+ ( dz z

P q n z −1 dq ∂ −n )(− n≥1 1−q ). n z −1 w q ∂X

n≥1 1−q n z −1 w

n,m≥1

−n

=

P

n≥1

q n z −1

=

P

P

q n z −1 dz −n n≥1 1−q n z −1 (− z w

P

We have P q n z −1

P

d(q n z −1 ) −n n≥1 1−q n z −1 w

L2 =

m≥0

q nm z −m w−n =

P

+ n dqq w−n ) =

n≥1,m≥0

q n(m+1) z −(m+1) w−n =

q nm z −m w−n .

To compute L3 we use the formula ∞

X Bn x = xn . ex − 1 n=0 n!

(3.4.28)

This formula implies the formula: ∞

1 1 X Bn+1 n − = x . ex − 1 x n=0 (n + 1)!

(3.4.29)

78

Thus Ba+1 a a≥0 (a+1)! f dz/z

P

L3 := − + ( dz z

dq ∂ 1 )(− w−1 q ∂X

−

1 f

−

Ba+1 a−1 dq/q a≥1 (a+1)! af

P

= ( dz + z

P Ba+1 a dq ∂ )(− a≥0 (a+1)! f ) q ∂X

). (3.4.30)

We deduce that ∇Liell = L1 + L2 + L3 = ( dz + z

dq ∂ )(− q ∂X

+ = ( dz z

1−

1 1−z

−

1 w−1

−

dq ∂ f 1 )(− 2πi F (ξ, 2πi )) q ∂X

P

m≥1,n≥1

q nm (z m wn − z −m w−n )) +

1 f

(3.4.31)

+ f1 ),

where the last equality which follows by section 2.2 in [61]. This gives the same result as above. Lemma 3.4.32. F ell has zero curvature. Proof. Since LogV has zero curvature, it is enough to show that curv(F 0 F ell ) =. Thus, the problem is reduced to proving that ∇2 (1 + Liell ) = 0. Since 1 is a flat section of F ell , and using Lemma 3.4.26 we need to show that ∇ −(F (ξ,

X ∂ X 1 1 , τ )dξ + F (ξ, , τ )dτ ) + 2πi dξ − 2 dτ . 2πi ∂X 2πi X X

We have ∇ = ∇S • − Y dξ = d −

∂ · Y dτ − Y dξ. ∂X

Using the “mixed heat” equation for F (ξ, η; τ ) (Equation 3.4.13), we see that all the terms are canceled (notice that dξdτ = −dτ dξ) and the result follows.

=

79

3.5

Algebraization of Liell

3.5.1

Kronecker theta function

We follow [7] section 1, which in turn follows [79]. This subsection contains nothing original. Given a lattice Γ ⊂ C, let A be the fundamental area of Γ divided by π. If Γ =< γ1 , γ2 >, with =(γ1 /γ2 ) > 0 then A =

γ1 γ¯2 −γ2 γ¯1 , 2πi

where =(z) denotes the

imaginary part of z. For τ ∈ H + , we let Γτ =< 1, τ >. Notice that A(Γτ ) =

τ −¯ τ . 2πi

Definition 3.5.1. We define θ(z; τ ) to be the reduced theta function on Eτ := C/Γτ corresponding to the divisor [0], normalized so that θ0 (0; τ ) = 1. For the definition of a reduced theta function corresponding to a line bundle, we refer the reader to the Theorem of Appell and Humbert (cf. [6] Theorem 1.7) and loc. cit. definition 1.8. Notice that this theta function is not the same one defined in section 2.1 (which was denoted by θJ ). We recall an explicit expression for θ(z; τ ) based on the Weierstrass σ-function σ(z; τ ) = z

Y γ∈Γτ \0

1−

z z2 z exp + 2 . γ γ 2γ

In [79], page 45 Chapter VI section 5, the numbers e∗a,b (Γτ ) are defined whenever b > a ≥ 0, and then on page 81 Chapter VIII section 15 there is another formula for these numbers. Then e∗2 (τ ) := e∗0,2 (Γτ ). By equations (26), (27) in [79] page 78, it P follows that e∗2 (τ ) = limu→+0 γ∈Γτ \0 γ −2 |γ|2u . The number e2 (τ ) defined in [79] page 20, equation (9), via the Eisenstein summation process is related to e∗2 (τ ) by equation (6) on page 43 in [79] which reads as e∗2 (τ ) = e2 (τ ) −

2πi . τ − τ¯

(3.5.2)

By [6], example 1.9 we have e∗ θ(z; τ ) = exp − 2 z 2 σ(z; τ ). 2

(3.5.3)

80

It is interesting to compare this equation with [79] section IV.3 just before equation (11). Θ(z; τ ) is a holomorphic function on C whose only zeros are simple zeros at z ∈ Γτ , and satisfies the transformation formula (cf. [6] equation (8), [7] equation (3)) γ γ¯ z+ θ(z; τ ) θ(z + γ; τ ) = (γ) exp 2πi τ − τ¯ 2

(3.5.4)

for any γ ∈ Γτ where (γ) = −1 if γ 6∈ 2Γτ and (γ) = 1 if γ ∈ 2Γτ . Definition 3.5.5. We define the Kronecker theta function Θ(z, w; τ ) to be the function Θ(z, w; τ ) :=

θ(z + w; τ ) . θ(z; τ )θ(w; τ )

(3.5.6)

This function is a reduced theta function (cf. [6] definition 1.8) associated to the Poincaré line bundle of C/Γτ . Since σ(z; τ ) and hence θ(z; τ ) are odd functions (in z), then θ00 (0; τ ) = 1. Let F1 (z; τ ) := limw→0 Θ(z, w; τ )−w−1 . Since θ(0; τ ) = θ0 (0; τ ) = 0 and θ(z; τ ) 6= 0 for 0 < |z| : η(w; Λ) = ζ(z + w; Λ) − ζ(z; Λ) for any z ∈ C (for which ζ(z) is defined). It is well known (cf. [70], page 240) that this is well-defined, i.e., this difference only depends Rτ R1 on w. This implies that 0 P(ξ)dξ = η(1; Λ), 0 P(ξ)dξ = η(τ ; Λ). We let e2 denote P0 the Eisenstein summation e ω −2 as defined in Chapter III §7 [79]. It is classical that e2 = η(1; Λ). Let ω ∗ = −η − e2 ω. We have the following lemma: f Lemma 3.6.15. −e = ω ∨ , 2πi = ω ∗∨ .

Proof. We need to show: 1. ω(e) = 1. Proof. ω(e) =

R1 0

ω=

R1 0

dξ = ξ|10 = 1.

2. ω(f ) = 0. Proof. ω(f ) = ω(f˜− τ e) = ω(f˜) − τ (ω(e)) =

Rτ 0

ω −τ

R1 0

ω=

Rτ 0

dξ − τ

R

01 dξ =

ξ|τ0 − τ ξ01 = 0. 3. ω ∗ (e) = 0. Proof. η(e) = result. 4. ω ∗ (f ) = 2πi.

R1 0

η =

R1 0

−dζ = η(1; Λ) = e2 . By linearity we deduce the

93

Proof. η(f˜) =

Rτ 0

η =

Rτ 0

−dζ = η(τ ; Λ). This implies: η(f ) = η(f˜ − eτ ) =

η(τ ; Λ) − τ η(1; Λ) = −2πi, where the last equality follows by Legendre relation (cf. page 241, 246 Lang). Since ω(f ) = 0, we deduce: ω ∗ (f ) = 2πi.

∗

ω ∨ ) . In fact, Proof of Lemma 3.6.14. By Lemma 3.6.15, we have: e = (−ω)∨ , f = ( 2πi ∗

∗

ω ω we need to show that −ω ⊗ Pe + 2πi ⊗ Pf = pol. We have: −ω ⊗ Pe + 2πi ⊗ Pf = −ω ⊗ f f E2 ω∗ dξ + −Ξ(ξ, 2πi )+ 2πi ⊗ −Ξ(ξ, 2πi )f dξ = ω ⊗ −E2 dξ + −Ξ(ξ, ω ∗∨ )+ edξ + 2πi 2πi f ∨ 1 ω dξ +ω ∗ ⊗ −Ξ(ξ, ω ∗∨ )ω ∗∨ dξ , which gives −pol in the representation of section ω ∗∨

1.5, lemma 1.28 of [7]. The explanation for the minus sign is the following: in [9] section 4.4.2 (page 162) there is the definition of Li(z), which is defined over C∗ in the classical situation. You dz see that already there, ∇Li = −dlog(1 − z) = − z−1 . This implies that its residue at

1 is -1. The meaning is that if one defines the classical logarithm, and adds to the logarithmic pro-sheaf a section p such that ∇p =

dz , 1−z

it is possible to construct p

as p := 1 − Li. This is different than p := 1 + Li by a factor of -1. In other words, one should have done the same in the elliptic case, and define 1 − Liell as the section defining the elliptic polylog, instead of 1 + Liell as we did following [9]. This would imply that one should define the other extension PV00 by defining its connection also to be the minus of the definition given by [9] (this system appears as a correction, that is, it is designed such that its Baer sum wth the PV0 will be a (q)Z equivariant, and hence descend to the elliptic curve.

94

3.7

Algebraic description of the elliptic polylogarithms

In this section we explicitly express the elliptic polylogarithms algebraically. Our exposition is influenced by [64] and [7], although our approach is based on the treatment of [9]. The following proposition appears in [64] proposition 5.2.3. Proposition 3.7.1. The differential forms dτ , 2πi

e2

dτ , 2πi

dξ +

1 E1 dτ. 2πi

are algebraic forms in Q(x, y, e4 , e6 ). Proof. Let g2 = 60e4 (τ ), g3 = 140e6 (τ ). We define the discriminant Λ = Λ(τ ) := g23 − 27g32 . Let e2 (τ ) e4 (τ ) e6 (τ ) , M= , N= . 2ζ(2) 2ζ(4) 2ζ(6) The Ramanujan identities are: L=

L2 − M ∂L = 2πi . (3.7.2) ∂τ 12 ∂M LM − N = 2πi . (3.7.3) ∂τ 3 ∂N LN − M 2 = 2πi . (3.7.4) ∂τ 2 These identities imply (via an easy computation) that dΛ(τ ) = 3g22 dg2 − 2 · 27g3 dg3 = dτ ). We let κ := −12Λ(τ )(e2 2πi

dΛ . 12Λ

It is clear that κ is algebraic in g2 , g3 and hence

dτ e2 2πi is algebraic. From the Ramanujan identity 3.7.3 and what we found it follows dτ that dg2 = 6g3 2πi − 4g2 κ. This implies that

dτ 2πi

is algebraic. From the Ramanujan

2 2 identity 3.7.2 we deduce that 2πi ∂e = (5e4 − e22 ). We also have 2πi ∂E = 3((E2 − ∂τ ∂τ

e2 )2 − 5e4 ) − E22 − 2E1 E3 . This implies dx = y dξ +

1 g2 dτ E1 dτ + 2x2 − − 2κx. 2πi 3 2πi

95

This implies that dξ +

1 E dτ 2πi 1

is also Algebraic, completing the proof.

We state without the proof well known properties of the Eisenstein functions and series. Lemma 3.7.5. For k ≥ 4, the e2k (τ ) are polynomials in e4 (τ ), e6 (τ ) with rational coefficients. The functions Ek (ξ, τ ) are polynomials in E2 (ξ, τ )−e2 (τ ), E3 (ξ, τ ), e4 (τ ), e6 (τ ) with rational coefficients. Proposition 3.7.6. The cohomology classes of ∇PU (Pe ) and ∇PU (Pf ) are algebraic. Proof. ∇PU (Pe ) ≡ f f ∂ )edξ − ( ∂f Ξ(ξ, 2πi )e + −Ξ(ξ, 2πi

+2πi

e dξ f

−

e dτ f2

+

E1 (ξ,τ ) f Ξ(ξ, 2πi )e)dτ 2πi

+

E2 dξ 2πi

−

∂E1 ∂τ

2πi

dτ

E1 e dτ, 0). f

We have f f ∂ −Ξ(ξ, 2πi )edξ − ( ∂f Ξ(ξ, 2πi )e +

E1 (ξ,τ ) f Ξ(ξ, 2πi )e)dτ 2πi

+

E2 dξ 2πi

−

∂E1 ∂τ

2πi

dτ + 2πi

e dξ f

−

e dτ f2

+ Ef1 e dτ = f E2 − Ξ(ξ, 2πi )e(dξ +

E1 (ξ,τ ) dτ ) 2πi

f ∂ − ( ∂f Ξ(ξ, 2πi ))edτ −

E3 dτ 2πi

+ 2πi

e dξ f

−

e dτ f2

+

E1 e dτ. f

(ξ,τ )f f f By Equation 3.5.21 we have Ξ(ξ, 2πi ; τ ) = exp(− E1 2πi )F (ξ, 2πi , τ ). Substituting f equation (13) in [64] for F (ξ, 2πi , τ) ∞

F (ξ,

X (− f )k f (2πi)2 2πi ; τ) = exp − (Ek − ek ) . 2πi f k k=1

(3.7.7)

96

We deduce that Ξ(ξ,

∞ f k X ) (− 2πi f (2πi)2 ; τ) = exp − (Ek − ek ) . 2πi f k k=2

(3.7.8)

We deduce ∞ 1 X 1 f f k−1 f ∂ Ξ(ξ, )= ( ) − (Ek − ek ) − Ξ(ξ, ), ∂f 2πi 2πi k=2 2πi f 2πi

We deduce that f E2 − Ξ(ξ, 2πi )e(dξ +

f = E2 − (Ξ(ξ, 2πi )−

f + 2πi )− Ξ(ξ, 2πi f

We notice that

2πi f

2πi f

E1 (ξ,τ ) dτ ) 2πi

2πi )e(dξ f

+

f ∂ − ( ∂f Ξ(ξ, 2πi ))edτ −

E1 (ξ,τ ) dτ ) − 2πi

P∞

k=2

E3 dτ 2πi

f − 2πi

+ 2πi

k−1

e dξ f

−

e dτ f2

+

E1 e dτ f

f dτ (Ek − ek ) Ξ(ξ, 2πi )e 2πi

dτ dτ . e 2πi − E3 2πi

f )− Ξ(ξ, 2πi

2πi f

e is algebraic by Equation 3.5.14 and Proposition

3.5.19. The result follows by Proposition 3.5.19 combined with Lemma 3.7.5 and Proposition 3.7.1. (ξ,τ ) f f f ∂ Similarly,∇PU (Pf ) ≡ −Ξ(ξ, 2πi )f dξ−( ∂f Ξ(ξ, 2πi )f + E12πi Ξ(ξ, 2πi )f )dτ −2πi 2πi pe , 1 ⊗ e dτ . We have f f ∂ −Ξ(ξ, 2πi )f dξ − ( ∂f Ξ(ξ, 2πi )f +

f −Ξ(ξ, 2πi )f (dξ +

E1 (ξ,τ ) dτ ) 2πi

f f 2πi −(Ξ(ξ, 2πi ) 2πi (dξ + f Ξ(ξ, 2πi )−

2πi dτ f 2πi

E1 (ξ,τ ) f Ξ(ξ, 2πi )f )dτ 2πi

1 f

dτ, 0)−

− 2πi f1 dτ =

f ∂ − ( ∂f Ξ(ξ, 2πi ))f dτ − 2πi f1 dτ =

E1 (ξ,τ ) dτ )− 2πi

P∞

k=2

f − 2πi

k−1

f f dτ (Ek − ek ) Ξ(ξ, 2πi ) 2πi + 2πi

.

The proof again follows by Proposition 3.5.19 combined with Lemma 3.7.5 and Proposition 3.7.1.

97

The C ∞ realization of the elliptic polylogarithm

3.8 3.8.1

Kronecker’s double serier

Let Γ be a lattice in C, and χ a character of the additive group Γ. In his later years, Kronecker dealt mainly with double series of the form Ka (x, w, s; Γ) :=

X

χ(w)(¯ x + w) ¯ a |x + w|−2s .

(3.8.1)

w∈Γ

The series appears in [79], Chapter VIII, equation (1). In [9] section 3, a C∞ realization of the elliptic polylogarithm is described. The main ingredient is some double series which is also described in [62], which describes the elliptic polylogarithm via C∞ currents. One of the questions that naturally arise is whether there is a relation between the C∞ and holomorphic realizations. In [81] and also subsequent papers of Wildeshaus, he deals with the C∞ realization of the elliptic polylogarithms. In this section we describe the relation between the Q- and R- Hodge structures found for the elliptic polylogarithm. For this we repeat the construction of [9], section 3.3, for the R-Hodge structure of the elliptic polylogarithm. In fact they even find the R-Hodge structure of F ell . We explain here the situation of pX : X → B a family of elliptic curves where B is a smooth C-scheme.

3.8.2

R-Hodge structures

Let V be a finite dimensional R-vector space. There are two ways to define a Hodge structure on V . Namely, a Hodge structure on V is either of the following linear algebra data: (i) An increasing finite filtration W on V and a decreasing finite filtration F • on VC = V ⊗ C, such that F, F¯ induce on grjW VC the j-complementary filtrations Fj• := F • ∩ Wj VC /F • ∩ Wj−1 VC , F¯j• := F¯ • ∩ Wj VC /F¯ • ∩ Wj−1 VC .

98

This means that grjW VC =

p+q=j

Fjp ∩ F¯jq .

V p,q and a linear operator N ∈ EndVC such that a,b ¯ = −N , and N (V p,q ) ⊂ L = V q,p , N . a, where < ∗, ∗ >: Γ × Γ → 2πiZ is the intersection pairing defined in Equation 3.8.3. Let H⊗C = H0,−1 ⊕H−1,0 be the Hodge decomposition. For γ ∈ H, denote by γ 0,−1 , γ −1,0 the components of γ. For a, b, ∈ Z P 0,−1 )a−1 (γ −1,0 )b−1 consider the series ga,b = 0 χγ (γ of (H0,−1 )⊗a−1 ⊗ (H−1,0 )⊗b−1 -valued a+b−1 P functions on X; we note that 0 denotes summation over Γ − {0}. For a + b > 2 this series converges absolutely; in [62], Levin works with distributions or currents, and in this context this series always converge. For our purposes, we need only the case

101

a + b > 2.

3.8.5

The Eisenstein-Kronecker series - continued

We rewrite ga,b in the case of the locally universal standard family of elliptic curves over the upper half-plane H. Let τ, ξ be the standard parameters on H, C, respectively. For τ ∈ H one has Γτ = Z · (0, 1) ⊕ Z · (0, τ ), Xτ ∼ = C/Z + Zτ . Let ∂ ∂ξ

∂ ∈ Hτ−1,0 , ∂ξ ∈ Hτ−1,0 be the base vectors dual to dξ, dξ, respectively. Write

∂ ∂ξ

= a · (0, 1) + b · (0, τ ) for a, b ∈ C. We know that dξ(a · (0, 1) + b · (0, τ )) = R1 Rτ 1, dξ(a · (0, 1) + b · (0, τ )) = 0. Since dξ((0, 1)) = 0 dξ = 1, dξ((0, τ )) = 0 dξ = τ . R1 Rτ Similarly, dξ((0, 1)) = 0 dξ = 1, dξ((0, τ )) = 0 dξ = τ . We get the equations: a + bτ = 1, a + bτ = 0. We deduce that b = ∂ ∂ξ

τ = − τ −τ · (0, 1) +

1 ,a τ −τ

1 τ −τ

τ = − τ −τ . Therefore,

· (0, τ ). Similarly, (3.8.4)

∂ ∂ξ

=

τ τ −τ

· (0, 1) −

1 τ −τ

· (0, τ ).

Consequently, (0, 1) =

∂ ∂ξ

+

∂ , (0, τ ) ∂ξ

∂ ∂ = τ ∂ξ + τ ∂ξ .

(3.8.5)

We notice that if γ = n(0, 1) + m(0, τ ) ∈ Γ for m, n ∈ Z, then γ −1,0 = (n + α

∂ ∂ mτ ) ∂ξ , γ 0,−1 = (n + mτ ) ∂ξ . We define an injection Γτ = Z · (0, 1) ⊕ Z · (0, τ ) → ∂ ∂ Z + Zτ ,→ C. We deduce that γ −1,0 = α(γ) ∂ξ , γ 0,−1 = α(γ) ∂ξ . That is, γ = ∂ ∂ . We notice that α extends to an isomorphism α : HR,τ → C, given α(γ) ∂ξ + α(γ) ∂ξ

by α(x(0, 1) + y(0, τ )) = x + yτ for x, y ∈ R. By what we showed we deduce that ∂ ∂ ξ ∂ξ +ξ ∂ξ is mapped via α to ξ. Thus, referring to the point (ξ, τ ) ∈ Xτ is equivalent to ∂ ∂ referring to the point ξ ∂ξ + ξ ∂ξ ∈ Hτ /Γτ . We also notice that
=

2πi τ −τ

=

1 , A

where A is the area of the fundamental domain of < 1, τ > divided by π. )) =< α(γ), ξ >Z+Zτ , in the notation of We deduce that χγ (τ, ξ) = exp(2πi( α(γ)ξ−α(γ)ξ τ −τ

102

[6]. We deduce that ga,b =

P

Γτ −{0}

Z+Zτ b

α(γ)a α(γ)

∂ b−1 ∂ a−1 ∂ξ ∂ξ ∂ ∂ < ∂ξ , >a+b−1 ∂ξ

= (3.8.6)

∂ b−1 ∂ a−1 (A(A ∂ξ ) ) ) (A ∂ξ

P

γ∈Z+Zτ −{0}

Z+Zτ γaγb

We define a complex structure on H ⊗ C, by deciding that H ⊗ R is fixed by the complex conjugation. That is, (0, 1) = (0, 1), (0, τ ) = (0, τ ). ∂ ∂ξ

By Equation 3.8.4 we deduce that

=

∂ . ∂ξ

For 2 ≤ n ∈ Z, we let

X

gn :=

(−1)a ga,b .

a,b≥1,a+b=n

Since < γ, ξ >Z+Zτ = < −γ, ξ >Z+Zτ , if follows that (−1)a ga,b = (−1)b gb,a , implying g g n = gn (cf. section 3.3.3 and corollary 3.3.5 in loc. cit.). The section g 0 := 2πi = P gn n≥2 2πi of Log|V defines by Lemma 3.8.2 a quasi-Hodge extension of R(0)V by

Log|V , which we call FRell . It is also called the Eisenstein-Kronecker series. We now define another complex structure on H ⊗ C, by deciding that (0, 1) and (0, τ ) are imaginary objects. That is, (0, 1) = −(0, 1) and (0, τ ) = −(0, τ ). ∂ ∂ ∂ By Equation 3.8.4 this implies that ( ∂ξ ) = − ∂ξ . Let η = A ∂ξ . since A = A, we

obtain: ga,b = Aη a−1 (−η)b−1

< γ, ξ >Z+Zτ γ aγ b γ∈Z+Zτ −{0} X

(3.8.7)

Let us rewrite g: g=

a a,b≥1 (−1) ga,b

P

−A ·

=

P∞

k l k,l=0 (−η) (−η)

a a−1 (−η)b−1 a,b≥1 (−1) Aη

P

P

P

γ∈Z+Zτ −{0}

Z+Zτ γaγb

=

Z+Zτ γ∈Z+Zτ −{0} γ k+1 γ l+1

By examining equation (13), p. 279 [61], we obtain that g = −A(K0 (η, ξ, 1)− |η|1 2 ). Let −Y := (0, 1) =

∂ ∂ξ

+

∂ ∂ξ

=

η−η , A

∂ ∂ −X := (0, τ ) = τ ∂ξ + τ ∂ξ =

τ η−τ η . A

Put

103

r :=

τ ξ−τ ξ ,s τ −τ

:=

ξ−ξ . τ −τ

Following p. 278, [61] we define the function: Ξ(ξ, τ, X, Y ) := esX+rY Λ(ξ, τ, X, Y ).

(3.8.8)

By [61], theorem 4.2, we deduce that Ξ(ξ, τ ; X, Y ) − Ξ(ξ, τ ; −X, −Y ) = − for η =

−Y τ +X . 2πi

τ −τ K0 (η, ξ, 1) (2πi)2

This implies

1 g 0 = Ξ(ξ, τ ; X, Y ) − Ξ(ξ, τ ; −X, −Y ) + A 2πi|η| 2 :=: Ξ(ξ, τ ; X, Y ) − Ξ(ξ, τ ; X, Y ) + τ −τ , (−Y τ +X)(Y τ −X)

where complex conjugation on the right hand side is the usual c.c on H ⊗ C.

3.8.6

The g/L correspondence in the C ∞ category

In Section 3.3 we introduced the notion of holomorphic quasi-Hodge sheaves following [9]. As we know from complex analysis, the analytic category is more rigid than the differentiable category. In this case, it is shown by the need to enlarge the notion of quasi-Hodge sheaves to the category or relative quasi-Hodge sheaves (cf. Section 3.3). This was essential in order to define the elliptic polylogarithms. In the differentiable category, it is enough to restrict attention to differentiable quasi-Hodge sheaves, and take extensions in this category. (cf. Lemma 3.8.2). The interested reader might wonder about the relation between the extensions established by Lemma 3.3.4 and those established by Lemma 3.8.2. In this section we investigate this relation. To do this, one has to reproduce Lemma 3.3.4 in the C∞ context, and for that one has to construct quasi-Hodge sheaves in the latter category. For this we reproduce Section 3.3 for the C∞ context. Replacing ∂ by ∇ and Q by R, the changes are minor. Lemma 4.1.2 should hold with the above mentioned modifications, but notice that the condition is ∇g ∈ F −1 FS˜ ⊗ ΩlS˜ , and ∂g = 0 (because 1 + g is a section of the corresponding holomorphic bundle (cf. the beginning of section 3.2 after the list of

104

axioms)), which is used afterwards in the proof of lemmas 3.2.1 and 4.1.2, and in the case of lemma 4.2.1 we also have ∇1 = 0). Let S be a connected complex analytic manifold. We defined the Tannakian category QH (S)R of quasi R-Hodge sheaf V on S. Let π : S˜ → S be a local homeomorphism such that S˜ is nonempty and connected. A quasi R=Hodge sheaf ˜ R , FS i ∈ F on S relative to S˜ is a collection (FS˜ , FS i , ϕi ) where FS˜ ∈ QH (S) ∼

QH (S)R (i ∈ Z), and ϕi : π ∗ FS i → griW (FS˜ ) are isomorphisms. Clearly FS i is a variation of pure Hodge structures of weight i on S, for each i ∈ Z. Such F ’s form a category QHS˜ (S)R which is a Tannakian category in an obvious manner. One has QHS (S) = QH (S)R . If π 0 : S˜0 → S is another homeomorphism and f : S˜0 → S˜ is an S-morphism then we have an exact tensor functor f ∗ : QHS˜ (S)R → QHS˜0 (S)R . We have Lemma 3.8.9. f ∗ is a fully faithful embedding. In particular, π ∗ : QH (S)R → QHS˜ (S)R is a fully faithful embedding. Corollary 3.8.10. We may consider QHS˜ (S)R as an enlargement of the category QH (S)R (and of H (S)R if S is algebraic). Lemma 3.8.11. Let V be a quasi-R Hodge sheaf. Then i

∇(F ) ⊂ Ω0,1 S ⊗F

i−1

i

+ ΩS1,0 ⊗ F .

Proof. We first prove that ∇(v) = ∇v for a section v ∈ VC . Write v = u + iw where u, w are C∞ sections of V . Since the connection is defined over V , we know that ∇u, ∇w belong to A 1 ⊗ V , where A 1 is the sheaf of real C∞ differential forms. (That is, S is viewed as a C∞ differentiable manifold whose dimension is twice its complex dimension as a complex manifold.) This implies that ∇(v) = ∇u + i∇w, i

and ∇(v) = ∇u − i∇w = ∇(u − iw) = ∇(v). Given g ∈ F , there exists f ∈ F i with 1,0 i i−1 g = f . We know that ∇(f ) ∈ ω 0,1 ⊗ fi + ω 1,0 ⊗ fi−1 ∈ Ω0,1 . Then S ⊗ F + ΩS ⊗ F

∇g = ∇f = ∇f = ω 0,1 ⊗ fi + ω 1,0 ⊗ fi−1 . Since Ω1,0 = Ω0,1 the claim follows.

105

The following is an analog of Lemma 3.3.4. Lemma 3.8.12. Let F ∈ QHS˜ (S)R be a quasi-Hodge sheaf such that F = W−2 F and F 0 F = 0. Then the R-vector space Ext1QHS˜ (S)R (R(0)S , F ) is canonically isomor−1 phic to the space of sections L of FOS˜ /FSR FS˜ ⊗ ΩS1,0 ˜ such that ∇L ∈ F ˜ .

˜ S Ui of S˜ by Proof. Given a section L of FOS˜ /FSR ˜ , it is given by a covering S = i open sets, sections li ∈ FOS˜ (Ui ) such that on intersections Ui ∩ Uj we have li − lj ∈ FSR ˜ (Ui ∩ Uj ). We say that li is a local lifting of L. The end result is an extension PL . It is described by extensions on each Ui together with compatible morphisms between them. 1. Each “local” extension Pi is defined as follows: there is a section li + 1i , which belongs to F 0 (Pi ). This 1i is flat and rational, and maps to 1Q ∈ QUi . We define F 1 (Pi ) = 0, F 0 (Pi ) = OUi · (li + 1i ), F i (Pi ) = F i (F |Ui ) + F 0 (Pi ) for i < 0. and W0 (Pi ) = Pi , W−i (Pi ) = W−i (F |Ui ), for i ≥ 1. 2. Over each Ui we obtain extensions Pi , and on them sections li + 1. Now, since we know that h := lj − li ∈ FSQ ˜ (Ui ∩ Uj ) there is a canonical isomorphism ϕi,j : Pi → Pj on the intersection, defined by ϕi,j |FUi ∩Uj = id, ϕi,j (1) = 1 + h from which follows that it sends li + 1 to lj + 1. Notice that 1 + h is also a flat rational section of Pj |Ui which maps to 1 in QUi ∩Uj . Altogether, we deduce that these extensions glue together to form an extension over S, and that there is a global section “L + 1” belonging to F 0 (PL ). 3. The condition which L satisfies guarantees that PL satisfies Griffiths’ transversality condition and that ∂ induces a holomorphic structure on F 0 (PL ).

106

4. We obtain F 1 (PL ) = 0, F 0 (PL ) = OS˜ · (L + 1), F i (PL ) = F i (F ) + F 0 (PL ) for i < 0. Since F i (F ) are holomorphic subbundles of F , it is clear that the Hodge filtration of PL is by holomorphic subbundles. 5. We obtain W0 (PL ) = PL , W−i (PL ) = W−i (F ), for i ≥ 1. 6. Let us clarify some subtle point. On a set Ui there is a section 1i ∈ PiQ . When one continues this section to Uj , we regard 1 as the section 1j + hi,j on Uj and one can then continue this section to Uk and obtain the section 1k + hi,j + hj,k = 1k + hi,k , etc. This implies that there is no global section “100 of PL . 7. We also remark that even though one may choose different coverings and liftings, by taking a refinement of both coverings, one obtains a canonical isomorphism between the two extensions. It is clear that L defines an extension PL ∈ Ext1QHS˜ (S)R (R(0)S , F ). Conversely, for an extension j

0 → FS˜ → P → R(0)S˜ → 0 the projection F 0 j : F 0 P → F 0 R(0)S˜ = OS˜ is an isomorphism of holomorphic bundles. Indeed, let ˜1F ∈ F 0 P denote the section of F 0 P such that j(1F ) = 1 ∈ OS˜ . It is shown in the course of the proof of Lemma 3.2.1 [9], that F 0 P = OS˜ · 1F . Since the morphisms of the short exact sequence respect the connection, we deduce that 0 j(∇1F ) = ∇(j(1F )) = 0. By Axiom 3.8.3.(iii) we deduce that ∇1F ∈ Ω0,1 S ⊗F P +

ΩS1,0 ⊗ F −1 P . Since F 0 P is generated by 1F it follows that ∇1F may be written as

107

∇1F = ω 0,1 ⊗ 1F + ω 1,0 ⊗ f . This implies that j(∇1f ) = ω 0,1 + ω 1,0 ⊗ j(f ). −1 Since j(∇1F ) = 0, it follows that ω 0,1 = 0. This implies that ∇1F ∈ Ω1,0 P . This S ⊗F

shows that indeed the isomorphism F 0 j is an isomorphism of holomorphic bundles. Denote by ˜1Q ∈ PQ any local section that projects to 1 ∈ Q. Then lP = ˜1F − ˜1Q ∈ FO ˜ , and L = lP mod FSR 1Q . ˜ ∈ FOS˜ /FSR ˜ is independent of the choice of ˜ S −1 Moreover, ∇L = ∇1F ∈ Ω1,0 P , hence L satisfies the condition appearing in the S ⊗F

statement of the lemma. It is clear that that the two correspondences thus depicted are mutually inverse of one another.

We now explain, given an extension in Ext1QHS˜ (S)R (R(0)S , F ) constructed via Lemma 3.8.2, how to define an extension in Ext1QH (S) ˜ R (R(0)S˜ , F ). Explicitly, we show the relation between g of Lemma 3.8.2 and L of Lemma 3.8.12. It is important to notice ˜ It is of course too optimistic that the extension thus constructed is defined over S. to hope that this extension would descend to S, and indeed only some extensions would descend to S. Examining the proofs of the above mentioned lemmas we see that 1 + L and 10 + g generate F 0 PS viewed as a holomorphic bundle. Thus, we would like to impose the condition 1 + L = 10 + g. Given L, a section of FOS˜ /FSR ˜ such that ∇L ∈ F −1 FS˜ ⊗ Ω1S˜ , ∂L = 0, ∂L = 0, we define g :=

L−L . 2

Then 10 = 1 +

L+L . 2

We need to check that this is well defined. We remind that L is given as a collection ˜ such that qi,j := li − lj are rational of sections li ∈ FOUi on some covering {Ui } of S, sections on the intersection, implying qi,j = q i,j . We deduce that li − li = lj − lj . Hence, g =

L−L 2

is indeed a well defined single-valued section. Another way to realize

this fact is by noticing that

L−L 2

=

(1+L)−(1+L) . 2

As 1 + L is a single valued section

of F 0 PS we deduce that so is its conjugate and therefore so is their difference. This also implies that 10 = 1 + L+L is a well defined single valued section being the sum of 2 two single valued sections. We notice that:

108

• ∂(g) =

∂(L)−∂(L) 2

• ∂(g) =

−∂(L) . 2

=

∂(L) . 2

We used the fact that ∇(L) = ∇L by Lemma 3.8.11, implying

that ∂(L) = 0 since ∇L ∈ F • ∇(10 ) = ∇(1) +

∇(L)+∇(L) 2

=

−1

FS˜ ⊗ Ω0,1 . S˜

∂(L)+∂(L) 2

= ∂g − ∂g.

This shows that the extension depicted by Lemma 3.8.2 is isomorphic to the one depicted by Lemma 3.8.12 by the above g/L correspondence.

3.8.7

The relation between the R- and Q-Hodge structures of the elliptic polylogarithm

In this section, we describe the relation between the two approaches that are contained in sections 3 and 4, Respectively, of [9]. Subsequent papers have used one of the approaches, but there was no systematic treatment, showing the relation between them. By Proposition 3.4.25 we have: Liell (ξ, τ, X, Y ) = −2πiΛ(ξ, τ, X + Y τ, Y )eY ξ over U × D. This Liell is a multi-valued section of the logarithmic extension over U × D.

Following [81], we construct the C∞ analog of the above; we define (cf. p.

284, [81]): Qn,m 1

:= −Λm,n +

n X i=0

(−1)i Bn Bl+1 rk−i (−s)l+i+1 τ i + (−1)n+1 . (k − i)!(l + i + 1)! n! (l + 1)! (3.8.13)

Definition 3.8.14. Let Q1 (ξ, τ, X, Y ) :=

P

m≥0,n≥1

n−1 m Qn,m X be a formal 1 (−Y )

power series over Eτ . Let R1 (ξ, τ, X, Y ) = exp(sX +rY )Q1 (ξ, τ, X, Y ) be the C ∞ section of the logarithmic extension over Eτ obtained by substituting −X := (0, τ ), −Y = (0, 1) the elements of the first homology group of the elliptic curve Eτ , with the im∂ l ∂ + ξ ∂ξ ). portant convention that X k Y m stand for liml X k Y m (γξ + ξ ∂ξ ←−

109

Proposition 3.8.15. Under the g/L correspondence, −g 0 found above corresponds to R1 . Proof. One should prove that the section R1 (ξ, τ, X, Y ) = exp(sX +rY )Q1 (ξ, τ, X, Y ) n as −Λm,n +Qm,n satisfies R1 (ξ, τ, X, Y )−R1 (ξ, τ, X, Y ) = −g 0 . We write Qn,m 2 (τ )+Q3 . 1

We notice that r, s are invariant under complex conjugation. As Qn3 is invariant under complex conjugation, it may be disregarded from the equation. Definition 3.8.16. Q2 (τ, X, Y ) :=

P

m≥0,n≥1

n−1 m Qm,n X . 2 (τ )(−Y )

That is, we have: R1 (ξ, τ, X, Y ) − R1 (ξ, τ, X, Y ) = −Λ(ξ, τ, X, Y ) exp(sX + rY ) + Λ(ξ, τ, X, Y ) exp(sX + rY ) + (Q2 (τ, X, Y ) − Q2 (τ , X, Y )) exp(sX + rY ). Following [81] we let: r1 = r, r2 = −s. We have: (s(−Y τ +X))−1 (sX)−1 Lemma 3.8.17. (Q2 (τ, X, Y )) exp(−r2 X − r1 (−Y )) = − exp(−Y + exp . )(−Y τ +X) (−Y )(X)

Proof. (Q2 (τ, X, Y )) exp(−r2 X − r1 (−Y )) =

P

m≥0,n≥1

P

k≥1,l≥0

(−1)i n−i m+i+1 i τ (−Y )n−1 X m i=0 (n−i)!(m+i+1)! r1 r2

Pn

Pk

p=0

Pl

q=0

(−1)p+q p q r1 r2 p!q!

p≥0,q≥0 (−1)

p+q q p q r2 r1 X (−Y )p

(∗) (−1)i k−p−i l−q+i+1 i r2 τ (−Y )k−1 X l = i=0 (k−p−i)!(l−q+i+1)! r1

Pk−p

−

P

k+l sk+l+1 τ k (−Y )k−1 X l k≥1,l≥0 l (k+l+1)!

−

P

sl+1 −1 l l≥0 (l+1)! (−Y ) X

1 = − −Y

P

=−

P

k≥0,l≥0

sm+1 m≥0 (m+1)! (−Y

P

k+l sk+l+1 τ k (−Y )k−1 X l l (k+l+1)!

τ + X)m +

P

sl+1 −1 l l≥0 (l+1)! (−Y ) X .

(3.8.18) Where equality (*) follows by [81] page 290 (beginning of proof). By the equality P∞ xn ex −1 we deduce that: k=0 (n+1)! = x

=

110

1 −Y

sm+1 m≥0 (m+1)! (−Y

P

sl+1 −1 l l≥0 (l+1)! (−Y ) X

P

exp (s(−Y τ +X))−1 and that: (−Y )(−Y τ +X) exp (sX)−1 . The result follows. (−Y )(X)

τ + X)m = =

Corollary 3.8.19. Notice that the latter term in the above equation is invariant under conjugation, hence we may omit it. We recall: Definition 3.8.20. Let Λ(ξ, τ, X, Y ) := Λ(ξ, τ, X, Y ) +

exp(−Y ξ) (−Y )(−Y τ +X)

+

1 . (exp Y −1)X

The latter term is also invariant under conjugation, so we may write Λ(ξ, τ, X, Y ) exp(sX + rY ) − Λ(ξ, τ, X, Y ) exp(sX + rY ) = Λ(ξ, τ, X, Y ) exp(sX + ξ) ξ) rY )−Λ(ξ, τ, X, Y ) exp(sX + rY )− (−Yexp(−Y exp(sX +rY )+ (−Yexp(−Y exp(sX + )(−Y τ +X) )(−Y τ +X) ξ) ξ) exp(sX+rY )+ (−Yexp(−Y exp(sX+ rY ) = Ξ(ξ, τ ; X, Y )−Ξ(ξ, τ ; −X, −Y )− (−Yexp(−Y )(−Y τ +X) )(−Y τ +X) ξ) ξ) τ −τ rY ) = g 0 − (−Y τ +X)(Y − (−Yexp(−Y exp(sX + rY ) + (−Yexp(−Y exp(sX + rY ). τ −X) )(−Y τ +X) )(−Y τ +X) τ −τ Re-writing R1 (ξ, τ, X, Y )−R1 (ξ, τ, X, Y ) = −g 0 + (−Y τ +X)(Y + exp(−Y ξ) exp(sX+ τ −X) (−Y )(−Y τ +X)

rY ) −

exp(−Y ξ) (−Y )(−Y τ +X)

exp(sX + rY ) −

exp (s(−Y τ +X))−1 (−Y )(−Y τ +X)

+

exp (s(−Y τ +X))−1 (−Y )(−Y τ +X)

The last equality is the following calculation: we first notice that: 1 τ −τ = − (−Y τ +X)(Y . We then notice that sτ (−Y )(−Y τ +X) τ −X) ξ) (s(−Y τ +X)) that: (−Yexp(−Y exp(sX + rY ) = exp and hence )(−Y τ +X) (−Y )(−Y τ +X)

conjugates.

= −g 0 . 1 (−Y )(−Y τ +X)

−

= ξ − r, which implies the same is true for their

Chapter 4 The syntomic theory Our main goal is to find the syntomic classes of the motivic elliptic polylogarithms. To this end we first need to define the category where these elements “live”; our approach is an amalgamation of the methods developed in [15] and [5]. In this chapter, we assume that K is a finite extension of Qp with ring of integers OK and residue field k. We let K0 be the maximal unramified extension of Qp in K, and W its ring of integers. σ is the lifting to K0 of the Frobenius automorphism on k. We also assume we are given a scheme X, smooth and of finite type over OK .

The philosophy behind the syntomic mechanism creates a sophisticated way to patch together two classical “worlds”.

The first of these two “worlds” is the algebraic world, which is generally referred to as the de Rham world. In a sense, one might say that this world is the classical one. Definition 4.0.1. A de Rham datum for XK is XdR = (XK , Y, j), satisfying: 1. Y is a smooth and proper K-scheme, such that the complement D = Y \ XK is a relative simple normal crossing divisor over K. 111

112

2. j : XK ,→ Y is an open immersion. We also denote a de Rham datum by Y , omitting the immersion j. 0 Definition 4.0.2. Define a morphism between de Rham data f : XdR → XdR to be

the collection (f, f ) such that: 0 (i) f : XK → XK is a morphism of schemes over K.

(ii) f : Y → Y 0 is a morphism of schemes over K compatible with j and f . Definition 4.0.3. Let Y be a de Rham datum for XK . We define CdR (Y ) to be the category of vector bundles with integrable connection on Y , having logarithmic singularity along D := Y \ XK . More explicitly, CdR (Y ) is the category consisting of objects the triple MdR := (MdR , ∇dR , F • ), where: (i) MdR is a coherent OY -module. (ii) ∇dR : MdR → MdR ⊗ Ω1 (log D) is an integrable connection on MdR with logarithmic poles along D. (iii) F • is a Hodge filtration, namely, a descending exhuative separated filtration on MdR by coherent sub-OY modules satisfying Griffiths’ transversality condition ∇dR (F m MdR ) ⊂ F m−1 MdR ⊗ Ω1Y (log D). We realize SdR (XK ) := limY CdR (Y ), where the limit is over all de Rham data for −→ XK . For an element MdR ∈ CdR (Y ), we say that Y is a de Rham datum for MdR . Remark 4.0.4. Given MdR ∈ CdR (Y2 ), and a morphism of de Rham data f : Y1 → Y2 , the triple (f ∗ MdR , f ∗ ∇dR , f ∗ F • ) belongs to CdR (Y1 ). This follows by [27] proposition II.4.3. Definition 4.0.5. We denote the category of de Rham data by DdR,K .

113

The second “world” is the rigid analytic world, which took more time to be discovered. We refer to [11] for a thorough reading on the subject. For a formal scheme P there is associated a rigid analytic space over K0 (cf. [11] section 0.2.2), called the generic fiber of P and denoted PK0 . There is a canonical specialization map sp : PK → P (loc. cit 0.2.2.1), which is continuous when PK0 is given its strong Grothendieck topology and P its Zariski topology (it is a morphism of ringed spaces). As in loc. cit. definitions 1.1.2(i), we define the tubular neighborhood of a locally closed subset Y of the special fiber of a formal OK -scheme P to be ]Y [P := sp−1 (Y ). Definition 4.0.6. A rigid datum for Xk is Xrig = (Xk , X k , j, PX , φX , ι), satisfying: 1. X k is a proper k-scheme. 2. j : Xk ,→ X k is an open immersion. 3. ι : X k ,→ PX is a closed immersion into a p-adic formal W -scheme, smooth in a neighborhood of Xk . 4. φX :]X k [PX →]X k [PX is a lifting of the absolute Frobenius of X k . 0 to be the Definition 4.0.7. Define a morphism between rigid data f : Xrig → Xrig

collection (f, f ) such that: (i) f : Xk → Xk0 is a morphism of k-schemes. 0

(ii) f : X k → X k is a morphism of k-schemes compatible with j and f . (iii) fP : PX → PX 0 is a morphism of formal schemes over W compatible with ι, f and φX . Definition 4.0.8. We denote the category of rigid data by Drig,k .

114

We say that a set V ⊂]X k [P is a strict neighborhood of ]Xk [P in ]X k [P if V ∪ (]X k [P \]Xk [P ) is a covering of ]X k [P for the Grothendieck topology. For any abelian sheaf M on ]X k [P , we let j † M := limαV ∗ αV∗ M, −→ V

where the limit is taken over strict neighborhoods V of ]Xk [P in ]X k [P with inclusion αV : V ,→]X k [P . Notice that if M has the structure of a O]X k [P -module then j † M has the structure of a j † O]X k [P -module. Also notice that a map φX : PX → PX induces a natural morphism of rigid analytic spaces φX :]X k [P →]X k [P . Definition 4.0.9. We define Srig (Xk ) to be the category of F -isocrystsals overconvergent over Xk /K0 , σ (cf. [11], Definition 2.3.6 and 2.3.7). Concretely, an element of Srig (Xk ) is defined via its specialization over a rigid datum Xrig = (Xk , X k , j, PX , φX , ι) over Xk , (cf. section 2.3.2 loc. cit) as a triple Mrig := (Mrig , ∇rig , ΦM ), where: (i) Mrig is a coherent j † O]X k [P -module. (ii) ∇rig : Mrig → Mrig ⊗ Ω1]X

k [P

is an integrable connection on Mrig .

(iii) ΦM is the Frobenius morphism, which is an isomorphism ∼ =

ΦM : φ∗X Mrig → Mrig of j † O]X k [P -modules compatible with the connection. We denote the category of rigid coefficients over a rigid datum Xrig by Srig (Xrig ). Remark 4.0.10. The mentioning of σ may be omitted from the definition, because by our assumption on K, we know that σ is unique. In general, one may take any lift of Frobenius to K for σ (cf. [11] section 2.3.7).

115

1. Let Mrig := (Mrig , ∇rig , ΦM ) be an object in Srig (Xrig ).

Remark 4.0.11.

By [11] theorem 2.5.7, the existence of the Frobenius ensures that the connection ∇rig on Mrig is indeed overconvergent (loc. cit. definition 2.2.5). In particular, the category Srig (Xrig ) is a realization, in the sense of loc. cit. p. 68, of the category F-Isoc† (Xk /K0 ) of overconvergent F -isocrystals on Xk . 2. By [11] Theorem (2.3.1) and Proposition (2.2.17), Srig (Xrig ) is independent up to canonical equivalence of categories of the choice of the auxiliary data (PX , φX , ι). By loc. cit section 2.3.6 it is also independent up to canonical equivalence of categories of the choice of (X k , j). In order to glue two objects, intersection face along which the objects are glued is required. In our case, this intersection is a third “world” which in a sense underlies both worlds described above.

Definition 4.0.12. A vector datum for Xk is Xvec = (Xk , X k , j, ι), satisfying: 1. X k is a proper k-scheme. 2. j : Xk ,→ X k is an open immersion. 3. ι : X k ,→ PX is a closed immersion into a p-adic formal OK -scheme, smooth in a neighborhood of Xk . 0 Definition 4.0.13. Define a morphism between vector data f : Xvec → Xvec to be

the collection (f, f ) such that: (i) f : Xk → Xk0 is a morphism of k-schemes. 0

(ii) f : X k → X k is a morphism of k-schemes compatible with j and f .

116

(iii) fP : PX → PX 0 is a morphism of formal schemes over OK compatible with ι and f . Definition 4.0.14. We denote the category of vector data by Dvec,k . Definition 4.0.15. We define Svec (X) to be the category of isocrystsals over X/K. Concretely, an element of Svec (Xk ) is defined via its specialization over a vector datum Xvec = (Xk , X k , j, ι) over Xk , as a pair Mvec := (Mrig , ∇rig ), where: (i) Mvec is a coherent j † O]X k [P -module. (ii) ∇vec : Mvec → Mvec ⊗ Ω1]X

k [P

is an integrable connection on Mvec .

We denote the category of vector coefficients over a vector datum Xvec by Svec (Xvec ). Definition 4.0.16. Consider a compactification j : X → X (over OK ). Let χ (resp. χ) be the formal (p-adic) completion of X (resp. X). Such a triple (X, X, j) is called a strong vector datum, as it gives rise to a vector datum (Xk , X k , jk , ι), where ι : X k → χ is the natural morphism. Remark 4.0.17. By Nagata, we know that a strong vector datum always exists. We observe an Lemma 4.0.18. XK is a strict neighborhood of ]Xk [χ inside ]X k [χ .

Proof. We have an an χ K ⊂ XK ⊂ χK ∼ = XK .

where the last isomorphism is [11] proposition 0.3.5. By loc. cit. proposition 0.3.3 we an deduce that XK is open in χK . The lemma follows by loc. cit. Exemples 1.2.4(ii).

Remark 4.0.19. Given a strong vector datum Xs,vec = (X, X, j), a realization of Svec (Xk ) over Xs,vec is given by a pair Mvec := (Mvec , ∇vec ), where:

117

(i) Mvec is a coherent j † OXKan -module. (ii) ∇vec : Mvec → Mvec ⊗ Ω1XKan is an integrable connection on Mvec . Remark 4.0.20. Notice that the above category is the category of isocrystsals over X/K. We omitted the requirement of overconvergence w.r.t. X, as we do not assume that DdR,K gives rise to an overconvergent isocrystal (cf. definition 4.0.23 below). It will be convenient to take (X, X, j) as above, and work with χ as PX . The patching is done via the following general homological algebraic construction: Definition 4.0.21. Given three categories A, B, C and functors F : A → B, G : C → B we define the gluing category GLUE(A, B, C, F, G) to be the category having triplets for objects (a ∈ A, c ∈ C, ϕ : F (a) → G(c)), where ϕ is a morphism in B, and morphisms Ψ : (a1 , c1 , ϕ1 ) → (a2 , c2 , ϕ2 ) are given by pairs (ΨA , ΨC ), where ΨA : a1 → a2 , ΨC : c1 → c2 are morphisms in A and C respectively, making the following diagram commutative: ϕ1

F (a1 ) −−−→ G(c1 )     F (ΨA )y G(ΨC )y ϕ1

F (a2 ) −−−→ G(c2 ). Definition 4.0.22. Let Y be a de Rham datum for XK , and let an →Y pan,Y : XK

be the natural map. Definition 4.0.23. Let Y be a de Rham datum for X. We define the functor Fan,Y : CdR (Y ) → Svec (X) by associating to the object MdR := (MdR , ∇dR , F • ) the object Fan,Y (MdR ) := (Mvec , ∇vec ), where Mvec = j † (p∗an,Y MdR ),

118

where the dagger is applied to any strong vector datum over X (cf. [13] section 1.2 for the independence of the strict neighborhood). ∇vec is the connection on Mvec induced from ∇dR . Fan,Y is an exact functor, as a composition of exact functors (cf. [11] proposition 2.1.3). The inductive limit limY ∈CdR (X) Fan,Y is well defined. Indeed, given a morphism of de Rham data for X, f : Y1 → Y2 , and MdR ∈ CdR (Y2 ) we have Fan,Y2 (MdR ) = j † (p∗an,Y2 MdR ) = j † (p∗an,Y1 f ∗ MdR ) = Fan,Y1 (f ∗ MdR ). Therefore, we may define Fan to be the inducitve limit of all Fan,Y , thus establishing the functor Fan : SdR (X) → Svec (X), which is exact as an inductive limit of exact functors. Remark 4.0.24. The strong vector datum uses in an essential way the fact that X is an OK -scheme. Definition 4.0.25. Let Xrig = (Xk , X k , j, PX , φX , ι) be a rigid datum for Xk . We define the functor FK : Srig (Xrig ) → Svec ((Xk , X k , j, PX ⊗W OK )) by associating to the object Mrig := (Mrig , ∇rig , ΦM ) the object FK (Mrig ) := (Mrig ⊗W OK , ∇rig ⊗W OK ) in Svec ((Xk , X k , j, PX ⊗W OK )). This functor is clearly exact. It induces the exact functor FK : Srig (Xk ) → Svec (Xk ). The definition for the syntomic world is given by: Definition 4.0.26 (compare [5] definition 4.9). We define the category of syntomic coefficients on X, Ssyn (X) to be GLUE(SdR (XK ), Svec (Xk ), Srig (Xk ), Fan , FK ). Remark 4.0.27. This definition uses the fact that X is an OK -scheme, since it works with both the generic and special fibers, and also the definition of the functor Fan works with an OK -scheme.

119

Definition 4.0.28. For each integer n ∈ Z let us define the Tate object K(n) in Ssyn (X) to be the triple K(n) := (K(n)dR , K(n)rig , p), where: 1. K(n)dR in SdR (X) is given by the free rank one OY -module generated by en,dR on any de Rham datum Y , with connection ∇dR (en,dR ) = 0 and Hodge filtration F m K(n)dR = K(n)dR if m ≤ −n and 0 otherwise. 2. K(n)rig in Srig (X) is given by the free rank one j † ]X k [P -module generated by en,rig , with connection ∇dR (en,rig ) = 0 and Frobenius Φ(en,rig ) := p−n en,rig . This is the overconvergent F -isocrystal denoted OXk /K0 (n) in [11] Exemples 2.3.8(i). 3. The isomorphism p is the isomorphism given by p(en,dR ) = en,rig . For M := (MdR , Mrig , ϕ) ∈ Ssyn (X), let Y 0 be a de Rham datum for MdR . For a • de Rham datum i : Y → Y 0 , let D = Y \ XK and define DRdR (M )Y by • DRdR (M )Y = i∗ MdR ⊗ Ω•Y (logD).

(4.0.29)

This is a filtered complex with filtration given on each term by q F m DRdR (M )Y = i∗ F m−q MdR ⊗ ΩqY (logD).

(4.0.30)

• DRrig (M ) = Mrig ⊗ Ω•]X k [P ,

(4.0.31)

We define X

where the complex Ω•]X

k [P X

is independent up to canonical isomorphism of the

choice of rigid datum.

• DRvec (M ) = FK (Mrig (X)) ⊗ j † Ω•XKan ,

(4.0.32)

120

where the dagger is independent up to canonical isomorphism of the choice of the strong vector datum. Definition 4.0.33. For M := (MdR , Mrig , ϕ) ∈ Ssyn (X) we define the de Rham and rigid cohomology with coefficients in M by • (M )Y ), RΓdR (XK /K, M ) := limY RΓ(Y, DRdR −→

• F k RΓdR (XK /K, M ) := limY RΓ(Y, F k DRdR (M )Y ), −→

i HdR (XK /K, M ) := H i (RΓdR (XK /K, M )).

(4.0.34)

• RΓrig (Xk /K0 , M ) = RΓ(]X k [PX , DRrig (M (Xrig ))),

i Hrig (Xk /K0 , M ) := H i (RΓrig (Xk /K0 , M )).

We also let • RΓvec (Xk /K, M ) = RΓ(]X k [PX , DRvec (M (Xvec ))),

(4.0.35) i (Xk /K, M ) := H i (RΓvec (Xk /K, M )). Hvec

Remark 4.0.36. The direct limit in definition 4.0.33 is taken over morphisms of de Rham data. When working with “admissible” coefficients (in the sense of definition 5.5.11), corollary 5.8.20 implies that all morphisms in the direct limit are filtered quasi-isomorphisms and the de Rham cohmology may be computed on any de Rham data for the given de Rham coefficient MdR . For other applications, one might define syntomic cohomology over a particular compactification, if one wishes to We notice that there are natural maps F k RΓdR (XK /K, M ) → RΓdR (XK /K, M ).

(4.0.37)

121

Let Xvec,s be a strong vector datum as above, and let Xvec be its corresponding vector datum. By definition we have • (M (Xvec ))). RΓvec (Xk /K, M (Xvec )) = RΓ(]X k [PX , DRvec

By [13] 1.2.5 we know that it can be computed on any strict neighborhood (instead an of ]X k [PX ), and in particular on XK an • RΓvec (Xk /K, M (Xvec )) = RΓ(XK , DRvec (M (Xvec ))).

4.0.8

(4.0.38)

Functoriality

Let f : Xk → Xk0 be a morphism of k-schemes, and let M ∈ Srig (Xk0 ). The discussion between corollaire 1.5 and corollaire 1.7 in [13] (and a bit after), together with the discussion from Theorem 2.3.5 until after Definition 2.3.6 in [11] shows that there is a natural inverse image functor f ∗ : RΓrig (Xk0 /K0 , M ) → RΓrig (Xk /K0 , f ∗ M ), where f ∗ M is defined in section 2.3.6 (loc. cit.) together with the Frobenius which is naturally induced by the treatment of section 2.3.7 (loc. cit.). Similarly, given M ∈ Svec (Xk0 ), we have a natural inverse image functor f ∗ : RΓvec (Xk0 /K, M ) → RΓvec (Xk /K, f ∗ M ). 0 The situation for the de Rham side is more involved. Assume f : XK → XK is 0 a morphism of K-schemes, and let M ∈ SdR (XK ) be a de Rham coefficient defined 0 over Y 0 via the embedding j 0 : XK → Y 0 . Let j : X → Y be any de Rham datum

for X. By [27] proposition II.4.6(3) we deduce that f ∗ M has logarithmic poles along Y \ XK (recall that by loc. cit proposition II.4.3 having logarithmic poles depends only on the restriction of M to X 0 ). This implies that f ∗ M ∈ CdR (Y /X). We have

122

the commutative digram XK

f

0 XK

>

.

j0

j

(4.0.39) ∨ 0

∨

Y Y . 0 We have an immersion i : XK → Y ×Y defined by j×j 0 ◦f . Let T be the schematic closure of the image of i in Y × Y 0 . (That is, the smallest proper subscheme that contains the image of i inside Y ×Y 0 , which exists as Y ×Y 0 is proper.) Let r : T → T be its resolution of singularities over X (cf. Janos Kollar section 1.2.2). This also implies that r is proper. We obtain a commutative diagram .... .... .... .... .

X ∨

>

T

i

Y × Y 0.

We have T \ ˜j(X) is a divisor with normal crossings. We deduce that there is a morphism f˜ : T → Y 0 , such that the following diagram is commutative: XK

f

>

˜j

0 XK

.

j0 ∨

T

f˜

(4.0.41)

∨ > Y 0.

Then f˜ induces a morphism 0 f˜∗ : RΓdR (XK /K, M (Y 0 )) → RΓdR (XK /K, f˜∗ M (T )).

(4.0.42)

Let us prove that f˜∗ is independent up to a canonical isomorphism of the choices made in the direct limit defining the de Rham cohomology. Assume that j1 : X → Y1 , j2 : X → Y2 are two compactifications of T like T above. That is, they are equipped with proper (!) morphisms r1 : Y1 → T, r2 : Y2 → T such that (see diagram 5.8.17) r1 ◦ j1 = r2 ◦ j2 = i.

(4.0.43)

123

By Equation 4.0.43 we deduce that j : X → Y1 ×T Y2 defined by j := j1 × j2 is an embedding. Let U be the schematic closure of X in Y1 ×T Y2 . This is possible since Y1 ×T Y2 is as proper over T as Y1 and Y2 are (cf. [37] 4.6(d)). Since T is proper, we deduce that Y1 ×T Y2 is also proper. Let Y be the resolution of singularities of U over X, together with jY : X → Y . Then (Y , jY ) is a de Rham datum for X. There exist maps g1 : Y → Y1 , g2 : Y → Y2 respecting the inclusions j1 , j2 , jY . We also let f˜Y1 , f˜Y2 denote the analogs of f˜ above. There is a morphism Y → Y1 ×T Y2 → T → Y × Y 0 → Y 0 , which can be written as f˜Y1 ◦ g1 = Y → Y1 ×T Y2 → Y1 → T → Y × Y 0 → Y 0 , or as f˜Y2 ◦ g2 = Y → Y1 ×T Y2 → Y2 → T → Y × Y 0 → Y 0 . We deduce that f˜Y1 ◦ g1 = f˜Y2 ◦ g2 . Define f˜Y := f˜Y1 ◦ g1 . Then there exist maps 0 • f˜V∗ : RΓdR (XK /K, M (Y 0 )) → RΓ(V, DRdR (f˜V∗ M )V ),

for V taken as either Y1 , Y2 or Y . We notice that the composition f˜∗

Y 0 • g1∗ ◦ f˜Y∗1 : RΓdR (XK /K, M (Y 0 )) →1 RΓ(Y1 , DRdR (f˜Y∗1 M )Y1 )

(4.0.44) g∗

1 • → RΓ(Y , DRdR (f˜Y∗ M )Y ),

is equal to the composition g2∗ ◦ f˜Y∗2 . So, in the direct limit of the de Rham complex (cf. definition 4.0.33) f˜∗ is well defined.

124

4.0.9

Comparison between de Rham and rigid complexes

For M = (MdR , Mrig , ϕ) as above, we will define a functorial map in the generalized sense RΓdR (X/K, M ) → RΓvec (X/K, M ). This map is not (!) a quasi-isomorphism in general. Let • DRdR (M )|XK = p∗dR MdR ⊗ Ω•XK ,

where pdR : XK → Y is the morphism defining the de Rham datum. For a de Rham datum pdR : XK → Y we have the composed morphism • • RΓ(Y, DRdR (M )Y ) → RΓ(Y, pdR ∗ DRdR (M )|XK )

• an , (p∗dR MdR )an ⊗ Ω•XKan ) → RΓ(XK , DRdR (M )|XK ) → RΓ(XK

(4.0.45)

an , (p∗an MdR ⊗ Ω•XKan )), = RΓ(XK an where the last morphism is induced by XK → XK . We have a morphism p∗an MdR →

j † p∗an MdR = Fan,Y (MdR ). Applying it to the above we obtain an an RΓ(XK , (p∗an MdR ⊗ Ω•XKan ) → RΓ(XK , Fan,Y (MdR ) ⊗ Ω•XKan ).

Taking the limit over all de Rham data Y we obtain a morphism RΓdR (X/K, M ) → an RΓ(XK , Fan (MdR ) ⊗ Ω•XKan ). We have a morphism

ϕ : Fan (MdR ) → FK (Mrig ). It induces a morphism an an RΓ(XK , Fan (MdR ) ⊗ Ω•XKan ) → RΓ(XK , FK (Mrig ) ⊗ Ω•XKan )

an • = RΓ(XK , DRvec (M )) = RΓvec (X/K, M ),

where the last equality follows by Equation 4.0.38. Combining these maps we obtain a morphism ν : RΓdR (XK /K, M ) → RΓvec (Xk /K, M ).

(4.0.46)

125

0 Let Xvec , Xvec be vector data for Xk . The discussion between corollaire 1.5 and

corollaire 1.7 in [13] (and a bit after), together with the discussion beginning with Theorem 2.3.5 until after Definition 2.3.6 in [11], shows that there exists a quasi isormorphism 0 f ∗ : RΓvec (X/K, M (Xvec )) → RΓvec (X/K, M (Xvec )).

Since both a strong vector datum and a rigid datum give rise to vector data, we obtain a quasi-isomorphism RΓvec (X/K, M ) → RΓrig (X/K0 , M ) ⊗W OK .

(4.0.47)

The base change from K0 to K is invariance algebrique(cf. [31] section 1.3 and [76] proposition 2.12). This quasi-isomorphism can also be explained via a generalization of equation f rig (Xk /X) (5.3) in [14], which uses a definition of RΓ X,PX (cf. loc. cit. definition 4.16), and indirectly uses proposition 4.7 (loc. cit.) in an essential way. We have by definition: Lemma 4.0.48. ν is functorial with respect to morphisms f : X → X 0 of OK schemes. By combining (4.0.37), (4.0.46), and (4.0.47) we obtain a (generalized) map θk : F k RΓdR (XK /K, M ) → RΓrig (X/K0 , M ) ⊗W OK ∼ = RΓrig (X/K, M ), (4.0.49) which is functorial in X, M . We notice that the isomorphism appearing on the right hand side of the above morphism is algebraic invariance (cf. [31] section 1.3 and [76] proposition 2.12). We now define the action of the Frobenius on RΓrig (X/K0 , M ). First, we give an explicit description of it. Let X = (Xk , X k , j, PX , φX , ι), and let Mrig ∈ Srig (Xrig )

126

with frobenius ΦM . Given an O]X k [P -module F , φX induces the adjoint morphism • ad : F → φX∗ φ∗X F (cf. [44] equation 4.9). Let DRrig (M ) ∼ = J • be an injective resolution, and let φ∗X J • ∼ = I • also be an injective resolution. Since φ∗X is exact • (M ) ∼ (cf. [44] equations 4.3 and 4.4), it follows that φ∗X DRrig = I is an injective ad resolution. Then, we obtain a morphism RΓ(]X k [P , DR• (M )) ∼ = Γ(]X k [P , J) → rig

Γ(]X k [P , φX∗ φ∗X J)

=

Γ(]X k [P , φ∗X J)

• → Γ(]X k [P , I) ∼ (M )), = RΓ(]X k [P , φ∗X DRrig

which we also denote by ad. There is also a morphism induced by ΦM , namely, ΦM ∗ : • • (M )) → RΓ(]X k [P , DRrig (M )). Then, we let ϕM = ΦM ∗ ◦ ad, RΓ(]X k [P , φ∗X DRrig

which is the induced Frobenius on the appropriate derived category. There is also a formal way to define the Frobenius, based on [11] section 2.3.7 (cf. [14] corollary 4.22); let π : Xk → k be the projection and let Xk

F rXK ×π

→

Xk ⊗k,F rk k be the relative

Frobenius map, where the map k → k in the last tensor product is the Frobenius map of k, which is the p-power map. σ induces a base change functor σ ∗ : Isoc† (Xk /K0 ) → Isoc† (Xk ⊗k,F rk k/K0 ), by sending E to E ⊗W OK (cf. [11] section 2.3.7). This induces a base change morphism: K0 ⊗ RΓrig (Xk /K0 , M ) → RΓrig (Xk ⊗k,F rk k/K0 , σ ∗ M ). We now write the Frobenius: 1⊗id

RΓrig (Xk /K0 , M ) → K0 ⊗σ RΓrig (Xk /K0 , M ) (F rX ×π)∗

→

base change

→

RΓrig (Xk ⊗k,F rk k/K0 , σ ∗ M )

Φ

M RΓrig (Xk /K0 , (F rX × π)∗ σ ∗ M ) → RΓrig (Xk /K0 , M ),

(4.0.50) where the morphism ΦM : (F rX × π)∗ σ ∗ M ) → M is the Frobenius as described in [11] section 2.3.7. The composed map is ϕM .

127

4.0.10

Absolute geometric cohomology

Definition 4.0.51. Let M be a syntomic datum in Ssyn (X). We define the ith geometric cohomology of X with coefficients in M , denoted HAi (X, M ) = (MdR , Mrig , ϕ) to be the following element of Ssyn (K): i 1. MdR = HdR (XK /K, M ) with Filtration defined in 4.0.33. i 2. Mrig = Hrig (Xk /K0 , M ) with Frobenius induced by ϕM . i i 3. ϕ : HdR (XK /K, M ) → Hrig (Xk /K0 , M ) ⊗W K induced by θ.(notice that Fan

is an isomorphism). Definition 4.0.52. Let f : X → X 0 be a morphism of OK schemes and let M = (MdR , Mrig , ϕ) ∈ Ssyn (X 0 ). Define f ∗ M = (f ∗ MdR , f ∗ Mrig , f ∗ ϕ) ∈ Ssyn (X). Lemma 4.0.53. Let f : X → X 0 be a morphism of OK schemes. Then the morphisms defined in Section 4.0.8 gives rise to a morphism f ∗ : HAi (X 0 , M ) → HAi (X, f ∗ M ). We also have (cf. [3] lemma 1.22) Lemma 4.0.54. Let π : X → OK be the structure morphism, and let M be an object in Ssyn (OK ). Then there exists a canonical isomorphism HAi (X, π ∗ M ) → HAi (X, K(0)) ⊗ M.

4.0.11

Syntomic cohomology

Definition 4.0.55. Suppose we are given complexes X • , Y • , and Z • with maps f : X • → Z • and g : X • → Z • . Following [14] we define the quasi-fibered product X • ×Z • Y • := Cone(f − g)[−1].

128

Since k is a finite field, k is perfect. The Frobenius ϕM (Equation 4.0.50) induces a map Cone(1 −

ϕM pn

: RΓrig (Xk /K0 , M ) → RΓrig (Xk /K0 , M ))[−1] → RΓrig (Xk /K0 , M )

→ RΓrig (Xk /K, M ). Definition 4.0.56. The syntomic complex of X with coefficient M twisted by n is defined to be RΓsyn (X, M ; n) := Cone(1 −

∼ ϕM )[−1] ×RΓrig (Xk /K,M ) F n (RΓdR (XK /K, M )). n p

i (X, M ; n). The i’th cohomology of RΓsyn (X, M ; n) will be denoted Hsyn

When n is omitted, we assume it is set to 0. We have functoriality with respect to morphisms of schemes over OK . Lemma 4.0.57. Let f : X → X 0 be a morphism of OK schemes. Then the mori phisms defined in Section 4.0.8 give rise to a morphism f ∗ : Hsyn (X 0 , M ; n) → i Hsyn (X, f ∗ M ; n).

We have an analog proposition to that of [14] proposition 6.3 and [5] proposition 6.7. Proposition 4.0.58. For M ∈ Ssyn (X) and n ≥ 0 we have the long exact sequence (1)

i−1 i−1 i−1 i−1 · · · → Hrig (Xk /K0 , M ) ⊕ F n HdR (XK /K, M ) → Hrig (Xk /K0 , M ) ⊕ Hrig (Xk /K, M )

i → Hsyn (X, M )

(2)

i i i i → Hrig (Xk /K0 , M ) ⊕ F n HdR (XK /K, M ) → Hrig (Xk /K0 , M ) ⊕ Hrig (Xk /K, M ) → · · · (4.0.59)

Where maps (1) and (2) are given in the appropriate degrees by (x, y) →

1−

ϕM x, x − θ (y) . n pn

(4.0.60)

129

Proof. By writing explicitly the quasi-fibered product in term of cones, one finds RΓsyn (X, M ; n) ∼ = Cone RΓrig (Xk /K0 , M ) ⊕ F n RΓdR (XK /K, M ) → RΓrig (Xk /K0 , M ) ⊕ RΓrig (Xk /K, M ) [−1], where the map defining the cone is given by 4.0.60. This gives the result (cf. [14] proposition 6.7 for a similar proof). We have as corollary Corollary 4.0.61. 0 Hsyn (X, M ) = HomSsyn (X) (K(0), M ). i The de Rham cohomology HdR (X, M ) has a filtration, called the Hodge filtration

induced by the Hodge to de Rham spectral sequence p+q • E1p,q = Rp+q ΓY Grp (DRdR (X, M (Y ))) ⇒ HdR (X, M ).

4.0.12

(4.0.62)

Admissible coefficients

We define a subcategory of the category of syntomic coefficients, which is of special interest to us. Definition 4.0.63. We define the category of admissible syntomic coefficients on ad X, Ssyn (X), to be the full subcategory of Ssyn (X) whose objects are the triplets

(MdR ∈ SdR (X), Mrig ∈ Srig (X), ϕ : Fan (MdR ) → FK (Mrig )), such that: 1. ϕ is an isomorphism. i i 2. θ : HdR (X/K, M ) → Hrig (X/K, M ) is an isomorphism, where θ is defined by

Equation 4.0.49.

130

i 3. Hrig (X/K0 , Mrig ) with the Frobenius induced by ϕM and the Hodge filtration

induced by the filtration on the de Rham cohomology over all de Rham data for MdR via θ is an object in M FKf , defined in [5] definition (1.1). 4. The Hodge de-Rham spectral sequence degenerates at E1 . We restate [3] proposition 1.19 with relaxed conditions. Proposition 4.0.64. For any exact sequence 0 → M 0 → M → M 00 → 0 in Ssyn (X), there is an associated long exact sequence · · · → HAi (X, M 0 ) → HAi (X, M ) → HAi (X, M 00 ) → · · · obtained by pasting together the corresponding sequences for de Rham and rigid cohomologies. The proof is essentially the same, so it is omitted. Remark 4.0.65. Notice that the rigid cohomology is defined as the sheaf cohomology • (M (Xrig )) over any strict neighborhood (cf. [31] section I.3, and [13] 1.2.5); of DRrig

the de Rham cohomology is of course computed as the direct limit over de Rham • data Y of the sheaf cohomology of DRdR (M )Y over Y . So these sequences exist by

classical sheaf cohomological arguments. θ is also functorial. We may work in the derived categories, and there you have triangles that induce the long exact sequences. We notice that although RΓrig (Xk /K0 , M ) are computed with respect to a choice of a rigid datum for X, the independence of the auxiliary data (see remark 2) implies that in the derived category these complexes are independent of the choices of rigid data for M , M 0 , M 00 .

131

ad Lemma 4.0.66. Let M be an object in Ssyn (X ), and assume S ∈ Ext1Ssyn (X ) (K(0), M ). ad Then S ∈ Ssyn (X ).

Proof. There is a short exact sequence 0 → M → S → K(0) → 0. We need to show that the three properties of definition 5.8.12 hold. We show the first two properties. The proof of the other properties is similar. 1. We have a commutative diagram 0

>

Fan (MdR )

>

ϕM o

Fan (SdR )

>

Fan (K(0)dR )

∨

FK (Mrig )

>

>

0

.

ϕK(0) o

ϕS ?

∨

0

>

(4.0.67)

∨

FK (Srig )

>

Frig (K(0)rig )

>

0

.

Since M , K(0) are admissible syntomic coefficients, ϕM , ϕK(0) are isomorphisms. The five lemma imply that ϕS is also an isomorphism. 2. By Proposition 4.0.64 we have the following commutative diagram i−1 HdR (X, K (0))

>

>

θiM o

K o θi−1 ∨

i−1 Hrig (X, K

i HdR (X, M )

(0))

i Hrig (X, M )

>

θiS ?

∨

>

i HdR (X, S )

i Hrig (X, S )

>

∨

>

i Hrig (X, K(0))

i+1 HdR (X, M ). M o θi+1

θiK o

∨

>

i HdR (X, K(0))

>

∨ i+1 Hrig (X, M ).

(4.0.68) Both horizontal sequences are exact. The vertical morphisms are all isomorphisms except possibly the middle one, since M , K(0) are admissible syntomic coefficients. The five lemma imply that the middle morphism is an isomorphism. This holds for all i.

132

Proposition 4.0.69. For any exact sequence 0 → M 0 → M → M 00 → 0 in Ssyn (X), there is an associated long exact sequence i i i · · · → Hsyn (X, M 0 ) → Hsyn (X, M ) → Hsyn (X, M 00 ) → · · ·

commuting with the exact sequence of theorem via the maps appearing in [4] proposition 2.17 when the syntomic coefficients are admissible. Proof. Let Y be a de Rham datum for X. We have a triangle for the de Rham cohomology over Y with logartihmic singularities along Y \ X by standard sheaf theoretical homological algebra 1

• • • RΓ(Y, DRdR (M 0 )Y ) → RΓ(Y, DRdR (M )Y ) → RΓ(Y, DRdR (M 00 )Y ) → .

By taking a direct limit over all de Rham datum for X, we obtain the triangle 1

RΓdR (XK /K, M 0 ) → RΓdR (XK /K, M ) → RΓdR (XK /K, M 00 ) → . We notice that although RΓrig (Xk /K0 , M ) are computed with respect to a choice of a rigid datum for X, the independence of the auxiliary data (see remark 2) implies that in the derived category these complexes are independent of the choices of rigid data for M , M 0 , M 00 . Thus, we obtain a triangle for the rigid cohomology (again, standard homological algebra) 1

RΓrig (Xk /K0 , M 0 ) → RΓrig (Xk /K0 , M ) → RΓrig (Xk /K0 , M 00 ) → . Given these triangles, we construct a triangle by taking the “mapping cone” of these triangles as in definition 4.0.56 to obtain the syntomic part. This is clearly a triangle. And of course this triangle commutes with the rigid part via the natural maps.

133

The following is a fundamental theorem in our theory. ad Theorem 4.0.70. Let M be an object in Ssyn (X). Then there exist canonical and

functorial isomorphisms, both denoted by Ξ, that is: 1 Ξ : Ext1Ssyn (X) (K(0), M ) → Hsyn (X, M ), 1 Ξ : Ext1Srig (X) (K(0), M ) → Hrig (X, M ),

such that the obvious diagram connecting these maps (via the forgetful functors) commute. Proof. The rigid part is [4] proposition 2.17 . Given S ∈ Ext1Ssyn (X) (K(0), M ), we write the corresponding short exact sequence of syntomic coefficients 0 → M → S → K(0) → 0. By Lemma 4.0.66 we deduce that S is admissible. There is a long exact sequence for syntomic cohomology by Proposition 4.0.69. Then Ξ(S) = δ0 (id), where δ0 is 0 the connecting homomorphism δ0 : Hsyn (X, K(0)) = HomSsyn (X) (K(0), K(0)) → 1 Hsyn (X, M ), where the first equality follows by corollary 4.0.61. Ξ(Srig ) is defined via

the long exact sequence for geometric cohomology 4.0.64, by Ξ(Srig ) := δ0,rig (id). It is clear that δ0 (id) is mapped to δ0,rig (id) under the forgetful functors. We therefore turn to the proof of the syntomic part.

For the proof of the syntomic part it suffices to prove: Theorem 4.0.71. With notations as in Theorem 4.0.70, there exists a short exact sequence of the form β

ζ

0 1 0 → Ext1Ssyn (K) (1, Hrig (E, M )) → − Ext1Ssyn (E) (1, M ) → − Ext0Ssyn (K) (1, Hrig (E, M )) → 0.

(4.0.72)

134

proof of Theorem 4.0.71. Indeed, proposition 6.8 [5] applied here, implies that the 1 same short exact sequence exists also for Hsyn (E, M ). By applying the five lemma,

we obtain that the middle terms are isomorphic, which finishes the proof.

Remark 4.0.73. Regarding S as a rigid coefficient, we know that there is a Frobenius ϕS : ϕ∗E S → S. This morphism can be thought of a ϕ∗E OE semi-linear morphism from S to itself, that is, as a morphism φS : S → S such that φS (f · s) = f p φS (s) ([11] p. 75). Notation: We write: i

j

0→M → − S→1→ − 0. We use the description of an admissible syntomic coefficient as a pair (SdR , Srig ) with an isomorphism ϕ : Fan (SDR ) → FK (Srig ) satisfying the properties of 5.8.12. In this case we have: SdR ∈ Ext1SdR (E) (1, M )), Srig ∈ Ext1Srig (E) (1, M )). 1 (E, M ) has a syntomic First, we define the map ζ. By definition 5.8.12(iii), Hrig 1 1 (E, Mrig ), ϕH ), where ϕ is induced structure on it, namely H 1 = (HdR (E, MdR ), Hrig

by the isomorphism of definition 5.8.12(i). To simplify notations, we may assume 1 1 that Hrig (E, Mrig ) ⊗K0 K is identified with its image inside Hvec (E, M ) (cf. [5] just 1 before definition 4.14). Similarly, HdR (E, MdR ) is identified via definition 5.8.12(i) 1 inside Hvec (E, M ). With this identification ϕH is idH 1 . We recall that for de Rham

coefficients, we have an isomorphism ζdR : ζdR 1 ∼ 1 Ext1SdR (E) (1, M )) −→ Ext0SdR (K) (1, HdR ) = F 0 HdR . ∼

Similarly, for rigid coefficients, we have an isomorphism ζrig 1 ∼ 1 Ext1Srig (E) (1, M )) −→ Ext0Srig (K) (1, Hrig ) = Hrig ∼

φH 1 =1

,

135

and for vector coefficients, we have an isomorphism ζvec 1 1 , )∼ Ext1Svec (E) (1, M )) −→ Ext0Svec (K) (1, Hvec = Hvec ∼

Lemma 4.0.74. 1 1 Ext0Ssyn (K) (1, H 1 ) ∼ ∩ Hrig = F 0 HdR

φH 1 =1

.

Proof. We have 1 1 Ext0Ssyn (K) (1, Hrig (E, M )) = HomSsyn (K) (1, Hrig (E, M )).

The object 1 is K(0) of definition 4.0.28. An element of HomSsyn (K) (1, H 1 ) is given 1 1 by Ψ = (ΨdR , Ψrig ), such that ΨdR ∈ HomSdR (K) (1, Hrig ), Ψrig ∈ HomSrig (K) (1, HdR ),

commute as in definition 4.0.21. We know that ΨdR respects the filtration. e0,dR ∈ 1 F 0 K(0), so ΨdR (e0,dR ) ∈ F 0 HdR . We know that Ψrig commutes with the Frobenius, 1 ◦ Ψrig . Since ΦK(0) (e0,rig ) = e0,rig , it implies that namely that Ψrig ◦ ΦK(0) = ΦHrig 1 ◦ Ψrig (e0,rig ) = Ψrig (e0,rig ). The condition of definition 4.0.21 implies that ΦHrig

ΨdR (e0,dR ) = Ψrig (e0,rig ). Combining these data together the result follows. We have a commutative diagram

Ext1SdR (E) (1, M )) >

Fan

ζdR ∼

>

1 ∼ 1 Ext0SdR (K) (1, HdR ) = F 0 HdR

∨ ∨ ζvec 1 1 > Ext0 Ext1Svec (E) (1, M )) (1, Hvec )∼ = Hvec Svec (K) ∼ .. .. ϕ ≡ id S ∈ Ext1Ssyn (E) (1, M )) idH 1 .. S . ∨ ∨ ζvec 1 1 > Ext0 Ext1Svec (E) (1, M )) (1, Hvec )∼ = Hvec Svec (K) ∼ ∧ ∧ FK > ζrig 1 1 ∼ 1 φH 1 =1 > Ext0 ExtSrig (E) (1, M )) (1, H ) H . = Srig (K) rig rig ∼ (4.0.75)

136

Starting from Ext1SdR (E) (1, M )), and using the commutativity of the composed maps, we define ζ(S) = (ζdR (SdR ), ζrig (Srig )). Using the commutativity of 4.0.75, and the proof of Lemma 4.0.74, we deduce that it belongs to Ext0Ssyn (K) (1, H 1 ). The surjectivity of ζ is proved via diagram chasing, given an element of α ∈ 1 Ext0Ssyn (K) (1, H 1 ) and viewing it as an element of Hvec , which is in the image of 1 1 φH 1 =1 both αdR ∈ F 0 HdR and αrig ∈ Hrig . Then it corresponds to the pair −1 −1 −1 (ζdR (αdR ), ζrig (αrig )). Then the commutativity of 4.0.75 implies that Fan (ζdR (αdR )) ≡ −1 FK (ζrig (αrig )) up to isomorphism, implying it defines an element of Ext1SdR (E) (1, M )). 1 (E, M )ϕ=1 we define We may also check the surjectivity explicitly; given ω ∈ F 0 Hrig

an extension S ∈ Ext1Ssyn (E) (1, M ) with ζ(S) = ω as follows; we let SdR be the natural de Rham extension deduced by ω via the natural transformation; ω gives an element 1 in F 0 HdR (E, M ) by definition 5.8.12.

It is equivalent to giving ωi ∈ F 0 (Ω1 (SdR , Ui )), i = 0, 1 and f ∈ SdR (U0 ∩U1 ) such that ∇ei = ω mod coboundary and df = ω0 − ω1 on the intersection. We let Srig = j † SdR and ϕ(ei ) = ei , ϕ(f ) = f . Clearly, S ∈ Ext1Ssyn (E) (1, M ), ζ(S) = ω. Next, let us describe the construction of β; assume that S ∈ Ext1Ssyn (E) (1, M ) is 1 (E, M )). What does it mean that S is mapped mapped to zero in Ext0Ssyn (K) (1, Hrig 0 1 (E, M ) is a (E, M ))? By definition 5.8.12(c), H 0 := Hrig to zero in Ext0Ssyn (K) (1, Hrig

syntomic coefficient over spec(K). • Since S is a syntomic coefficient, it contains both de Rham and rigid data. 1 The assumption on S implies that both ζdR (SdR ) ∈ Ext0Srig (K) (1, Hrig (E, M )) 1 and ζrig (Srig ) ∈ Ext0SdR (K) (1, Hrig (E, M )) are zero. Since ζdR and ζrig are iso-

morphisms, we first deduce that the hodge and Frobenius structures on S are trivial. There is also a relation between the de Rham and rigid componenets. Namely, one should describe a morphism ϕS : Fan (SDR ) → FK (Srig ) in Svec (E). When restricting this morphism to M , one should obtain the isomorphism ϕM according to definition [5] 4.9(b), and definition 5.8.12(a). We explain all these

137

ingredients below. By the triviality of the Frobenius structure there is a globally defined section on Srig , say x, mapping to 1, such that φS (x) = x, ∇(x) = 0. By the triviality of the Hodge structure there is a globally defined section y ∈ F 0 (SDR ) mapping to 1, such that ∇(y) = 0. We now use ϕ above to compare between ϕ(Fan (y)) and FK (x). Notice that they are not the same, as S need not be trivial as a syntomic coefficient ! However, can we say something about their difference? It maps to 0 under the projection, implying there exists a global section v ∈ Mvec with v = ϕ(Fan (y))− FK (x). However, as we know that ∇(x) = 0, ∇(y) = 0 we deduce that ∇(v) = 0, 0 0 0 implying v ∈ Hvec (E, M ) = Hvec . Thus, S is determined by v ∈ Hvec , i.e., we 0 established a surjection between Hvec → {kernel of the short sequence above},

which we denote by F . Did we have freedom in our choices of v? Below we prove the following lemma: Lemma 4.0.76. The following sequence is exact: 0 Hrig

φH 0 =1

α

0 0 ⊕ F 0 (HDR ) → Hvec → {kernel above} → 0, where α is defined by

the rule α : (a, b) → ϕ(Fan (b)) − FK (a). Lemma 4.0.77. We have the following exact sequence: 0 Hrig

φH 0 =1

α

0 0 ⊕ F 0 (HDR ) → Hvec → Ext1Ssyn (K) (1, H 0 ) → 0.

Definition 4.0.78. we define β to be the induced map Ext1Ssyn (K) (1, H 0 ) → {kernel above} composed with the inclusion {kernel above} → Ext1Ssyn (E) (1, M ). By Lemma 4.0.76 the first map in the composition is an isomorphism, so the proof of the theorem is complete.

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Proof of Lemma 4.0.76: Proof. The only non-trivial part is exactness in the second component; we first show 0 0 that the image under α of any two elements a ∈ Hrig , b ∈ F 0 (HDR ) gives a trivial

element in the kernel of 4.0.72. If we denote S := F (α(a, b)) ∈ Ext1Ssyn (E) (1, M ), then S is described by taking x, y as above to satisfy : ϕ(Fan (y)) − FK (x) = ϕ(Fan (b)) − FK (a)). This implies that ϕ(Fan (y − b)) = FK (x − a). Hence, the pair x − a, y − b defines a trivialization of S as a syntomic coefficient (notice also that ∇(x − a) = 0, ∇(y − b) = 0). We should show the other direction of exactness, that is, we assume F (v) defines a syntomic coefficient S, with x and y satisfying the properties in the proof of the theorem with their relation to v, such that it is isomorphic to the trivial coefficient, and show that it lies in the image of α. By the assumption, we may find x1 ∈ Srig and y1 ∈ F 0 (SDR ) trivializing the Frobenius and Hodge Structures, respectively, as above, but also satisfying ϕ(Fan (y1 )) = FK (x1 ), which trivializes the syntomic ”glue” between the two structures. We summarize all the data in 1. φS (x) = x 2. ϕ(Fan (y)) − FK (x) = v 3. φS (x1 ) = x1 4. ϕ(Fan (y1 )) = FK (x1 )

139

Since x − x1 is mapped to 0, and also ∇(x) = ∇(x1 ) = 0, φS (x) = x, φS (x1 ) = x1 , 0 φH 0 =1 there exists a ∈ Hrig such that x − x1 = a. Since y − y1 is mapped to 0, and also ∇(y) = ∇(y1 ) = 0 there exists b ∈ F 0 (HDR ) such that y − y1 = b ⇒ y1 = y − b ⇒ ϕ(Fan (y1 )) = ϕ(Fan (y − b)). By the last equality we deduce that it equals FK (x1 ) = FK (x − a). We deduce that ϕ(Fan (b)) − FK (a) = ϕ(Fan (y)) − FK (x) = v. This implies that α(−a, b) = v. Proof of Lemma 4.0.77: Proof. Let S ∈ Ext1Ssyn (K) (1, H 0 ). We first observe that the Frobenius structure on S is trivial. So, let x ∈ Srig , such that it projects to 1. Then x − φS (x) projects to 0. 0 This implies the existence of a ∈ Hrig such that x − φS (x) = a. By [41] lemma 5.3,

we deduce that 1 − ϕH 0 : H 0 → H 0 is surjective. Hence, there exists a0 ∈ H 0 such that a0 − ϕH 0 (a0 ) = a. By letting x0 = x − a0 we obtain φS (x0 ) = x0 , thus trivializing the Frobenius structure. The proof continues following the same lines as in Lemma 4.0.76. Remark 4.0.79. We give here another way to realize ker(ζ). By the considerations above, we know that SdR is isomorphic to MdR ⊕ 1dR , say via χdR , and that Srig is isomorphic to Mrig ⊕ 1rig , say via χrig . These isomorphisms are obviously noncanonical. We omit the mentioning of Fan and FK to simplify notations. We have a commutative diagram: χ

dR SdR −−− → MdR ⊕ 1dR    ϕ ⊕? ϕS =?y y M

χrig

Srig −−−→ Mrig ⊕ 1rig . Thus, given χdR and χrig , in order to define ϕS we should describe ϕS (1x ), where 1x ∈ F 0 (1dR ). This may also be described by ϕS (1x ) − 1y , where 1y ∈ 1ϕ=1 rig as before. In other words, by the component of ϕS (1x ) belonging to Mrig in the direct sum decomposition, it is clear that this element vanishes under the action of ∇Mrig , thus

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0 it belongs to Hvec in the above notations. But, and there is a big but here, χdR and

χrig are not prescribed to us. For the same choice of ϕS , different χs yield different 0 . Therefore, one must take a quotient by commutative diagrams of elements in Hvec

the form

idM

⊕?

MdR ⊕ 1dR −−−dR −−→ MdR ⊕ 1dR    ϕ ⊕? ϕM ⊕idy y M idMrig ⊕?

Mrig ⊕ 1rig −−−−−→ Mrig ⊕ 1rig . 0 This easily implies the quotient by Hrig

φH 0 =1

0 ), which is what we wanted ⊕F 0 (HDR

to show. ad (K). Since M is admissible, we may endow M with a Remark 4.0.80. Let M ∈ Ssyn

filtration and a Frobenius (exactly as Bannai defined a filtered Frobenius module [4] 1 remark 2.8). By remark 2.13 [4] we know that Hsyn (OK , M ) = coker(1−φM : F 0 M →

M ). On the other hand, by the just-proven theorem and Lemma 4.0.77, we know α that it should be isomorphic to Ext1Ssyn (K) (1, M ) ∼ = coker((M φM =1 ⊕ F 0 (M )) → M ).

The mapping between the two modules is given by 1 − ϕM . To show that the two are isomorphic we use lemma 5.3 [41] to deduce that 1−ϕM : M → M is a surjection. All that remains is to show that (1 − φM )−1 (1 − φM )(F 0 M ) = M φM =1 ⊕ F 0 (M ). Indeed, given x ∈ (1 − φM )−1 ((1 − φM )(F 0 M )), we may write (1 − φM )(x) = (1 − φM )(y) for y ∈ F 0 M . Then h := x − y ∈ M φM =1 , so we can write x = h + y ∈ M φM =1 ⊕ F 0 (M ). A simpler proof: actually, the morphism 1 − φM is a map of abelian groups (it is only σ-linear, so it is not a map of K modules). Thus, we deduce that M/ ker(1 − φM ) ∼ = Im(1−φM ) = M . We have ker(1−φM ) = M φM =1 . Since this is a morphism of abelian groups, it is immediately apparent that (1−φM )−1 (1−φM )(F 0 M ) = M φM =1 ⊕F 0 (M ) (a preimage of an element is just some element in the preimage directly summed with the kernel).

Chapter 5 The Leray spectral sequence for syntomic cohomology Remark 5.0.1. Saito’s work ([40] remark after definition 1.2) implies the existence of a Leray spectral sequence for absolute cohomology. In particular, there exists a Leray spectral sequence for de Rham cohomology, respecting the filtration. There is also a Leray spectral sequence for rigid cohomology (no problems arise there). It is not obvious, however, that the two spectral sequences commute with the comparison morphism θ between de Rham and rigid cohomologies. A challenging future project would be to find an analog of Saito’s work that gives the Leray spectral sequence for syntomic cohomology. The work of this chapter does not use Saito’s work.

5.1

Paranjape’s work and homological algebraic background

In this section, we give the homological background needed for the proof of the Leray spectral sequence for syntomic cohomology. We freely use notations and propositions from [67]. Let C be an abelian category. All objects, morphisms, etc. will be with respect to

141

142

this category. We recall a couple of results from homological algebra Lemma 5.1.1. Let α : P • → A, β : Q• → B be two projective resolutions. Suppose f : A → B and g : B → A are morphisms, and denote ϕ = f ◦ g. Suppose further that f˜ : P • → Q• , g˜ : Q• → P • , ϕ˜ : P • → P • lift f, g, ϕ, respectively. Then f˜ ◦ g˜ ≡ ϕ˜ up to homotopy equivalence. Proof. The proof is omitted. Lemma 5.1.2. Let I • be an injective (projective) complex such that H n (I) = 0. Then im(∂n ) is injective (resp. projective) for all n. Proof. By proposition 1, we know that there exists a morphism h : I → I[−1] such that id = ∂ ◦ h + h ◦ ∂.

(5.1.3)

To prove injectivity of im(∂n ) take a monomorphism i : X ,→ Y and a mapping f : X → im(∂n ). Let g = h ◦ f . Since im(g) ⊆ I n , and I n is injective, it follows that there exists a function g˜ : Y → I n lifting g. Let f˜ = ∂n ◦ g˜. Then for x ∈ X, f˜(x) = ∂n ◦ h ◦ f (x). Since im(f ) ⊆ im(∂n ), we may write f (x) = ∂n x0 . By Equation 5.1.3 we deduce that f˜(x) = ∂n ◦ h ◦ ∂n x0 = ∂n x0 = f (x).

5.1.1

First proposition

Proposition 5.1.4. Let P • , Q• be two complexes of projectives, such that P n , Qn = 0 for n < 0, and let f : P • → Q• be a quasi-isomorphism between them. There exists a morphism of complexes g : Q• → P • such that f ◦ g and g ◦ f are homotopic to the identity. Proof. We recall [44] theorem 6.2o

143

Theorem 5.1.5. Let R• be a bounded above complex of projectives. Any quasiisomorphism c : X • → Y • induces an isomorphism [1, c] : [R• , X • ] ∼ = [R• , Y • ]. By letting R• := Q• , X • := P • , Y • := Q• and f := c we deduce: [1, f ] : [Q• , P • ] ∼ = [Q• , Q• ]. This implies that there exists g ∈ Hom[Q• , P • ] such that [1, f ](g) = f ◦g is homotopic to idQ• . We claim that g ◦ f is homotopic to idP • . Indeed, by using theorem 6.2o loc. cit. similarly, we deduce an isomorphism [1, g] : [P • , Q• ] ∼ = [P • , P • ]. This implies that there exists f˜ ∈ Hom[P • , Q• ] such that [1, g](f˜) = g ◦ f˜ is homotopic to idP • . We deduce that f˜ = idQ• ◦ f˜ ∼ f ◦ g ◦ f˜ ∼ f ◦ idP • = f. This implies that g ◦ f ∼ g ◦ f˜ ∼ idP • as claimed. Remark 5.1.6. One may write C := M C(f ), and then C is quasi-isomorphism to 0. Then the theorem follows by the proof of theorem 6.2 [44].

5.1.2

Second proposition

Definition 5.1.7. We say that (K, F ) is a good filtered complex in C if K is a cochain complex in C that is bounded below and F is a filtration on it that is compatible with the differential and is finite on K n for each integer n. Proposition 5.1.8. Let A• , B • be good filtered complexes whose graded objects are projective. Then the filtered pieces are projective. Moreover, given fn : Grn A• → Grn B • , n ∈ Z, there exist fñ : F n A• → F n B • , n ∈ Z whose graded induce fn .

144

Proof. Let n0 ∈ Z such that F n A• , F n B • = 0 for n > n0 . The proof is done via induction. The claim is true for n = n0 (since Grn0 = F n0 by assumption). Suppose the claim is true for general n. Write the short exact sequence of complexes i

A 0 → F n+1 (A• ) → F n (A• ) → Grn (A• ) → 0.

Since Grn (A• ) is a complex of projectives, the sequence is termwise split, and the same is true for B. Thus we have a termwise decomposition F n (A• ) ∼ = F n+1 (A• ) ⊕ Grn (A• ), F n (B • ) ∼ = F n+1 (B • ) ⊕ Grn (B • ).

(5.1.9)

We use this identification later. As the sum of projectives is projective, the first part of the proposition follows. We have the following cummutative diagram of complexes F n+1 (A• ) fñ+1 F

∨ n+1

(B • )

F n (A• ) .. . ? .... fñ . ∨ > F n (B • ) >

>

Grn (A• )

>

∨ n

iA

iB

>

0

>

0.

fn Gr (B • )

We inductively construct, given fñ+1 and fn as in the diagram, the morphism fñ : F n (A• ) → F n (B • ). Let F n+1 A be the complex . . . → PkA → . . . → P1A → P0A → 0. Let F n A be the complex A A . . . → QA k → . . . → Q1 → Q0 → 0.

Then the complex Grn A is given by A A A A A . . . → QA k /Pk → . . . → Q1 /P1 → Q0 /P0 → 0.

145

We use the same notations for B. B We define fñ by induction on k. Basis of induction: we let fñ0 : QA 0 → Q0 be defined 0 ⊕ fn0 under identification 5.1.9. We assume fñ0 , . . . , fñk−1 have been by fñ0 = fñ+1

defined, and we define fñk . Using the identification of 5.1.9 once more, an element of A A A QA k is represented as a pair (pk , qk ), with pk ∈ Pk , qk ∈ Qk /Pk . k It is clear that fñk (pk , 0) = (fñ+1 (pk ), 0).

Therefore, it suffices to define fñk (0, qk ). A

B

Define gnk (qk ) := fñk−1 (∂kQ (0, qk )) − ∂kQ (0, fnk (qk )). A B B It is easy to verify that gnk : QA k /Pk → Qk−1 vanishes modulo Pk−1 . Thus, we obtain

a morphism A B gnk : QA k /Pk → Pk−1 .

Since PkB is projective, it follows that there exists a morphism PB k A B gñk : QA ñ = gnk . k /Pk → Pk , such that ∂k g k (pk ) + gñk (qk ), fnk (qk )). By definition of gñk , gnk we easily We define fñk (pk , qk ) = (fñ+1

deduce that B A ∂kQ ◦ fñk = fñk−1 ◦ ∂kQ .

This completes the inductive step for the construction of fñk for all k. The second induction clearly follows, and the proof is complete. Corollary 5.1.10. Let A• be a good filtered complex whose graded object Grn (A• ) is a complex of projectives for all n. Then A• is a complex of projectives. Proof. The proof follows by the finiteness of the filtration.

5.1.3

Third proposition

Proposition 5.1.11. Let P • , Q• be two bounded above complexes of projectives, and let fn : H n (P • ) → H n (Q• ) be a morphism between their n’th cohomology groups.

146

Further assume that im∂nP , im∂nQ are projectives. Then, there exists a morphism of complexes f : P • → Q• such that fn induces the map H n (f ) on the nth cohomology groups for n ≥ 0. Proof. The proof is via induction on n. We notice that H 0 (P ) ∼ = = P0 /im(∂1P ), H 0 (Q) ∼ Q0 /im(∂1Q ). Since P0 is projective, there exists a morphism f0 making the following diagram commutative: > P0 /im(∂ P ) >0 P0 1 .. . f0 .... f˜0 .∨. > ∨ ∨ Q > Q0 /im(∂ ) > 0. Q0 1 This is the 0-step of the induction. For simplicity we prove the transition 0 → 1

which translates verbatim to the case n → n + 1. Remark 5.1.12. We notice here that f0 (im(∂1P )) ⊆ im(∂1Q ) since f˜0 (0) = 0. We have the short exact sequences: 0

>

ker(∂1P )

0

>

ker(∂1Q )

iP

iQ

>

P1

>

Q1

∂1P

>

∂1Q

>

im(∂1P )

>

0.

(5.1.13)

im(∂1Q )

>

0.

(5.1.14)

We deduce the s.s.e.: 0

>

ker(∂1P )/im(∂2P )

0

>

ker(∂1Q )/im(∂2Q )

iP

>

iQ

>

P1 /im(∂1P )

Q1 /im(∂2Q )

∂1P

>

∂1Q

>

im(∂1P )

>

0.

(5.1.15)

im(∂1Q )

>

0.

(5.1.16)

By our assumption im(∂1P ), im(∂1Q ) are Projective, so the sequences split. Therefore there exists a morphism f10 making the following diagram commutative. 0

>

ker(∂1P )/im(∂2P )

iP

>

f10

H 1 (f ) ∨

0

>

ker(∂1Q )/im(∂2Q )

P1 /im(∂2P )

iQ

>

∨

Q1 /im(∂2Q )

∂1P

>

im(∂1P ) f0

∂1Q >

>

0. (5.1.17)

∨

im(∂1Q )

>

0.

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Since P1 is projective, there exists a morphism f1 making the following diagram commutative: P1 .. . f1 .... .. ∨ Q1

>

P1 /im(∂2P ) >

>

∨

>

0

>

0.

f10

Q1 /im(∂2Q )

∨

Clearly f1 induces the map H 1 (f ) on the first cohomology groups. This is the end of the 1-step of the inductive process, which is what we tried to prove.

5.1.4

Fourth proposition

Lemma 5.1.18. A direct summand of an injective (projective) object is injective (resp. projective). Lemma 5.1.19. Let I • be a bounded below complex of injectives. Assume that for any n ∈ Z, the nth cohomology group H n (I • ) is injective. Then, for any n ∈ Z, im∂nP is injective. Proof. Assume w.l.o.g. that I n = 0 for n < 0. We first notice that Z 0 (I • ) = H 0 (I • ) and hence Z 0 (I • ) is injective. We have the short exact sequence 0 → Z 0 (I • ) → I0 → B 1 (I • ) → 0. Since Z 0 (I • ) is injective, the sequence splits. Thus, B 1 (I • ) is a direct summand of I0 which is injective, and by the above lemma, we deduce it is also injective. We have the short exact sequence 0 → B 1 (I • ) → Z 1 (I • ) → H 1 (I • ) → 0. Since B 1 (I • ) is injective the sequence splits. Since H 1 (I • ) is also injective, we deduce that Z 1 (I • ) is injective. We proceed via induction.

148

Proposition 5.1.20. Let P • , Q• be two bounded above complexes of projectives, and let fn : H n (P • ) → H n (Q• ) be a morphism between their n’th cohomology groups. Further assume that H n (P ), H n (Q) are projectives for all n. Then, there exists a morphism of complexes f : P • → Q• such that fn induces the map H n (f ) on the nth cohomology groups for n ≥ 0. Proof. This is a corollary of Proposition 5.1.11 together with Lemma 5.1.19.

5.1.5

Fifth proposition

Lemma 5.1.21. Let f : A• → B • be a morphism of complexes, and let ι : A• → I • , κ : B • → J • be injective resolutions of A• , B • , respectively. Then there exists a unique morphism u : I • → J • such that u ◦ ι = κ ◦ f . Proof. Let g := κ ◦ f . By the injective analog of [44] theorem 6.2o we know that ι : Hom[I • , J • ] → Hom[A• , J • ] is an isomorphism. Thus there exists a unique morphism u ∈ Hom[I • , J • ] such that ι ◦ u = g completing the proof of the lemma.

5.1.6

Sixth proposition

Proposition 5.1.22. Let C • be a complex of π-acyclic objects. Let I • be an injective resolution of C • . Then π∗ C • is quasi-isomorphic to π∗ I • . In other words π∗ C • = Rπ∗ C • in the derived category. Proof. This is a simple corollary of the Acyclicity theorem 7.5 in [44].

149

5.2

The spectral sequence of a mapping cone

In this section we devise a new method in homological algebra. This is the central ingredient in the proof of the Leray spectral sequence Theorem 5.9.1.

5.2.1

The spectral sequence of a good filtered complex

We recall that for any good filtered complex (K, F ) (cf. 5.1.7 there is a spectral sequence E0p,n−p ⇒ H n (K), such that the filtration on H n (K) induced by this spectral sequence coincides with that induced by F . This spectral sequence is defined as follows; E0p,q (K, F ) = GrFp K p+q . d0 : E0p,q (K, F ) → E0p,q+1 (K, F ) is induced by d. We define inductively a bigraded complex Erp,q (K, F ), together with a morphism dr : Erp,q (K, F ) → Erp+r,q−r+1 (K, F ). induction step: r → r + 1. Define p,q p−r,q+r−1 (K, F ) := ker(dp,q ), and dr+1 is induced by d. Er+1 r )/image(dr

We recall (cf. [28] section I.3.I) that we have Erp,q = Im{Zrp,q → K p+q /Brp,q }, where Zrp,q := Ker{d : F p (K p+q ) → K p+q+1 /F p+r (K p+q+1 )}, K p+q /Brp,q := coker{d : F p−r+1 (K p+q−1 ) → K p+q /F p+1 (K p+q )}.

(5.2.1)

150

Then the spectral sequence converges, which is written in standard notation E0p,n−p (A, F ) ⇒ H n (A). We let Er (K, F ) abbreviate the complex defined by Erp,q (K, F ). Namely Er (K, F )n := L

p+q=n

Erp,q .

Definition 5.2.2 ([67] definition 3.1). A morphism f : (K, F ) → (L, G) of good filtered complexes is said to be a level-r filtered quasi isomorphism if the morphism Er−1 (f ) induces quasi-isomorphisms Er−1 (K, F ) → Er−1 (L, G), i.e., Erp,q (f ) are isomorphisms for all integers p and q. Lemma 5.2.3. Let f : (K, F ) → (L, G) be a level-r filtered quasi isomorphism. Then f : K → L is a quasi isomorphism. Proof. We have the spectral sequences: E0p,n−p (K, F ) = GrFp K n ⇒ H n (K), p n E0p,n−p (L, G) = GrG L ⇒ H n (L).

Since Erp,q (K, F ) ∼ = Erp,q (L, G), it implies that H n (K) → H n (G) is an isomorphism, finishing the proof. Following [28] proof of 1.3.4, and [67] 2.3-4 we let Definition 5.2.4. Let (K • , F • ) be a good filtered object. Then Bac(F )p K n := F p−n K n . Definition 5.2.5. Let (K • , F • ) be a good filtered object. Then Dec∗ (F )p K n := Im(F p+n−1 K n−1 → K n ) + F p+n K n .

151

We have p−ln,(l+1)n−p

Erp,q (K, Bacl (F )) = Er−l

(K, F ),

(5.2.6)

for all integers r. This is [67] equation (1). For a detailed discussion of Bac, Dec∗ we refer to [67] 2.3-4. Definition 5.2.7 ([67] definition 3.2). A level-r filtered injective resolution of a good filtered complex (K, F ) is a level-r filtered quasi-isomorphism (K, F ) → (L, G), such that the terms Erp,q (L, G) are injective for all r0 < r and all integers p and q. Lemma 5.2.8 ([67] Lemma 3.5). If C has enough injectives then any good filtered complex (K, F ) has a level-r injective resolution for any r ≥ 1. We restate this lemma for later use. Lemma 5.2.9. Assume C has enough injectives. Let (K, F ) be a good filtered complex, and r ∈ N+ . Let (L, G) be a level-1 injective resolution of (K, (Dec∗ )r−1 (F )), which is well known to exist. Then (L, Bacr−1 (G)) is a level-r injective resolution of (K, F ). For the following we assume that C has enough injectives. Lemma 5.2.10. Let (L, G) be a level-r filtered injective resolution of a good filtered complex (K, F ). Then (L, G) is an injective resolution of (K, F ) (in the usual sense). Proof. This follows by Lemma 5.2.3 together with corollary 5.1.10. We need the following proposition, whose flavor is similar to [67] Lemma 3.6. Proposition 5.2.11. Let (Li , Gi ) be a level-r filtered injective resolution of the good filtered complexes (Ki , Fi ), respectively. Let f : (K1 , F1 ) → (K2 , F2 ) be a morphism of good filtered complexes. Then there is a morphism g : (L1 , G1 ) → (L2 , G2 ) of good filtered complexes such that g is quasi-isomorphic to f , that is H n (f ) = H n (g) under the identification of H n (Ki ) with H n (Fi ), i = 1, 2 of Lemma 5.2.3.

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p,q p,q Proof. We have a morphism fr−1 : Er−1 (K1 , F1 ) → Er−1 (K2 , F2 ). We know that p,q p,q Er−1 (Li , Gi ) is an injective resolution of Er−1 (Ki , Fi ), i = 1, 2. By Proposition 5.1.21 p,q p,q we deduce that there exists a morphism gr−1 : Er−1 (L1 , G1 ) → Er−1 (L2 , G2 ) inducing

the same morphism on the level of Er . Inductively applying Proposition 5.1.20 we obtain morphisms fi : Eip,q (L1 , G1 ) → Eip,q (L2 , G2 ) for all i < r such that H i (fi ) = p p fi+1 . Applying Proposition 5.1.8 to g0 : GrG Lq → GrG Lq we obtain a morphism 1 1 2 2

g0 : Gp1 Lq1 → Gp2 Lq2 whose graded morphism is g0 . By construction, we know that on the r’th sheet, Er , this morphism will induce the same morphism as f . Since we have the spectral sequences of good filtered complexes, which are identical from the r’th sheet, we get that the morphism on cohomologies will be the same. We restate [67] Lemma 3.7 as a theorem. This is the main result of [67] and is a critical ingredient in our proof of the Leray spectral sequence Theorem 5.9.1. Theorem 5.2.12. Let (L, G) be a level-r filtered injective resolution of the good filtered complex (K, F ). Let Er (K, F ) (resp. Er (L, G) denote the complex deduced by the spectral sequence associated to (K, F ) (resp. (L, G)). Let D be a left exact functor. Then there is a spectral sequence Erp,q = Rp+q DEr−1 (K, F ) ⇒ Rp+q D(K). Moreover, we have H p+q DEr0 −1 (L, G) = DEr0 (L, G) for 0 < r0 < r. Moreover, Rp+q DEr−1 (K, F ) = H p+q DEr−1 (L, G) = DEr (L, G), and the spectral sequence Erp,q = Rp+q DEr−1 (K, F ) ⇒ Rp+q D(K), is the spectral sequence associated to the good filtered complex (DL, DG).

153

5.3

The l’th mapping cone spectral sequence corresponding to a mapping cone of two filtered complexes

The result of this section is a new result in homological algebra. To be more specific, it introduces a new spectral sequence for the mapping cone of two complexes. As stated before, it turns out to be the central ingredient in the proof of the Leray spectral sequence Theorem 5.9.1. Theorem 5.3.1. Let (A• , F ), (B • , F ) be good filtered complexes, together with a filtered morphism ϕ : A• → B • . Let D• := M C(ϕ) be their mapping cone. That is, Dn = An ⊕ B n−1 and dD (a, b) = (−da, db − ϕ(a)). Given l ∈ N+ , there exists a filtration F on D• such that the spectral sequence associated to the good filtered complex (D• , F ), Erp,q (D• , F ) ⇒ H p+q (D), satisfy p,q p,q Elp,q (D• , F ) ∼ = M C(ϕ : El (A, F ) → El (B, F )).

Proof. We define the l’th twisted filtration F on D by F p D• := M C(ϕ : F p (A) → F p−l (B)). Then: Grp D• = M C(ϕp : Grp (A) → Grp−l (B)), and E0p,∗ (D, F ) = M C(ϕp : Grp (Ap+∗ ) → Grp−l (B p+∗ )) (viewed as a complex on the second term, denoted by ∗). The important point is that ϕp : Grp (A) → Grp−l (B)) is 0. Hence, E0p,∗ (D, F ) ∼ = p,q p−l,q+l−1 Grp (Ap+∗ )⊕Grp−l (B p−1+∗ ). We deduce that E1∗,q (D, F ) ∼ (B, F ) = E1 (A, F )⊕E1

with the differential induced by the mapping cone, i.e., E1∗,q (D, F ) ∼ = M C(ϕ : E1∗,q (A, F ) → E1∗−l+1,q+l−1 (B, F )).

154

We recall that E1p,q (A, F ) = H p+q (Grp Ap+q ). This implies again that ϕ : E1∗,q (A, F ) → E1∗−l+1,q+l−1 (B, F ) is 0. Thus the same argument shows that p,q p−l+2,q+l−2 E2p,q (D, F ) ∼ (B, F )). = M C(ϕ : E2 (A, F ) → E2

To continue inductively, we need the following: For a good filtered complex (K • , F, d), by Equation 5.2.1 we have Erp,q = Im{Zrp,q → K p+q /Brp,q }, where Zrp,q := Ker{d : F p (K p+q ) → K p+q+1 /F p+r (K p+q+1 )}, K p+q /Brp,q := coker{d : F p−r+1 (K p+q−1 ) → K p+q /F p+1 (K p+q )}. We claim that for s ≥ 1 the identity map on K • induces a well defined map id : Erp,q → Erp−s,q+s , which is identically zero. Let e ∈ Erp,q . Let x ∈ Zrp,q belong to the cohomology class of e. Then x ∈ F p (K p+q ), dx ∈ F p+r K p+q+1 . Since F is a decreasing filtration, we deduce that x ∈ F p−s (K p+q ), dx ∈ F p−s+r K p+q+1 , so x ∈ Zrp−s,q+s . Decompose the morphism Zrp−s,q+s → K p+q /Brp−s,q+s as Zrp−s,q+s ,→ K p+q → K p+q /F p−s+1 (K p+q )} → K p+q /Brp−s,q+s . Since x ∈ Zrp,q then x ∈ F P (K p+q ). Since s ≥ 1 and F is decreasing, it implies that x ∈ F p−s+1 . This implies that the image of x under the above morphism is 0. We deduce inductively that for s < l, ϕ : Erp,q (A, F ) → Erp−l+r,q+l−r (B, F ), which is 0 as the decomposition Erp,q (A, F ) → Erp−l+r,q+l−r (A, F ) → Erp−l+r,q+l−r (B, F ), since the first map is the one depicted above and therefore is zero. This implies inductively (the last step where the induction p,q p,q should hold) that Elp,q (D, F ) ∼ = M C(ϕ : El (A, F ) → El (B, F )). We denote this

spectral sequence the l’th mapping cone spectral sequence corresponding to ϕ. The proof is complete.

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5.4

Algebraic differential equations with regular singular points

Following [27], we introduce the notion of regularity. From now on, we assume that K ⊂ C is a number field, so that complex analytic methods are applicable here. We intend to relax this condition in the future.

5.4.1

Definitions

We follow [51] very closely. Let X be a smooth K-scheme. An algebraic differential equation is a pair (M, ∇) with M , a coherent sheaf on X and ∇ integrable connection on X (the existence of ∇ implies that M is also locally free). ∇ is a homomorphism of abelian sheaves ∇ : M → Ω1X ⊗OX M

(5.4.1)

satisfying as usual Leibniz rule. The integrability of ∇ induces a structure of a complex on Ω•X , the “absolute de Rham complex of (M, ∇)”. Let X be a proper and smooth K scheme, D = ∪Di a union of connected smooth divisors in X with normal ∼

crossings, such that X → X − D, which we will refer to as a compactification of X. Let DerD (X/K) denote the (locally free) sheaf on X of derivations that preserve the ideal sheaf of each branch Di of D. The sheaf of differentials on X with logarithmic singularities along D is defined by Ω1X (log D) := HomOX (DerD (X/K), OX ), (5.4.2) ΩpX (log D) =

Vp

OX

Ω1X (log D).

Clearly Ω•X (log D) is a subcomplex of j∗ Ω•X where j : X → X. Definition 5.4.3. An algebraic differential equation (M, ∇) has regular singular points if, for every compactification X = X − D as above, there exists a pair (M , ∇)

156

consisting of a locally free sheaf M on S which prolongs M (i.e., the restriction of M to X is isomorphic to M ) and a homomorphism ∇ of abelian sheaves 1 (log D) ⊗OX M , ∇ : M → ΩX

(5.4.4)

which prolongs ∇.

5.4.2

Remarks on the definition of regular singular points

1. By [27] proposition 5.7 (which uses the local monodromy around D), it follows that the underlying analytic differential equation (M an , ∇an ) always admits a unique analytic extension (M an , ∇an ) as in definition 5.4.3, which by GAGA, is uniquely algebrifiable since X is proper. Restricting this algebraic data to X, we get a second algebraic differential equation (M 0 , ∇0 ) on X, which depends functorially on (M, ∇), and an isomorphism ∼

(M an , ∇an ) → (M 0an , ∇0an ). The condition that the differential equation (M, ∇) has regular singular points is that the above isomorphism comes from an isomorphism of (M, ∇) and (M 0 , ∇0 ). 2. Using 1 (cf. [27] proposition 4.4), we deduce that (M, ∇) has regular singular points if and only if for every morphism f : V → X with V a smooth curve over K, the inverse image f ∗ (M, ∇) on V has regular singular points. 3. If X is a connected curve, smooth over K, and X = X − D its canonical compactification, then (M, ∇) has regular singular points if there exists an extension (M , ∇) as in definition 5.4.3 with M coherent. 4. Combining 2 and 3, it follows that (M, ∇) has regular singular points if for one compactification X = X − D there exists an extension (M , ∇) as in definition 5.4.3 with M coherent.

157

5. This also implies that for MdR ∈ CdR (Y2 ) (cf. definition 4.0.3) and any morphism of de Rham data f : Y1 → Y2 , one has f ∗ MdR ∈ CdR (Y1 ). (This fact is also proven in [27] proposition II.4.3). 6. If (M, ∇) has regular singular points, then [27] proposition II.4.6(3) implies that for a morphism g : X → X 0 of schemes, smooth over C, the algebraic differential equation (g ∗ M, g ∗ ∇) also has regular singular points.

5.4.3

Relative de Rham cohomology revisited

Let π : X → S be a proper and smooth morphism over K, and (M, ∇) an algebraic differential equation on X. Composing ∇ with the projection Ω1X ⊗OX M → Ω1X/S ⊗OX M we obtain an integrable S-connection, still noted, ∇ : M → Ω1X/S ⊗OX M,

(5.4.5)

which provides a structure of a complex to Ω•X/S ⊗OX M , the “relative de Rham complex of (M, ∇)”. The relative de Rham cohomology sheaves on S of (M, ∇) are defined by q HdR (X/S, (M, ∇)) := Rq π∗ (Ω•X/S ⊗OX M ).

(5.4.6)

These sheaves are coherent, as π is proper, and are endowed with an integrable connection, which is explained in detail in [54]. We recall its construction here. We first filter the absolute de Rham complex of (M, ∇) by the subcomplexes i Kosi = Kosi (Ω•X ⊗OX M ) = image(π ∗ ΩiS ⊗OX Ω•−i X ⊗OX M → ΩX ⊗OX M ). (5.4.7)

The associated graded objects are given by Gri = Kosi /Kosi+1 = π ∗ ΩiS ⊗OX (Ω•−i X/S ⊗OX M ).

(5.4.8)

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q (X/S, (M, ∇)) is the differential d0,q The integrable connection on HdR 1 in the spec-

tral sequence of the filtered complex Ω•X ⊗OX M and the functor R0 π∗ , or, in other words, the coboundary map δq , in the long cohomology sequence Rq arising from the short exact sequence 0 → Gr1 → Kos0 /Kos2 → Gr0 → 0. By 5.4.8 we have q Rq π∗ (Gr0 ) = HdR (X/S, (M, ∇)).

(5.4.9) Rq+1 π∗ (Gr1 ) = Ω1S ⊗OS

q HdR (X/S, (M, ∇)).

Locally freeness of the relative de Rham cohomology groups q We deduce that (HdR (X/S, (M, ∇)), δq ) is an algebraic differential equation on X. In q particular, HdR (X/S, (M, ∇)) is locally free; this being so for all q, it follows that the q formation of the HdR (X/S, (M, ∇)) is compatible with arbitrary change of base.

Remark 5.4.10. In the case (M, ∇) = (OU , d), the connection thus constructed on HdR (X/S) := Rπ∗ (ΩX/S ) is the Gauss-Manin connection, and the resulting algebraic differential equation is called the Picard-Fuchs equation. q Here, (HdR (X/S, (M, ∇)), δq ) is defined when π : X → S is assumed to be proper. q (X/S, (M, ∇)) is not necessarily locally If π : X → S is no longer proper, HdR

free. However, there is a weaker condition that guarantees the locally freeness of q HdR (X/S, (M, ∇)). For this we state

Proposition 5.4.11. Let π : X → S be a morphism between schemes smooth of finite type over C, which admits a compactification ⊂

>

π

π >

˜ X

j1

>

Y π

π ∨

S

i

∨ >

T

.

such that ˜ := π −1 S. 1. X ⊆ X 2. Y is a proper scheme, and the complement D := Y \ X is a simple normal crossing divisor on Y . 3. j1 is an open embedding. 4. i : S → T is an open immersion into a proper scheme. 5. DT := T \ S is a reduced simple normal crossing divisor on T . ˜ \ X is a relative simple normal crossing divisor over S. 6. X 7. We assume S is separated. Then π : Y → T is proper. By base change we deduce ˜ → S is proper. 8. π : X ˜ → (T, DT ) is a morphism of pairs of generalized 9. We assume that π : (Y, Y \ X) semi-stable type. Lemma 5.6.3 implies ˜ → S is smooth. 10. π : X

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In fact, Abramovich and Karu conjecture that these conditions are implied by the conditions of proposition 5.4.11 (cf. [1]). However, as this conjecture is not yet settled, we have to assume them, and we abbreviate it by saying that π admits a good compactification. Let M ∈ Ssyn (X), and assume M = MdR is defined over Y , and ∇ = ∇dR is such that (M , ∇) has regular singular points on Y \ X. That is, M has logarithmic singularities along Y \ X. We denote by j : X → Y the open embedding that is the ˜ Assume further that the eigenvalues of the residue composition of j1 and X ,→ X. matrix associated to (M , ∇) are not positive integers (e.g., (M , ∇) is taken as the canonical extension of an algebraic differential equation (M, ∇) on X with regular singular points). By [27] corollaire 3.14(i), we deduce M ⊗ Ω•Y (log D) ∼ = j∗ M |X ⊗ Ω•X . Since π : X → S is smooth we have a short exact sequence 0 → π ∗ (Ω1S ) → Ω1X → Ω1X/S → 0.

(5.7.1)

We define Ω1Y /T (log D) to be the subsheaf of j∗ Ω1X/S having logarithmic singularities along D. It is the image of Ω1Y (log D) inside j∗ Ω1X/S . Since π is of generalized semi-stable type, we know that π ∗ (Ω1T (log(T \ S))) is a subbundle of Ω1Y (log D). Thus, there is a short exact sequence 0 → π ∗ (Ω1T (log(T \ S))) → Ω1Y (log D) → Ω1Y /T (log D) → 0.

(5.7.2)

Moreover, by definition 5.6.2 this sequence is locally split, and Ω1Y /T (log D) is locally free. We define the locally free OY -modules Ω•Y /T (log D)

:=

p ^

Ω1Y /T (log D).

166

The bundle ΩiY /T (log D) is a subbundle of j∗ ΩiX/S , and a quotient of ΩiY (log D). Thus, it is induced with a differential operator, and Ω•Y /T (log D) becomes a complex of locally free OY -modules. The following theorem is an important generalization of the regularity theorem of the Gauss Manin connection (cf. [51] Theorem V) and the monodromy theorem (cf. loc. cit. Theorem VII). Theorem 5.7.3. If (M, ∇) ∈ QR(X), then the algebraic differential equation q (HdR (X/S, (M, ∇)), δq ) ∈ QR(S). Moreover, if M is induced with a filtration F • q satisfying Griffiths’ transversality, then (HdR (X/S, (M, ∇)), δq ) admits a natural fil-

tration satisfying Griffiths’ transversality. Proof. Definition 5.7.4. For V equals T or specK we let • (M /V ) := M ⊗ Ω•Y /V (log D), DRdR

equipped with the filtration q F m DRdR (M /V ) = F m−q M ⊗ ΩqY /V (log D).

Introducing the Koszul filtration on the filtered de Rham complex Definition 5.7.5. Let • Kosi F m DRdR (M /K) :=

•−i • • image[DRdR (M /K) ⊗OY π ∗ (ΩiT )(log(T \ S)) → DRdR (M /K)] ∩ F m DRdR (M /K).

We now use the convention that, given a complex C • with a filtration F • , we have: F m C •−i = (F m (C • ))−i . Together with the definition q F m DRdR (M /V ) = F m−q M ⊗ ΩqY /V (log D),

167

we deduce that F m−i DR•−i (M /V )

k

= F m−k M ⊗ Ωk−i Y /V (log D).

This implies that • Kosi F m DRdR (M /K) :=

•−i • image[F m−i DRdR (M /K) ⊗OY π ∗ (ΩiT )(log(T \ S)) → DRdR (M /K)].

Combining this with Equation 5.7.2 we deduce that the associated graded objects of this filtration are given by: i • GrKos F m (DRdR (M /K)) = Kosi /Kosi+1 F m =

(5.7.6) •−i π ∗ (ΩiT (log(T \ S))) ⊗OY F m−i DRdR (M /T ).

We define sheaves on T by q • (M /T )), HdR (Y /T, (M , ∇)) := Rq π∗ (DRdR

(5.7.7)

q which are prolongations of the locally free sheaves HdR (X/S, (M, ∇)) on X.

The extensions of δq to homomorphisms of abelian sheaves q δ q : H q (Y /T, (M , ∇)) → Ω1T (log D) ⊗OT HdR (Y /T, (M , ∇))

(5.7.8)

are provided by the coboundary maps of the long cohomology sequence of the Rq π∗ arising from the short exact sequence 1 0 0 → GrKos → Kos0 /Kos2 → GrKos → 0,

where Kos is defined by 5.7.6. By 5.7.6 we have q Rq π∗ (Gr0 ) = HdR (Y /T, (M , ∇)).

(5.7.9) q Rq+1 π∗ (Gr1 ) = Ω1T (log D) ⊗OT HdR (Y /T, (M , ∇)).

168

q (Y /T, (M , ∇)), δ q ) provide the desired extension of the Thus, the (HdR q (HdR (X/S, (M, ∇)), δq ),

if we show it is locally free. Moreover, by defining q • (Y /T, (M , ∇)) := Rq π∗ F m DRdR F m HdR (M /T ) ,

(5.7.10)

q equations 5.7.6 and 5.7.8 imply that (HdR (Y /T, (M , ∇)), δ q ) is a filtered object

satisfying Griffiths’ transversality condition. We call this filtration the log Hodge filtration on the relative log de Rham cohoi mology HdR (Y /T, (M , ∇)). This filtration coincides with the filtration induced by

the (relative) log Hodge to de Rham spectral sequence q i E1p,q = Rp+q π∗ Grp (DRdR (Y /T, (M , ∇)). ((M , ∇)/T )) ⇒ HdR

(5.7.11)

This filtration extends the Hodge filtration on the relative de Rham cohomology i HdR (X/S, M ), which is induced by the (relative) Hodge to de Rham spectral sequence p+q ˜ ⇒ HdR (X/S, M ). E1p,q = Rp+q π∗ Grp (MdR |X˜ ⊗ Ω•X˜ (log(D ∩ X)))

(5.7.12)

Theorem 5.7.13. With the above assumptions, if (M, ∇) ∈ QR(X), q (Y /T, (M , ∇))), δ q ) is locally free, then it is the canonical extension of and (F p (HdR q (F p (HdR (X/S, (M, ∇))), δq ).

Proof. It suffices to check the assertion after pullbacks to curves that are suitably transverse to the divisors at infinity. This reduces to the problem to the case dim S = 1. Let S be a smooth connected curve, T its canonical compactification, π : X → S a smooth morphism, and π : Y → T be a good compactification of π : X → S.

169

˜ := π −1 S. In this case, having a good compactification is equivalent to having Let X DT := T \ S a finite set of points, a Cartesian diagram ˜ X

j1

>

Y

π ∨

S

π i

∨ >

T

.

˜ → S is smooth, X ˜ \ X is a union of connected smooth RELATIVE such that π : X ˜ and Y \ X ˜ is a union of connected smooth normal normal crossing divisors in X, ˜ = π −1 DT . Let D := Y \ X. crossing divisors Ni on Y , such that Y \ X We use an analog of the monodromy theorem VII in [51] for the open case. The proof applies verbatim, so it is omitted. Theorem 5.7.14. Let π : S → T be a smooth morphism of schemes over K, where T is a curve, admitting the above good compactification. Let (M, ∇) be an algebraic differential equation on U with regular singular points, and let (M , ∇) be an extension as in 5.4.3. Denote by Pi (x) the indicial polynomial of (M , ∇) around Ni . Let Pr y ∈ T \ S, and π −1 y = i=1 ai Ni its fiber. Then the indicial polynomial at y of q (HdR (Y /T, (M , ∇)), δq ) divides a power of r aY i −1 Y

Pi (ai x − ji ).

i=1 ji =0

˜ That is, what First, we explain why we only investigate divisors Ni of Y \ X. ˜ \ X? about the divisors of X Although ∇ may have logarithmic singularities there, when taking the higher direct images Rq π∗ these singularities vanish. Indeed, by [27] proposition 6.14 we have q ˜ \ X)) ∼ Rq π∗ M |X˜ ⊗ ΩqX/S (log( X = HdR (X/S, (M, ∇)). ˜

(5.7.15)

˜ denote the connection corresponding to the coboundary map Moreover, let δ q (X) ˜ instead arising from the short exact sequence 5.7.6, or more precisely its analog for X

170

˜ coincides with δq under the isomorphism 5.7.15. Thus, as δq has of Y . Then, δ q (X) no singularities along S, we deduce that the same applies for ˜ Rq π∗ M |X˜ ⊗ ΩqX/S ˜ (log(X \ X)) . Since q q ˜ ˜ (Y /T, (M , ∇)), δ q )|S ∼ (HdR = Rq π∗ M |X˜ ⊗ ΩX/S ˜ (log(X \ X)) , δ q (X) , ˜ \ X is trivial. we deduce that the indicial polynomial of any divisor of X

Let (M , ∇) be the canonical extension of (M, ∇). By definition 5.5.6, we may write Pi (x) =

ni Y

(x − αi,ki )

ki =1

for αi,ki ∈ Q with 0 ≤ αi,ki < 1. Then r aY i −1 Y i=1 ji =0

Pi (ai x − ji ) =

ni r aY i −1 Y Y

(ai x − ji − αi,ki ).

i=1 ji =0 ki =1

Thus, roots x of this polynomial are of the form x =

αi,ki +ji ai

for some i, ji , ki as above.

Since 0 ≤ ji ≤ ai − 1 and 0 ≤ αi,ki < 1, we deduce that such root x satisfy 0 ≤ x < 1. Given the rationality of αi,ki , the rationality of

αi,ki +ji ai

follows, and therefore the proof

is complete. Theorem 5.7.16. With the above notations, if (M, ∇) ∈ QR(X), then the sheaf q HdR (X/Y, (M , ∇)) is locally free for all q. Moreover, if π is flat, then the sheaves q F p (HdR (X/Y, (M , ∇))) are locally free for all p, q.

5.7.2

Logarithmic geometry

For the proof of the theorem, we use techniques from logarithmic geometry. We give here a review of the theory needed for the proof of the theorem. Throughout we work

171

within the structure of log geometry in the sense of J.-M. Fontaine, L. Illusie, and K. Kato. Our main reference of log geometry is [49], which is nicely compiled in [47]. Following loc. cit., our treatment is complex analytic in nature. A log analytic space is a complex analytic space with a logarithmic structure on it (cf. [49] definitions 1.1–1.2). We keep the notation of [47]. 1. We often abbreviate the log analytic space (X, M ) to X. The underlying ana◦

lytic space is denoted by X. The same convention is used for morphisms of log analytic spaces. 2. For a log analytic space X we denote ◦

• OX is the structure sheaf of X, • MX is the log structure of X, and × • CX is the characteristic of X, namely CX := MX /OX .

The homomorphism MX → OX defining the log structure M of X is denoted by αX . 3. For a morphism f : X → Y of log analytic spaces (cf. [49] definition 1.1), we define CX/Y := Coker(f −1 CY → CX ) and call it the relative characteristic of f. We notice that the characteristic CX has a torsion-free stalk at every point. Moreover, it is easily seen that every stalk of CX has no invertible elements other than the neutral element. (This is because the invertible elements of a log structure are identified with × OX via αX .) We repeat the examples and definitions as they appear in [47] for the

sake of completion.

172

Example. Canonical log structure Let P be a monoid and consider the complex analytic space (Spec C[P ])an . The monoid homomorphism P → C[P ] induces the isomorphism P(Spec C[P ]) an → O(Spec C[P ]) an of sheaves of monoids, that is, a prelog structure on (Spec C[P ])an .

Then the

associated log structure (cf. [49] 1.1) is called the the canonical log structure on (SpecC[P ])an . We always regard (Spec C[P ])an as a log analytic space endowed with the canonical log structure, unless otherwise specified. Note that, in particular, the analytic space (Spec C[P ])an is equipped with the trivial log structure. Definition 5.7.17 (Strict morphisms).

1. Let f : X → Y be a morphism of an-

alytic spaces and let N be a log structure on Y . Then the pull-back of N to X is the log structure, denoted by f ∗ N , which is the associated log structure of the prelog structure f −1 N → f −1 OY → OX . 2. A morphism f : X → Y of log analytic spaces is called strict if f ∗ MY → MX is an isomorphism of log structures on X. Definition 5.7.18 (Chart).

1. Let X be a log analytic space and P a monoid.

Then a chart of X modeled on P is a strict morphism X →(Spec C[P ])an of log analytic spaces. 2. Let x ∈ X be a point. Then the chart X →(Spec C[P ])an is said to be good at x if the induced homomorphism P → CX,x is an isomorphism of monoids. 3. A chart (resp., good chart) of a morphism f : X → Y of log analytic spaces (resp., at x ∈ X) modeled on a homomorphism h : Q → P of monoids is a commutative diagram of log analytic spaces X

>

(Spec C[P ])an h∗

f ∨

∨

Y

>

(Spec C[P ])an

173

where the horizontal arrows X →(Spec C[P ])an and Y →(Spec C[Q])an give charts (resp., good charts at x ∈ X and f (x) ∈ Y ) and the right vertical arrow is induced by h. ◦

Any open subset of X has the pullback log structure so that the inclusion U ,→ X is strict. We have therefore the obvious concept of local charts. The definition of local charts of morphisms is similar. In what follows we discuss the category of fs log analytic spaces defined as follows. Definition 5.7.19 (fine and fs log analytic space).

1. A monoid P is said to be

fine if it is finitely generated and the natural homomorphism P → P gp is injective. 2. A monoid is said to be fs if it is fine and for every x ∈ P gp , if xr ∈ P for some r > 0 then x ∈ P . 3. A log analytic space X is called fine (resp., fs) if locally it admits charts modeled on a fine (resp., an fs) monoid. Remark 5.7.20. For a fine (resp., fs) log analytic space X and x ∈ X, the characteristic CX,x at x is a torsion-free fine (resp., fs) monoid. Remark 5.7.21. It is known that an fs log analytic space has a good local chart at any point (cf. loc. cit. 2.1.6). Example 2.1.8 defines the canonical structure that corresponds to a pair (X, D) where D is a reduced normal crossing divisor on X. The resulting log structure is proved to be an fs log analytic structure. Definition 5.7.22 (Log smoothness). Let f : X → Y be a morphism of fine log analytic spaces. Then f is said to be log smooth if the following condition is satisfied.

174

The infinitesimal lifting property: for any commutative diagram T

0

s0

>

X f

t ∨

∨

T

>

s

Y

of fine log analytic space with t a thickening of order ≤ 1, there exists locally on t a morphism g : T → X such that s0 = g ◦ t and s = f ◦ g. Log smoothness is stable under base change in the category of fine log analytic spaces. Kato also gives a local criterion for smoothness using charts (cf. loc. cit. Theorem 2.2.4). Next is the notion of “Real blow-up” of log analytic spaces. This is an idea presented by Kato and Nakayama [50] and uses the fact that we work over C. We will not reproduce the construction of X log here. The assignment X → X log ◦

is functorial in X. There is a morphism τX : X log → X of ringed spaces that is continuous and topologically proper (that is, the inverse image of a compact subspace is compact). Following [49] example 1.5.1, for a pair (X, D) where X is a complex manfiold (or a regular scheme), and D is a reduced divisor with normal crossings, then L(D) = {g ∈ OX : g is invertible outside D} ⊂ OX . Algebraically, this is the reason for working with the étale topology and not with the Zariski topology. We let 1 ωX (D) := (Ω1X ⊗ (OX ⊗Z L(D)gp ))/N,

where the notation gp is for the group of fractions of a monoid, and N is the OX submodule of the direct sum generated by local sections of the form (dα(f ), 0) − (0, α(f ) ⊗ f ), f ∈ L(D).

175

1 For a local section of L(D), the class of (0, 1 ⊗ f ) ∈ ωX (D) is denoted by d log(f ) we

denote by 1 1 (D)/f ∗ ωY1 (T ). (D/T ) := ωX ωX/Y • From now on, we replace the complex ωX/Y with • • ωX/Y,(M (D/T ) := ωX/Y (D/T ) ⊗ M , ,∇)

and set 1 M Z := ker(∇ : M → ωX/Y (D/T ) ⊗ M ),

M Y := M Z ⊗Z f −1 OY . This definition implies • • ωX/Y,(M (D/T ) = DRdR (M /Y ). ,∇)

(5.7.23)

We have Proposition 5.7.24 (cf. loc. cit. proposition 3.2.3). Let f : X → Y be a log smooth morphism of f s log analytic spaces with Y log smooth over (Spec C)an such that the induced morphism CY → CX is injective and CX/Y has torsion-free stalks. Assume further that f is exact. Then there exists a canonical isomorphism gp • • Hq (ωX/Y,(M (D/T )) = H0 (ωX/Y,(M (D/T )) ⊗Z CX/Y . ,∇) ,∇)

Proposition 5.7.25 (cf. loc. cit. proposition 3.2.5). Let f : X → Y be as in Proposition 5.7.24. We have • (D/T )) = M Y . H0 (ωX/Y,(M ,∇)

Notation: let f : X → Y be a morphism of f s log analytic spaces and τX : X log → X be the real blow-up. We define the topological space XYlog by XYlog := X ×Y Y log .

176

We fix the same notation as in loc. cit. notation 3.3.3, namely X log

τX/Y

>

fYlog

f log Y

∨ log

XYlog

=

>

Y

τXY

>

X f

∨ log

(5.7.26)

∨ >

τY

Y,

with the square in the right-hand side cartesian, so that τX = τX/Y ◦ τXY . We have Proposition 5.7.27 (proposition 3.3.5 [47]). Let F be a sheaf of abelian groups on XYlog . Then there exists a canonical isomorphism R

q

−1 τX/Y ∗ τX/Y F

∼ = F (−q) ⊗Z

q ^

gp τX−1Y CX/Y

of abelian sheaves on XYlog for all q. To state the relative log Ponicaré lemma we need the following notation Definition 5.7.28. Let log M log := OX ⊗τ −1 OX τX−1 M , X

q,log log q ωX/Y,(M (D/T ) := OX (D/T ) ⊗τ −1 OX τX−1 ωX/Y,(M ,∇) ,∇) X

for all q. There exists a natural morphism 1,log ∇log : M log → ωX/Y,(M (D/T ) ,∇) 1 ⊗ M. induced by ∇ : M → ωX/Y q+1,log q,log (D/T ) → ωX/Y,(M (D/T ) for all q, such that ∇log extends easily to ∇log : ωX/Y,(M ,∇) ,∇) •,log (ωX/Y,(M (D/T ), ∇log ) is a complex of (f log )−1 OYlog -modules. ,∇)

177

Theorem 5.7.29 (cf. [47] Theorem 3.4.2). Let f : X → Y be a log smooth morphism of f s log analytic spaces with Y log smooth over (spec C)an . Assume that the induced morphism of characterstics f −1 CY → CX is injective and, for every x ∈ X, the relative characteristic CX/Y,x is torsion-free. Then the natural morphism •,log • (D/T )) → ωX/Y,(M (f log )−1 OYlog ⊗τ −1 f −1 OY τX−1 H0 (ωX/Y,(M (D/T ) ,∇) ,∇) X

is an isomorphism in the derived category D+ (X log , (f log )−1 OYlog ). We now arrive at the integral structure of the variation of mixed Hodge structures in question. For this we first need an auxiliary lemma, which is a special case of [46] II.2.6.6. Lemma 5.7.30. Let f : X → Y be a continuous map of topological spaces, and let R be a sheaf of rings on Y . Let F be an f −1 R-module and G an R-module. Then, if f is proper and G is flat over R, the natural morphism Rf∗ F ⊗R G → Rf∗ (F ⊗f −1 R f ∗ G ) is an isomorphism in D+ (Y, R). We now have Lemma 5.7.31 (cf. loc. cit. lemma 4.1.1). Let f : X → Y be as in Theorem 5.7.29. Assume further that f is exact (cf. loc. cit. definition 3.2.4). Then there exists a canonical isomorphism ∼

• (D/T )⊗(f log )−1 (τY )−1 OY (fYlog )−1 OYlog → RτX/Y ∗ ((f log )−1 OYlog )⊗τ −1 f −1 OY τX−1Y M Y τX−1Y ωX/Y,(M ,∇) Y

in the derived category D+ (XYlog , (fYlog )−1 OYlog ).

XY

178

Analog to loc. cit. proof of lemma 4.1.1. Since the inverse image functor on sheaves is exact, it commutes with the cohomological functor H. Thus, due to Proposition 5.7.27 and 5.7.24, together with the commutativity of 5.7.26 and the projection formula (Lemma 5.7.30), there exists a canonical isomorphism • (D/T )) ∼ Hq (τX−1Y ωX/Y,(M = Rq τX/Y ∗ (f log )−1 (τY )−1 OY ⊗τ −1 f −1 OY (τX−1Y M Y ) (5.7.32) ,∇) XY

for all q. We first notice that OYlog is a τY−1 OY module. The commutativity of 5.7.26 then implies that (f log )−1 OYlog is a τX−1 f −1 OY module. We shall construct a morphism • (D/T )⊗(f log )−1 (τY )−1 OY (fYlog )−1 OYlog → RτX/Y ∗ ((f log )−1 OYlog )⊗τ −1 f −1 OY τX−1Y M Y τX−1Y ωX/Y,(M ,∇) XY

Y

of complexes that yields the isomorphisms 5.7.32 tensored by (fYlog )−1 OYlog . By Theorem 5.7.29 and Proposition 5.7.25, there exists a morphism of complexes •,log τX/Y ∗ ωX/Y,(M (D/T ) → RτX/Y ∗ ((f log )−1 OYlog ⊗τ −1 f −1 OY τX−1 M Y ). ,∇) X

(5.7.33)

On the other hand, we have a natural morphism •,log • (D/T ). (5.7.34) (D/T )⊗(f log )−1 (τY )−1 OY (fYlog )−1 OYlog → τX/Y ∗ ωX/Y τX−1Y ωX/Y,(M ,∇) (M ,∇) Y

By the projection formula (Lemma 5.7.30) we have a canonical isomorphism log RτX/Y ∗ ((f log )−1 OYlog ⊗τ −1 f −1 OY τX−1 M Y ) ∼ = RτX/Y ∗ ((f log )−1 OY ) ⊗τ −1 f −1 OY τX−1Y M Y . X

XY

(5.7.35) By combination of the morphisms 5.7.33, 5.7.34, and 5.7.35 we deduce the desired morphism of complexes, which is clearly a quasi-isomorphism by 5.7.32. We are now ready to state the main theorem, which is an analog of [47] theorem 4.1.5.

179

Theorem 5.7.36. Let f : X → Y be as in Theorem 5.7.29. Assume further that f ◦

is exact and that the underlying morphism f is proper. Then there exists a canonical isomorphism log • (D/T ) ⊗τ −1 OY OYlog ∼ τY−1 Rf∗ ωX/Y,(M = Rf∗log (τ −1 M Y ) ⊗Z OY ,∇) Y

in D+ (Y log , OYlog ). Proof. The proof of [47] theorem 4.1.5 applies verbatim here, substituting the new lemmata for the old ones.

5.7.3

The logarithmic Riemann Hilbert correspondence (after [50] 0.4)

Let X be an f s log analytic space over C. Assume that there exists an open covering P (Uλ )λ of X, f s monoids Pλ , and an ideal λ of Pλ for each λ, such that Uλ is P isomorphic to an open analytic subspace of Spec (C[Pλ ]/( λ ))an endowed with the log structure associated to Pλ → OUλ . Define the categories Dnilp (X) and Lunip (X log ) ◦

as follows. Let D(X) be the category of vector bundles V on X endowed with an integrable connection with log poles 1 ∇ : V → ωX/C ⊗OX V,

and let L(X log ) be the category of local systems of finite C-vector spaces on X log . Let Dnilp (X) (resp. Lunip (X log )) be the full subcategory of D(X) (resp. L(X log )) consisting of objects V (resp. L) satisfying the following condition locally on X: there exists a finite family of OX -subsheaves (Vt )0≤t≤n of V satisfying 1 ∇(Vt ) ⊂ ωX/C ⊗OX Vt

(resp. finite family of C−subsheaves (Lt )0≤t≤n of L) such that 0 = V0 ⊂ Vt ⊂ · · · Vn = V (resp. 0 = L0 ⊂ L1 ⊂ · · · ⊂ Ln = L),

180

and such that for each 1 ≤ i ≤ n, Vt /Vt−1 is a vector bundle and the connection induced on Vt /Vt−1 does not have a pole (that is the image of ∇ is contained in the 1 ⊗OX V ) (resp. Lt /Lt−1 is isomorphic to the inverse image of Ω1X/C ⊗OX V → ωX/C

image of a local system of finite dimensional C-vector spaces on X). Theorem 5.7.37 ([50] theorem 0.5, and see also [78] 1.4.3). Let the notation be as in (0.4). Then there exists an equivalence of categories ∼

Φ : Dnilp (X) → Lunip (X log ) such that: 1. If V is an object of Dnilp (X) whose connection does not have a pole, Φ(V ) is 1 the inverse image of Ker(∇ : V → ωX/C ⊗OX V ) under τX , or equivalently, 1,log ⊗Olog τX∗ V ). Ker(τX∗ ∇ : τX∗ V → ωX/C X

log 2. The inverse of Φ is given by L → τX∗ (OX ⊗C L).

Being an equivalence of categories, Φ is also exact. We refer the reader to remark 5.7.41 for a generalization of the logarithmic Riemann Hilbert correspondence.

5.7.4

Proof of Theorem 5.7.16

Proof. By Theorem 5.7.36, we deduce that Lq := τT−1 Rq π∗ ωY• /T,(M ,∇) (D/DT ) ⊗τ −1 OT OTlog T

are locally free over T log for all q. By the log Riemann-Hilbert correspondance (Theorem 5.7.37) we deduce that V q := τT ∗ (τT−1 Rq π∗ ωY• /T,(M ,∇) (D/DT ) ⊗τ −1 OT OTlog ) are T

locally free over T for all q, provided we show L admits a filtration Lqt , such that q

Lqt /Lqt−1 is isomorphic to the inverse image of a finite dimensional vector bundle on Y

181

whose connection has no poles. There is a weight filtration on Rq π∗ ωY• /T,(M ,∇) (D/DT ) ∼ = q HdR (Y /T, (M , ∇)), defined similarly to [75] subsection 5.7. Its construction is omit-

ted here. This weight filtration naturally induces a weight filtration on Lq , whose graded objects have the desired property. Therefore, Theorem 5.7.37 implies that V q is locally free. The projection formula (Lemma 5.7.30) implies that these sheaves are q isomorphic to HdR (Y /T, (M , ∇)). Thus, these sheaves are locally free.

We now turn to prove that the filtered Hodge pieces are also locally free. For this we show that the graded Hodge pieces are locally free. The proof is similar to the treatment in [84] section 2. Following similar argument to that of [75] Theorem 5.6 including subsection 5.7, and also [32] Theorem 4.8, loc. cit. corollary 5.15, we deduce that the spectral sequence E1p,q = H q (Zt , GrFp (ωY /T,(M ,∇) (D/DT )⊗OY OZt )) ⇒ H p+q (Zt , ωY• /T,(M ,∇) (D/DT )⊗OY OZt ) (5.7.38) degenerates at E1 , where Zt is the fiber of π : Y → T over t ∈ DT = T \ S. On the other hand, π : X = Y \ D → S = T \ DT is smooth and therefore 5.7.38 holds also for fibers Zt where t ∈ S (cf. [30] Theorem D.2). Using the identification 5.7.23, we show that the dimension function • t −→ dimk(t) H q (Zt , GrFp DRdR (M /T ) t ) is a constant function of t. For t ∈ S, the dimension function is constant. We denote this constant by rp,q . By [66] corollary (a) on page 50 the dimension function is upper semi-continuous. This implies that for any t ∈ T we have • dimk(t) H q (Zt , GrFp DRdR (M /T ) t ) ≥ rp,q .

(5.7.39)

182

On the other hand, the degeneration of 5.7.38 at E1 implies that dimk(t) H n (Zt , ωY• /T,(M ,∇) (D/DT ) ⊗OY OZt ) = (5.7.40) P

p+q=n

dimk(t) H q (Zt , GrFp (ωY /T,(M ,∇) (D/DT ) ⊗OY OZt ))

for all t ∈ T . q We already proved that HdR (Y /T, (M , ∇)) is locally free for all q. This implies that

that dimk(t) H n (Zt , ωY• /T,(M ,∇) (D/DT ) ⊗OY OZt ) is a constant function of t ∈ T , and we denote this constant by Rn . Equation 5.7.40 implies that Rn =

X

rp,q .

p+q=n

Together with 5.7.39 it implies that • dimk(t) H q (Zt , GrFp DRdR (M /T ) t ) = rp for all t ∈ T . We deduce that the dimension function is constant on T . q By [66] corollary (ii) the locally freeness of GrFp (HdR (Y /T, (M , ∇))) for all p, q folq lows, if π is assumed to be flat. Standard induction implies that F p (HdR (Y /T, (M , ∇)))

is also locally free for all p, q assuming the flatness of π. This completes the proof Theorem 5.7.16. Remark 5.7.41. Only after this proof was written, was it brought to our attention through an email correspondence with Professor Fujisawa that Theorem 5.7.16 is proved in [48] Theorem 1. Studying this result we also learned of a generalization of the log Riemann Hilbert correspondance (cf. [42] and also [43]), using which our proof of Theorem 5.7.16 is simplified, as one need not define a weight filtration on Lq satisfying certain properties.

183

Corollary 5.7.42. With the above notations, let (M, ∇) ∈ QR(X). Then q (X/S, (M, ∇)), δq ) ∈ QR(S). (HdR q (X/S, (M, ∇)), δq ) is a filtered object satisfying GrifIf, moreover, f is flat, then (HdR

fiths’ transversality condition. Proof. The proof follows by Theorems 5.7.3 and 5.7.13. We recall [55] proposition 1. Proposition 5.7.43. Let f : X → Y be a morphism of non-singular proper algebraic varieties and let C and D be divisors of normal crossings on X and Y , respectively, such that f (X0 ) ⊂ Y0 , where X0 = X \ C and Y0 = Y \ D. Put f0 = f |X0 . Let H0 be a variation of Hodge structures on Y0 . We assume that all the local monodromies of H0 around D are unipotent. Let H and H 0 be the canonical extensions of sim

H0 and f0∗ H0 on Y and X, respectively. Then there is an isomorphism f ∗ H → H 0 that is compatible with the Hodge filtration. Proof. Let (U ; z1 , . . . , zd ) and (V ; w1 , . . . , wd0 ) be local coordinate neighborhoods on Q Q X and Y as above, respectively. If wk = ziaik , then γzi = γwaikk , where the γzi and the γwk are local monodromies corresponding to the zi and the wk . Since all the local monodromies are unipotent we have log Thus

P

log γzi log zi =

extensions coincide.

P

Y

γwaikk =

X

aik log γwk .

log γwk log wk . By the proof of [27] proposition 5.2, the

184

5.8

Relative geometric cohomology and π−admissible coefficients

Let πrig : Xrig → Srig be a rigid datum for Mrig in the sense of O-triples. (We assume π is quasi-projective, so it is always possible.) Let K(0)rig = (Speck, Speck, OK ). From now on, X, S are schemes over OK . When working in the de Rham case, X, S refer to the generic fibers XK , SK , and Y, T are defined over K. We also let V be either T or OK for the de Rham side, and S or K(0)rig as described above for the rigid side. Definition 5.8.1. Let • DRrig (M /V ) := Mrig ⊗ Ω•]X k [/]Vk [ .

The Koszul filtration and graded objects on the rigid (or vector) parts are defined similarly, without the filtered de Rham parts. Definition 5.8.2. Let i i i HA (X/S, M ) := (HdR (X/S, M ), Hrig (X/S, M ), θX/S,M ),

where i • HdR (X/S, M ) is a filtered object, whose filtration is defined by the (relative)

Hodge to de Rham spectral sequence 5.7.12. i • Hrig (X/S, Mrig ) has a Frobenius induced by ϕM .

• θX/S,M is defined as follows. Let i i (X/S, M ), HdR (X/S, M )|SKan = p∗an HdR

185

an where pan : SK → S is the natural morphism of ringed spaces. We have the

natural adjunction morphism i i HdR (X/S, M ) → pan ∗ HdR (X/S, M )|SKan .

(5.8.3)

By classical rigid geometry, we deduce an i i ∼ (X/S, M ) (X/S, M )|SKan = HdR HdR = H q (X an /S an , (M an , ∇an )). Since the dagger operation commutes with cohomological functors, it follows that † † jS† H q (X an /S an , (M an , ∇an )) ∼ = H q (X an /S an , (jX M an , jX ∇an )).

We have a morphism induced by the adjoint map (direct limit of such maps) for an OS -module V p∗an V → jS† p∗an V = Fan (V ). This yields a morphism i HdR (X/S, M )|SKan → jS† H q (X an /S an , (M an , ∇an )) ∼ =

† † H q (X an /S an , (jX M an , jX ∇an )) = H q (X an /S an , (Fan,Y (M ), Fan,Y (∇))). (5.8.4)

Combining this with the morphism 5.8.3 yields a morphism i HdR (X/S, M ) → pan ∗ H q (X an /S an , (Fan (M ), Fan (∇))) , where we used the natural morphism Fan,Y (M ) → Fan (M ). We have a morphism ϕ : Fan (MdR ) → FK (Mrig ). It induces a morphism i HdR (X/S, M ) → pan ∗ H q (X an /S an , (FK (Mrig ), FK (∇rig ))) = q pan ∗ (Hrig (X/S, M ) ⊗ K).

(5.8.5)

186

This morphism induces a morphism q i i Fan (HdR (X/S, M )) = jS† p∗an HdR (X/S, M ) → jS† p∗an pan ∗ (Hrig (X/S, M ) ⊗ K) →

q q q jS† (Hrig (X/S, M ) ⊗ K) = Hrig (X/S, M ) ⊗ K = FK (Hrig (X/S, M )),

(5.8.6) where we used the natural morphism in the category of coherent O-modules F → f ∗ f∗ F q and the fact that Hrig (X/S, M ) is a jS† -module.

Thus, we have a morphism i Fan (HdR (X/S, M ))

θX/S,M

→

q (X/S, M )), FK (Hrig

(5.8.7)

which is the desired θX/S,M . Following Section 4.4.0.8 step by step, one may define all morphisms there also in the relative case, thus establishing functoriality. That is, we have Lemma 5.8.8. Let f : X → X 0 be a morphism of S schemes, and M ∈ Ssyn (X 0 ). Then the relative analogs of the morphisms defined in Section 4.0.8 give rise to a morphism f ∗ : HAi (X 0 /S, M ) → HAi (X/S, f ∗ M ). We also have (cf. Lemma 4.4.0.54) Lemma 5.8.9. Let π : X → S, and let M be an object in Ssyn (S). Then there exists a canonical isomorphism HAi (X/S, π ∗ M ) → HAi (X/S) ⊗ M. q Corollary 5.8.10. HA (X/S, M ) is functorial with respect to proper morphisms of

good compactifications up to base change.

187

Proof. A morphism of two compactifications Y1 /T1 → Y2 /T2 of X/S induces a horizontal morphism between the usual de Rham complexes (over S). Since the canonical extension is functorial with respect to horizontal morphisms the result follows. We also have the relative analog of Proposition 4.0.64. Proposition 5.8.11. For any exact sequence 0 → M 0 → M → M 00 → 0 in Ssyn (X), there is an associated long exact sequence · · · → HAi (X/S, M 0 ) → HAi (X/S, M ) → HAi (X/S, M 00 ) → · · · obtained by pasting together the corresponding sequences for relatve de Rham and rigid cohomologies The proof is essentially the same, so it is omitted. Definition 5.8.12. We define the category of π−admissible syntomic coefficients on ad π−ad (X) whose objects are the triplets (X), to be the full subcategory of Ssyn X, Ssyn

(MdR ∈ SdR (X), Mrig ∈ Srig (X), ϕ : Fan (MdR ) → FK (Mrig )), such that: 1. ϕ is an isomorphism. i i 2. θX/S,M : Fan (HdR (X/S, M )) → FK (Hrig (X/S, M )) is an isomorphism, where θ

is defined by 5.8.7. ad 3. HAi (X/S, M ) ∈ Ssyn (S).

Ideally, the syntomic cohomology would be independent of the particular choice of de Rham datum for X. Unfortunately, this is not always the case. However, it turns out that when the de Rham coefficient has regular singular points with non-integral exponents and local

188

unipotent monodromy this is true. This would allow us to define the syntomic cohomology for coefficients that have regular singular points with non-integral exponents and local unipotent monodromy over X, independent of any particular choice of de Rham datum for X. We then state a strong form of the Leray spectral sequence Theorem 5.9.1 for syntomic coefficients whenever the de Rham side is as mentioned above. Proposition 5.8.13.

1. Let M ∈ Ssyn (X) such that MdR has regular singular

points with non-integral exponents. Assume MdR is defined over the de Rham datum Y 0 . Let j : X → Y be a de Rham datum for X contained in Y 0 as in the definition. Then j induces an isomorphism • j ∗ : RΓ(Y, DRdR (M )Y ) → RΓ(X, j ∗ MdR ⊗ Ω•XK /K ).

2. Let µ : Y1 → Y2 be a morphism of de Rham data for X. That is, µ ◦ j1 = j2 . ad Let M ∈ Ssyn (X) such that MdR has regular singular points with non-integral

exponents. We assume MdR is defined over Y2 . Then µ naturally induces a filtered isomorphism • • RΓ(Y2 , DRdR (M )Y2 ) → RΓ(Y1 , DRdR (µ∗ M )Y1 ).

3. More generally, given two de Rham data j1 : X → Y1 , j2 : X → Y2 for X, and admissible de Rham coefficients M1 , M2 over Y1 , Y2 , respectively, such that j1∗ M1 = j2∗ M2 and have local unipotent monodromies. Then, there exists a filtered isomorphism • • RΓ(Y2 , DRdR (M2 )Y2 ) → RΓ(Y1 , DRdR (M1 )Y1 ).

Proof.

1. By [27] proposition 3.14(i), we deduce that MdR ⊗ Ω•Y /K (log D) ∼ = j∗ (j ∗ MdR ⊗ Ω•XK /K ).

189

Thus • RΓ(Y, DRdR (M )Y )

j∗

∼ = RΓ(Y, j∗ j ∗ MdR ⊗ Ω•XK /K ) ∼ = RΓ(X, j ∗ MdR ⊗ Ω•XK /K ),

where j ∗ is an isomorphism as its higher direct images vanish. 2. The canonical morphism • • (µ∗ M )Y1 (M )Y2 → DRdR µ∗ DRdR

is a filtered morphism. It induces a filtered morphism µ∗

• • • RΓ(Y2 , DRdR (M )Y2 ) → RΓ(Y1 , µ∗ DRdR (M )Y2 ) → RΓ(Y1 , DRdR (µ∗ M )Y1 ).

(5.8.14) • Let GdR (M ) denote the Godement resolution of DRdR (M )Y2 (cf. [38] defini-

tion 5.1.2). This also appears in the proof of Theorem 5.9.1, item 2. Since the Godement resolution is functorial (cf. [38] proposition 5.2.1), we deduce µ∗ GdR (M ) = GdR (µ∗ M ). This implies that • RΓ(Y2 , DRdR (M )Y2 ) = Γ(Y2 , GdR (M )),

• RΓ(Y1 , DRdR (µ∗ M )Y1 ) = Γ(Y1 , µ∗ GdR (M )),

in the derived category. Under this identification, the filtered morphism 5.8.14, is given via µ∗ : Γ(Y2 , GdR (M )) → Γ(Y1 , µ∗ GdR (M )).

(5.8.15)

This is a morphism of filtered Hodge modules, which is also an isomorphism. By the principle of two filtrations (cf. [28] Théorème 2.3.5 we deduce that this is a filtered quasi-isomorphism (compare also with loc. cit. 3.2.II.C). 3. We have

X

Y1

j2

(5.8.16)

>

T

i

We have T \ ˜j(X) is a divisor with normal crossings. We deduce that there are morphisms fi : T → Yi , i = 1, 2, such that the diagrams X ˜j

ji

T

(5.8.18)

>

fi

Yi ,

are commutative. By the previous item fi , induce filtered isomorphisms • • fi∗ : RΓ(Yi , DRdR (Mi )Y1 ) → RΓ(T , DRdR (fi∗ Mi )T ).

(5.8.19)

By Proposition 5.7.43, fi∗ Mi , i = 1, 2 are both isomorphic to the canonical extension of j1∗ M1 = j2∗ M2 over T . Under this identification, we deduce that f2∗ −1 ◦ f1∗ is the desired filtered isomorphism.

Corollary 5.8.20.

ad,QR 1. Let M = (MdR , Mrig , Mvec ) ∈ Ssyn (X). Assume MdR is

defined over some de Rham datum Y . Then, we have • RΓ(Y, DRdR (M )Y ) → RΓdR (XK /K, M )

is a filtered quasi-isomorphism, where RΓdR (XK /K, M ) is defined in Equation 4.0.33.

191

2. With the above notations, if MdR |X has local unipotent monodromies, then M is independent of the de Rham datum Y . In other words, when we restrict attention to coefficients having local unipotent monodromies, it is enough to define the de Rham part of a sytomic coefficient over X. Proof.

1. By Equation 4.0.33, we have RΓdR (XK /K, M ) =

lim −→

• (M )Y ). RΓ(Y, DRdR

Y ∈CdR (X)

By Proposition 5.8.13.(2) we deduce that all morphisms on the right hand side are filtered isomorphisms. The lemma follows. 2. This is a result of Lemma 5.7.43 and Proposition 5.8.13.(3).

We deduce Corollary 5.8.21. Let M = (MdR , Mrig , Mvec ) ∈ Ssyn (X) and assume MdR has regular singular points. Assume MdR is defined over some de Rham datum Y . Then the i (X, M ) may be computed by replacing RΓdR (XK /K, M ) syntomic cohomology Hsyn • in definition 4.0.56 with RΓ(Y, DRdR (M )Y ).

5.9

The Leray spectral sequence theorem π−ad,QRf

Theorem 5.9.1. Let M ∈ Ssyn

(X). There is a Leray spectral sequence for

syntomic cohomology q p p+q (X, M ), E2p,q = Hsyn (S, HA (X/S, M )) ⇒ Hsyn

(5.9.2)

q where the connection on the complex Ω•S,syn ⊗ HA (X/S, M )) is induced by the Gauss-

Manin connection (cf. [52], [77], [33]).

192

The same theorem holds, if M ∈ Ssyn (X), and one takes the syntomic cohomology for a specific choice of a de Rham datum for M . π−ad,QRf

Remark 5.9.3. If M 6∈ Ssyn

(X), there is an analogous statement of the the-

orem, but then the coefficient is defined on a particular compactification, and the definition of syntomic cohomology doesn’t involve direct limit over de Rham data for this coefficient. Proof.

1. We notice that if f : A• → B • is a map of complexes, then RΓ(M C(f )) = M C(f∗ : RΓ(A• ) → RΓ(B • ))

in the derived category, and similarly ∼

∼

RΓ(A• ×C • B • ) = RΓ(A• ) ×RΓ(C • ) RΓ(B • ), for quasi-fibered product. If I • , J • , K • are injective resolutions of A• , B • , C • respectively, then the quasi-fibered product of them is an injective resolution of the quasi-fibered product. 2. By Lemma 5.2.8, we take (Gtyp , Gtyp ), typ = rig, vec to be filtered injective • resolutions of (DRtyp (X, M ), Kostyp ), typ = rig, vec (a 1-level filtered injective

resolution in definition 5.2.7) in the category typ, where typ = rig, vec. For • the de Rham side, the situation is more involved. We notice that DRdR (X, M )

is a bi-filtered object (the Koszul filtration and the Hodge filtration). In this case, choosing the Godement resolution gives a bi-filtered injective resolution (cf. also [38] Lemma 3.2.1), which we denote by (GdR , GdR , FdR ). We have • By Proposition 5.1.8 the graded and hence the filtered pieces of both resolutions are injective. • Since (Gtyp , Gtyp ), typ = rig, vec are 1-level filtered injective resolutions, Proposition 5.2.11 implies the existence of a filtered morphism between the

193

de Rham and vector resolutions, which we also denote by θn . Similarly, the Frobenius ϕM induces a filtered morphism which we also denote by ϕM on the resolutions. We recall a classical fact; it appears in the paper of [52] equation 3.2.5, and also appears in the proof of [67] corollary 4.1; the Koszul filtration on the de Rham complex (or its filtered injective resolution) induces a spectral sequence, which is the spectral sequence for a functor applied to a complex with filtration. p,q p • E1,typ = Rp+q π∗ (GrKos (DRtyp (X, M ))) ⇒ H p+q (π∗ Gtyp ).

This is also the spectral sequence associated to the filtered complex (π∗ Gtyp , π∗ Gtyp ). By 5.7.6 we notice that p • Rp+q π∗ GrKos F n (DRdR (M )) =

•−p Rp+q π∗ π ∗ (ΩpT (log(T \ S))) ⊗OY F n−p DRdR (M /S) = •−p ΩpT (log(T \ S)) ⊗OT Rp+q π∗ F n−p DRdR (M /S) =

(5.9.4)

• (M /S) = ΩpT (log(T \ S)) ⊗OT F n−p Rq π∗ DRdR q (X/S, M ) . F n ΩpT (log(T \ S)) ⊗OT RdR For the vector and rigid parts there is a similar description. Apply Theorem 5.2.12 to these spectral sequences, and choose DdR := ΓT , Drig := Γ]Sk [ , Dvec := ΓSKan to obtain , E2p,qtyp = Rp+q Γtyp (E1p,qtyp (π∗ Gtyp , π∗ Gtyp )) ⇒ Rp+q Γtyp (π∗ Gtyp ).

(5.9.5)

Let (Ityp , Ityp ) be a level-2 filtered injective resolution of (π∗ Gtyp , π∗ Gtyp ) (for the de Rham part, take a level-(2,1) bi-filtered injective resolution in the following sense:

194

W F W F (IdR , IdR , IdR ) is constructed as follows: First let (Jtyp , Dec∗ (Jtyp ), Jtyp ) be a bi-filtered

injective resolution (cf. [38] Lemma 3.2.1) of (π∗ GdR , π∗ Gtyp , π∗ FdR ). Then similarly F W F W ). Thus, with respect to ), JdR ) := (JdR , Bac(JdR , IdR to Lemma 5.2.9, take (IdR , IdR

the weight filtration this is a level-2 injective resolution on each GrFp (IdR ).

By

Equation 5.2.6, we deduce that −q p+q W W E1p,q (IdR , IdR ) = E0−q,p+2q (JdR , JdR ) = GrW JdR .

Since this Bac operation applies to each GrFp (IdR ) (and F p (IdR )), it implies that −q n p+q −q p+q W W E1p,q (F n IdR , IdR ) = E0−q,p+2q (F n JdR , JdR ) = GrW F JdR = F n GrW JdR =

W F n E1p,q (IdR , IdR ).

(5.9.6)

Below, when working with typ = dR, IdR means the weight filtration, and F • would denote the Hodge filtration; we know that the latter commutes with the first sheet of the spectral sequence. By Theorem 5.2.12 it follows that: Rp+q Γtyp E1p,q (π∗ Gtyp , π∗ Gtyp ) = H p+q Γtyp E1p,q (Ityp , Ityp ). Moreover, the new spectral sequence is the one associated to the good filtered complex (Γtyp Ityp , Γtyp Ityp ), and E1p,q (Γtyp Ityp , Γtyp Ityp ) = H p+q Γtyp E0p,q (Ityp , Ityp ) = Γtyp E1p,q (Ityp , Ityp ). According to Theorem 5.3.1, we take the “mapping cone spectral sequence” (or more precisely, the quasi-fibered spectral sequence) corresponding to (Γtyp Isyn , Γtyp Isyn ) according to the definition of syntomic cohomology. For this one should notice that there is an inner cone (for the Frobenius) and an outer cone, for the quasi-fibered

195

product. The construction of the mapping cone spectral sequence is compatible with nested cones (in particular check the first term in the quasi-fibered product). Hence the mapping cone (quasi-fibered) spectral sequence reads as: E1p,q =

Cone(1 −

ϕM pn

∼

: E1p,q (Γrig Irig , Γrig Irig ))[−1] ×E1p,q (Γvec Ivec ,Γvec Ivec ) E1p,q (ΓdR F n IdR , ΓdR IdR )). (5.9.7)

But this is equal to E1p,q = Cone(1 −

ϕM pn

∼

: Γrig E1p,q (Irig , Irig ))[−1] ×Γvec E1p,q (Ivec ,Ivec ) ΓdR E1p,q (F n IdR , IdR )). (5.9.8)

The corresponding E2p,q term is given by (using Equation 5.9.6) E2p,q = H p+q Cone(1 −

ϕM pn

∼ : Γrig E1 (Irig , Irig ))[−1] ×Γvec E1 (Ivec ,Ivec ) ΓdR F n E1 (IdR , IdR ) =

p q Hsyn (S, Rsyn (X/S, M )).

(5.9.9) Indeed, the E1 (Ityp , Ityp ) are injective resolutions of E1 (π∗ Gtyp , π∗ Gtyp ) = E1,typ , which by Equation 5.9.4 are the de Rham complexes in the corresponding categories. Thus q (X/S, M )), and the last equality follows by Γtyp E1 (Ityp , Ityp )) = RΓtyp (DRtyp (Rsyn

corollary 5.8.21 and item 1 in the proof. This sequence converges to H p+q Cone(1 −

∼ ϕM : Γrig Irig )[−1] ×Γvec Ivec ΓdR F n IdR . n p

By Lemma 5.1.22 we deduce • • Γtyp Ityp = RΓtyp (π∗ Gtyp ) = RΓtyp (Rπ∗ DRtyp (X, M )) = RΓY,typ DRtyp (X, M ),

where these isomorphisms are compatible with the Hodge filtration when typ = dR. Thus, H p+q Cone(1 −

∼ ϕM p+q : Γrig Irig )[−1] ×Γvec Ivec ΓdR F n IdR = Hsyn (X, M ), n p

196

where the last equality follows by corollary 5.8.21 and item 1 in the proof. This completes the proof. Equation 5.9.5 implies Corollary 5.9.10. Let Mtyp ∈ Styp (X), where typ = dR, rig. There is a Leray spectral sequence for de Rham and rigid cohomologies p q p+q (S, Htyp (X/S, Mtyp )) ⇒ Htyp (X, Mtyp ), E2p,q = Htyp

(5.9.11)

q (X/S, Mtyp )) is induced by the Gausswhere the connection on the complex Ω•S,typ ⊗Htyp

Manin connection. Moreover, for a syntomic datum M ∈ Ssyn (X), there is a Leray spectral sequence for absolute cohomology q E2p,q = HAp (S, HA (X/S, M )) ⇒ HAp+q (X, M ).

(5.9.12)

Chapter 6 The syntomic elliptic polylogarithm Following the previous chapter, we assume that K ⊂ C is a number field, so that complex analytic methods are applicable here. We intend to relax this condition in the future.

6.1

Gysin isomorphism and localization sequences

We refer to the proof of Proposition 1.2.14 for the notion of cohomological purity. For the rigid case, we refer to [76] theorem 4.1.1 or [60] page 303 for more references. Let i : U ,→ X be an open embedding, and let j : Z := X \ U → X be of codimension d. Let M ∈ Ssyn (X). There exists a long exact localization sequence for absolute geometric cohomology i+1 i i (X, M ) → HA (U, i∗ M ) → HA HA ,Z (X, M ) → · · ·

(6.1.1)

Purity, also known as the Gysin isomorphism, means that i ∼ i−2d (Z, j ∗ M )(−d), for all i ≥ 0. HA ,Z (X, M ) = HA

The Gysin isomorphism holds for the absolute syntomic cohomology since it holds for both the de Rham and rigid cohomologies and clearly commutes with the morphism 197

198

θ. For more details on this we refer to the proof of [4] proposition 2.19. This principle also holds for the relative case π : X → S, where d denotes the relative codimension of Z/S inside X/S. (The morphism always exists, and since it is an isomorphism in each fiber, it follows that the morphism itself is an isomorphism.)

We also need another result for the definition of the syntomic elliptic polylogarithm. Let j : X → X be such that X is a proper curve over S. Let D = X \ X. Theorem 6.1.2. (Gysin exact sequence for rigid cohomology) Let M be a syntomic coefficient on X. Then there exists an isomorphism 0 0 HA (X/S, j ∗ M ) ∼ (X/S, M ) = HA

and an exact sequence 1 1 0 2 0 → HA (X/S, M ) → HA (X/S, j ∗ M ) → HA (D/S, i∗ M )(−1) → HA (X/S, M ) → 0.

The proof of [4] theorem 2.19 applies to this more general setting verbatim.

6.2

The first logarithm Log (1)

Let U := E \e(S), D = e(S). Let j : U ,→ E and e˜ : D → E denote the corresponding inclusions. Since e is a section of π, the morphism e : S → D is in fact an isomorphism. 1 1 Definition 6.2.1. Let H∨ = HA (E/S, K(0)), and let H = HA (E/S, K(1)) de-

note its dual. We also let H∨ , H denote the corresponding pullbacks to E, namely 1 1 π ∗ HA (E/S, K(0)) (resp. π ∗ HA (E/S, K(1))) ambiguously. π−ad We notice that since π : E → S is proper, it is easy to verify that K(0) ∈ Ssyn (E).

This implies in particular that H, H∨ ∈ Ssyn (S) and similarly for the pullbacks. By Lemma 4.0.54 we deduce

199

Corollary 6.2.2. 0 HA (E/S, H) = H,

1 (E/S, H) = H ⊗ H∨ , HA

(6.2.3)

2 HA (E/S, H) = H(1).

using corollary 4.0.61 we deduce Corollary 6.2.4. 0 Hsyn (S, H ⊗ H∨ ) = End(H).

(6.2.5)

Lemma 6.2.6. We have the short exact sequence: 1 1 0 → Hsyn (S, H) → Hsyn (E, H) → End(H) → 0

(6.2.7)

Proof. This is a corollary of the Leray spectral sequence Theorem 5.9.1 which clearly applies here. Let e : S → E be the zero section of E. In the above sequence, the first map is π ∗ where π : E → S is the structure morphism. Therefore, this sequence is split by 1 1 e∗ : Hsyn (E, H) → Hsyn (S, H). 1 Definition 6.2.8. We define l to be the element in Hsyn (E, H) characterized by

properties so that 1. the image of l in End(H) is the identity map. 2. the pullback e∗ (l) is zero. Definition 6.2.9. Let Log (1) be the extension 0 → H → Log (1) → K(0) → 0

(6.2.10)

in Ssyn (E), whose extension class in Ext1Ssyn (E)(K(0), H) corresponds to l through the canonical and functorial isomorphism Ξ given in Theorem 4.0.70.

200

The extension Log (1) is determined up to unique isomorphism. Since the morphism Ξ is functorial, the definition of l implies that the pullback of Log (1) by e∗ is split. We now prove a syntomic analog of Proposition 1.2.14. For this, we need a base change theorem in absolute geometric cohomology for proper morphisms. Theorem 6.2.11. Given a Cartesian diagram q

Y −−−→ X     gy fy p

T −−−→ S where X,S,T,Y are smooth schemes over OK . Then for any F ∈ Ssyn (X), one has a canonical morphism

k k p∗ H A (X/S, F ) → HA (Y /T, q ∗ F )

which is an isomorphism if f is proper. Proof. The de Rham side follows by [71] Thèoréme 4.3, and the rigid side follows by [77] Theorem 4.1.1. These morphisms clearly respect the comparison morphism between de Rham and rigid cohomologies. We recall the notations U¯ := (E − e(S)) ×S E and V¯ := U¯ − ∆(E). Let p : E × E → E is the second projection and will also denote the second projection of U, U¯ , V, V¯ . Proposition 6.2.12. We have 1 ¯ HA (V /E)(1) ∼ = Log (1) .

Proof. We first apply purity to the inclusion of V¯ in U¯ over E \ e(S). Here, the relative codimension is maximal, namely 1, and purity means that i i HA (U¯ /E)(1) = HA (V¯ /E)(1) for i < 1,

201

and the above long exact sequence becomes: 1 ¯ 1 ¯ (V /E)(1) → K(0) → 0. (U /E)(1) → HA 0 → HA

(6.2.13)

This short sequence is exact by the analog argument in the proof of Proposition 1.2.14. We now apply purity in the following situation: the closed set (denoted by Z above) corresponds to the embedding e × id : E → E ×S E, that implies that in the above notation X = E ×S E, U = U¯ . Purity therefore implies i i HA (E × E/E)(1) = HA (U¯ /E)(1) for i < 1,

and we obtain the exact sequence (it is not short exact): 1 1 ¯ 0 → HA (E ×S E/E)(1) → HA (U /E)(1) → Q.

(6.2.14)

Since E ×S E is proper, the residue theorem (in each fiber) implies that there is no residue (it is not possible to have residue only at one point, since by the residue theorem the sum of residues is zero); hence we in fact have an isomorphism 1 1 ¯ HA (E ×S E/E)(1) ∼ (U /E)(1). = HA

Applying the base change theorem for absolute geometric cohomology (Theorem 6.2.11 above) corresponding to the cartesian square p

E ×S E −−−→ E     py πE y E

(6.2.15)

π

E −−− → S

1 implies it is equal to πE∗ HA (E/S)(1) ∼ = πE∗ HE , where the last isomorphism is Poincaré

duality. We recall that the Poincaré duality in rigid cohomology was proved in [12] Lemma 2.1, and the relative case follows by applying the Poincaré duality on the 1 ¯ fibers. We have therefore shown that HA (V /E)(1) sits in the same short exact

202

(1)

sequence as LogE . It remains to show that it satisfies the conditions of Lemma 1.2.12. The first step is to show that the sequence 6.2.13 is split after applying e∗ . We use base change in cohomology with respect to the cartesian square id×e

E ×S E ←−−− E     py πE y E

e

←−−− S

restricted to V¯ , and we obtain a base change morphism: 1 ¯ 1 α : e∗ HA (V /E)(1) → HA (U0 /S)(1).

(6.2.16)

We notice that in this case the map p is not proper, so the induced map need not be an isomorphism. We also notice that by the above argument (Poincaré duality and the residue theorem) 1 we have HA (U0 /S)(1) ∼ = HE . Notice that the horizontal maps of the Cartesian square

6.2.15 commutes with composition with the horizonal maps of this Cartesian square, implying that we obtain a commutative diagram p

id×e

E −−−→ E ×S E −−−→ E       py πE y πE y S −−−→ e

E

π

E −−− → S

all of whose resulting squares are Cartesian. Base change for the right square provides a morphism 1 1 ¯ (V /E)(1). β : πE∗ HA (U0 /S)(1) → HA

(6.2.17)

The composition of the two inner squares gives the identity square. By functoriality of base change we obtain that id = α ◦ e∗ (β). Thus, α provides a split1 ¯ ting of the exact sequence as claimed. We now turn to prove that HA (V /E)(1) ∈ ∨ Ext1Ssyn (E) (K(0), πE∗ HE ) is mapped to the standard map in HomSsyn (S) (K(0), HE ⊗ 1 HE ). By Proposition 1.2.14 we deduce that HdR (V¯ /E)(1) ∈ Ext1SdR (E) (K(0), πE∗ HE )

203

∨ is mapped to the standard map in HomSdR (S) (K(0), HE ⊗ HE ). By functoriality we

deduce that the same holds for the rigid part (since θE/S must preserve the identity). The claim is therefore proved.

6.3

The nth logarithm Log (n)

Definition 6.3.1. We define the elliptic logarithmic sheaf Log (n) in Ssyn (E) by Log (n) = Symn Log (1) The splittiing e∗ Log (1) induces ∗

e Log

(n)

=

n Y

Symj H.

(6.3.2)

j=0

There is a natural projection map p : Log (n+1) → Log (n) defined to be the composition Symn+1 Log(1) → Symn+1 (Log (1) ⊕ K(0)) → Symn Log (1) . Here the first map is induced from the sum of the identity map and the projection Log (1) → K(0). The second map is the canonical projection in the symmetric algebra of a direct sum. By definition, we have an exact sequence 0 → Symn+1 H → Log (n+1) → Log (n) → 0. We refer here to Equation 2.5.1 for another description of the same map. Lemma 6.3.3. (cf. also 2.5.5) 1. The geometric cohomology of Log (n) is as follows: 0 HA (E/S, Log (n) ) = Symn H, 1 HA (E/S, Log (n) ) = (Symn+1 H)(−1), 2 HA (E/S, Log (n) ) = K0 (−1).

204

2. The morphisms 0 0 HA (E/S, Log (n+1) ) → HA (E/S, Log (n) ), 1 1 HA (E/S, Log (n+1) ) → HA (E/S, Log (n) )

induced from the projection Log (n+1) → Log (n) (cf. definition 2.5.1 chapter 2.5) are zero maps. Proof. The proof of [39] Lemma A.1.4 applies here. The case of a single elliptic curve is treated in [4] lemma 3.4.

6.4

Definition of the elliptic syntomic polylogarithm

Let U0 := E \ e(S), D = e(S). Let j : U0 ,→ E and e˜ : D → E denote the corresponding inclusions. Since e is a section of π, the morphism e : S → D is in fact an isomorphism. The purpose of this subsection is to give the definition of the elliptic polylogarithm Pol(n) in Ssyn (U0 ) (see definition 6.4.11), following [4] section 3.3, which in turn follows the method of [39] Sec.A.3. We start by constructing the map 1 δ (n) : Hsyn U0 , H∨ ⊗ Log (n) (1) → End(H) for each integer n ≥ 1 (see definition 6.4.7). First, by Theorem 6.1.2 and Lemma 6.3.3, we have an isomorphism 0 HA U0 /S, H∨ ⊗ Log (n) (1) = H∨ ⊗ Symn H(1). Hence, the Leray spectral sequence Theorem 5.9.1 for U0 /S and the coefficient

205

H∨ ⊗ Log (n) (1) apply here, and give the short exact sequence 1 1 0 → Hsyn S, H∨ ⊗ Symn H(1) → Hsyn U0 , H∨ ⊗ Log (n) (1) (6.4.1) 0 → Hsyn

1 (U0 /S, H∨ ⊗ Log (n) (1)) → 0. S, HA

The Gysin exact sequence in Theorem 6.1.2 gives an exact sequence 1 1 0 → HA E/S, H∨ ⊗ Log (n) (1) → HA U0 /S, H∨ ⊗ Log (n) (1) (6.4.2) 0 → HA D/S, H∨ ⊗ e˜∗ Log

(n)

u

2 → HA

E/S, H∨ ⊗ Log (n) (1) → 0.

Identifying D with S, Equation 6.3.2 implies that we deduce that ∗

e˜ Log

(n)

∼ =

n Y

Symj H.

j=0

Thus, 0 HA

∨

∗

D/S, H ⊗ e˜ Log

(n)

=

n Y

H∨ ⊗ Symj H.

j=0

By Lemma 6.3.3.(a), we have 2 HA E/S, H∨ ⊗ Log (n) (1) = H∨ . Thus, the kernel of u in 6.4.2 is isomorphic to n Y

H∨ ⊗ Symj H.

j=1

Applying Lemma 6.3.3.(a) again, we deduce that 1 HA E/S, H∨ ⊗ Log (n) (1) = H∨ ⊗ Symn+1 H.

(6.4.3)

Hence, the sequence 6.4.2 reduces to the sequence 1 0 → H∨ ⊗ Symn+1 H → HA U0 /S, H∨ ⊗ Log (n) (1) (6.4.4) →

Qn

j=1

H∨ ⊗ Symj H → 0.

206

By corollary 4.0.61 we deduce 0 S, H∨ ⊗ Symn+1 H = 0 (n ≥ 1), Hsyn (6.4.5) 0 Hsyn S,

Qn

j=1

H∨ ⊗ Symj H = F 0 (H ⊗ H∨ )ΦH⊗H∨ =1 .

Applying Proposition 4.0.69 to the short exact sequence 6.4.4 of coefficients in Ssyn (S), together with 6.4.5, yields the long exact sequence 1 0 (U0 /S, H∨ ⊗ Log (n) (1)) → F 0 (H ⊗ H∨ )ΦH⊗H∨ =1 S, HA 0 → Hsyn (6.4.6) 1 → Hsyn S, H∨ ⊗ Symn+1 H → · · · .

Definition 6.4.7 (The residue maps). For each integer n ≥ 1, we define the map 1 δ (n) : Hsyn U0 , H∨ ⊗ Log (n) (1) → End(H) by the composition of the maps in (6.4.1),(6.4.6) and the canonical isomorphism F 0 (H ⊗ H∨ )ΦH⊗H∨ =1 = EndH. We have: Proposition 6.4.8. The maps δ (n) give rise to an isomorphism 1 δ : lim Hsyn (U0 , H∨ ⊗ Log (n) (1)) ∼ = End(H). ←− n

cf. [4] proposition 3.9. By definition of δ (n) , we have an exact sequence 1 1 0 → Hsyn S, H∨ ⊗ Symn H(1) → Hsyn U0 , H∨ ⊗ Log (n) (1) (6.4.9) δ

n 1 → End(H) → Hsyn

S, H∨ ⊗ Symn+1 (H) → . . . .

Equation 6.4.3 together with Lemma 6.3.3.(b) implies that the morphisms 1 1 Hsyn S, H∨ ⊗ Symn+1 (H)(1) → Hsyn S, H∨ ⊗ Symn H(1) ,

207

1 1 Hsyn S, H∨ ⊗ Symn+2 (H) → Hsyn S, H∨ ⊗ Symn+1 (H) induced by the projection pr: Log (n+1) → Log (n) are zero-maps. Hence the assertion follows by taking the limit. Definition 6.4.10. We define the syntomic elliptic polylogarithm to be the system (n)

1 (U0 , H∨ ⊗ Log (n) (1)) such that of elements polE ∈ Hsyn

δ(lim pol(n) )E = id ∈ End(H). ←− n

(n)

Definition 6.4.11. We define the syntomic elliptic polylogarithmic extension PolE to be any syntomic coefficient in Ssyn (U0 ) representing the extension class (n)

Ξ−1 (polE ) ∈ Ext1Ssyn (U0 ) (K(0), H∨ ⊗ Log (n) (1)), where Ξ is the map defined by Theorem 4.0.70.

6.5

Comparison with the motivic elliptic polylogarithms

We follow Kings’s definition of the motivic polylogarithm, where the abelian variety is an elliptic curve. The setup is as follows: E is an elliptic curve over a base S. Denote by e : S → E its zero section, and by π the projection from E to S. We define U = (E − e(S)) ×S (E − e(S)) = {(p1 , p), πp1 = πp, p1 , p 6∈ e(S)} . Now we define V ⊂ U to be the complement of the diagonal section ∆ : (E − e(S)) → U , so that V = {(p1 , p) ∈ U, p1 6= p} .

208

We consider these as schemes over E − e(S) via the second projection. For every scheme W/T , we define its n-th fiber power W n := W ×T . . . ×T W n-times. We can write U n as follows: U n = {(p1 , . . . , pn , p), πpi = πp, pi , p 6∈ e(S)}. Next, for every I ⊂ {1, . . . , n} we define V I ⊂ U n by V I = {(p1 , . . . , pn , p) ∈ U n , pi = p iff i ∈ / I}. Let

P

be the permutation group of I. We also write P P sgnI resp. sgnn for the sign character of I resp. n . I

Definition 6.5.1. For any Q-vector space H with

P

P

n

I -action

for I = {1, . . . , n} and

we denote by

HsgnI P the largest quotient on which I acts via sgnI , and by H sgnI the largest subspace of P H on which I acts via sgnI . Lemma 6.5.2. For any Q-vector space H with

P

I -action

HsgnI ∼ = H sgnI .

Proof. Decomposing H into direct summands of irreducible

P

I

modules is possible

as char Q = 0, and the result follows.

6.5.1

The motivic polylogarithm

Let U := (E − e(S)) ×S (E − e(S)) be as above. Fix an integer a ≥ 2 and let [a](x, y) = (ax, ay) be the a-multiplication on the abelian scheme E ×S E over S. Then [a]−1 U ⊂ U and [a]−1 V ⊂ V are open subschemes.

209

Definition 6.5.3.

1. For W one of the schemes E ×S E, U, V let tr[a] be the com-

position j∗

[a]∗

• • • (W, ∗) ([a]−1 W, ∗) → HM (W, ∗) → HM tr[a] : HM

where j : [a]−1 W → W is the inclusion. 2. For W as above, let n = tr[a] × . . . × tr[a] n-times tr[a] • be the induced endomorphism of HM (W n , ∗), where the fibre product is over

E or E − e(S) depending on the choice of W . 3. Let for r ∈ Z • HM (W n , ∗)(r) n be the generalized eigenspace of tr[a] for the eigenvalue ar , that is, the subspace n of all vectors which are in the kernel of (tr[a] − id · ar )k for some k ≥ 0.

For more details, we refer to the paper [29]. Proposition 6.5.4. For V = U − Λ(E) as above, n > 1 and I with |I| = n − 1 the residue map induces an isomorphism • (1) ∼ •−1 I HM (V n , ∗)(1) sgnn = HM (V , ∗ − 1)sgnI .

Proof. See [56] corollary 2.2.4. Corollary 6.5.5. There exist isomorphisms (1) • ∼ •−1 n−1 , ∗ − 1)(1) ∼ ∼ •−n (6.5.6) HM (V n , ∗)(1) sgnn = HM (V sgnn−1 = . . . = HM (E − e(S), ∗ − n)

given by residue maps as in Proposition 6.5.4.

210

Thus, we have the composed isomorphism (1) •−n • (V n , ∗)signn ∼ (E \ e(S), ∗ − n)(1) . κE/S : HM = HM

(6.5.7)

Since all of the above is valid over an arbitrary base, we can base change E/S to E ×S E/E considered as an elliptic curve over E, and obtain: (n)

(1)

n+2 Definition 6.5.8. The n0 th motivic polylogarithm PolM,E ∈ HM (E×S V n , n+1)sgnn

corresponding to E/S is the element of the motivic cohomology group (1)

n+2 HM (E ×V n , n+1)signn mapping under the isomorphism (the isomorphism 6.5.7 base

changed to E ×S E/E) (1) n+2 2 κE×S E/E : HM (E × V n , n + 1)signn ∼ (E × (E \ e(S)), 1)(1) = HM 2 to the class of the diagonal cl(∆(E \ e(S))) ∈ HM (E ×S (E \ e(S)), 1)(1) .

It remains to establish the fact that the elliptic polylogarithm defined in 6.4.10 coincides with the syntomic realization of the motivic elliptic polylogarithm defined by definition 6.5.8. To this end, following the method of [56] corollary 2.3.4, we need to prove: Theorem 6.5.9. There is an isomorphism n+2 Ext1Ssyn (E\e(S)) (H, Log (n) (1))(1,1) ∼ (E ×S V n , n + 1)(1,1) = Hsyn sgnn ,

and the residue maps, which appear in corollary 6.5.5, give isomorphisms Ext1Ssyn (E\e(S)) (H, Log (n) (1))(1,1) → Ext1Ssyn (E\e(S)) (H, Log (n) (1))(1,1) induced by Log (n) (1) → Log (n−1) (1). Based on Lemma 6.3.3 and Levin in his proof [62] of propositions 2.5.1–2, we have Lemma 6.5.10.

(n)

1 1. HA (U0 /S, LogU0 ) =

Qn+1 i=1

Symn H (−1).

211

(n) k k 2. HA (U0 /S, LogU0 ) ∼ (E/S, Log (n) ), for k < 1. = HA (n)

k 3. HA (U0 /S, LogU0 ) = 0 for k > 1.

Proof of Theorem 6.5.9. We make some notations. Define the schemes U¯ := (E − e(S)) ×S E, V¯ := U¯ − ∆(E), U := (E − e(s)) ×S (E − e(S)), V := U − ∆(E). Let p : E × E → E be the second projection and also denote the second projection of U, U¯ , V, V¯ . We comment here that the proof of admissibility for the syntomic coefficients appearing here is omitted. We prove the theorem in several steps. 1. 1 ¯ HA (V /E)(1) ∼ = Log (1) .

Proof. This is the content of Proposition 6.2.12. 2. (1) (1) 1 HA (V /U0 )(1) ∼ = LogE |U0 :=: LogU0 .

This is a syntomic coefficient on U0 , as the restriction to U0 of a syntomic coefficient on E (flat base change). In the following, the subscript U0 is omitted. i 3. We notice that for i ≥ 0, HA (V /U0 ) is a syntomic coefficient on U0 , being the i restriction to U0 of the syntomic coefficient HA (V¯ /E) on E (flat base change).

The K¨ unneth formula for absolute geometric cohomology gives (the K¨ unneth formula for rigid cohomology is [12] Thèoréme 3.2, and the commutation with the de Rham part is clear) · i Hsyn (U0 , HA (V n /U0 )(∗)) =

M i1 +···+in =i

i1 in · Hsyn (U0 , HA (V /U0 ) ⊗ · · · ⊗ HA (V /U0 )(∗))

212

4. i1 in · (V /U0 ))(1) = 0, (V /U0 ) ⊗ · · · ⊗ HA Hsyn (U0 , HA

if for at least one index j, we have ij < 1 or ij ≥ 2. i

i1 i1 (V /U0 ) is a (V /U0 ) ⊗ · · · ⊗ HAj (V /U0 ). Since each HA Proof. Let F = HA

syntomic coefficient on U0 as explained above, if follows that F is also a syntomic coefficient on U0 . π−ad,QR (U0 ). Lemma 6.5.11. F ∈ Ssyn k ad Proof. The non-trivial part is to show that HA (U0 /S, F) ∈ Ssyn (S).

Let i i1 ˜ F˜ = HA (V /E) ⊗ · · · ⊗ HAj (V˜ /E).

There is a localization sequence βi αi i+1 i i ˜ γi ˜ → ˜ → HA HA (U0 /S, i∗ F) · · · → HA (E/S, F) ,e(S) (E/S, F) → · · ·

(6.5.12)

Flat base change implies i∗ F˜ ∼ = F. Purity implies i+1 ∗ ˜ ˜ ∼ i−1 HA ,e(S) (E/S, F ) = HA (e(S)/S, e F))(−1).

We deduce the short exact sequence α

βi

k 0 → coker γi−1 →i HA (U0 /S, F) → ker γi → 0.

(6.5.13)

k ad ˜ ∈ Ssyn Since π : E → S is proper, we deduce that HA (E/S, F) (S). Clearly, i−1 ˜ ∈ S ad (S). This implies the coherence of the de Rham and HA (e(S)/S, e∗ F) syn

rigid parts for both the left and right terms of the short exact sequence 6.5.13 together with the fact that the Frobenius and corresponding θ are isomorphisms on them. Using this short exact sequence, we deduce that

213

k • The de Rham and rigid parts of HA (U0 /S, F) are coherent (cf. [37] propo-

sition II.5.7 for the de Rham side, and similarly for the rigid side). k (U0 /S, F) is an isomorphism by the five lemma. • The Frobenius of HA

• The five lemma also implies that ΘU0 /S,F is an isomorphism. The result follows. By the Leray spectral sequence for absolute cohomology (cf. Corollary 5.9.10) k HAi (S, HA (U0 /S, F)) ⇒ HAi+k (U0 , F)

(6.5.14)

The following lemma is proven below. Lemma 6.5.15. If r < 0 then F (r) = 0. Using this lemma, we prove item 4. Assume w.l.o.g. that j = 1 is an index with i1 ij 6= 1. First assume that i1 ≥ 2. Then HA (V /U0 ) = 0. The case i1 > 2 is clear

by de Rham theorem as the real dimension of U0 is 2. Thus, we assume that i1 = 2. It suffices to show that Ri1 πU0 ∗ KU0 = 0. Notice that the map induced by the inclusion V ,→ U induces a surjection R2 p∗ KU → R2 p∗ KV , as the next element in the localization sequence is 0. We now claim that R2 p∗ KU = 0. Indeed, by base change we deduce that R2 p∗ KU = πU∗ 0 R2 πU0 ∗ KU0 . We conclude by noticing that the 2’th cohomology of U0 is 0, as it is isomorphic to the dual of the 0’th cohomology with compact support, which is 0. Hence, the result follows in this case. Next assume that i1 = 0 and that all ij are either 0 or 1. We first show that 0 HA (V /U0 )(l) = 0, for l < 2.

By purity (for the inclusion of V ,→ U , where U − V is of relative codimen0 0 0 sion 1) implies that HA (V /U0 ) ∼ (U/U0 ) ∼ (U0 /S), where the last = HA = π U0 ∗ H A

isomorphism is flat base change (the morphism is smooth and locally of finite

214

type and hence flat by [65] proposition 3.24). The action of tra on the relative homology groups Hk (E/S) ∼ = Hk (U0 /S), k 6= 0 0 (U0 /S) via multiis by multiplication by ak . This implies that tra acts on HA 0 plication by a2 , and therefore HA (U0 /S)(l) = 0 for l < 2 as claimed. i2 i1 in (V /U0 ). By (V /U0 ) ⊗ · · · ⊗ HA (V /U0 ) ⊗ G, for G = HA Next, write F = HA

Lemma 6.5.15 applied to G, we deduce that G (r) = 0 for r < 0. 0 0 We can therefore write F = HA (V /U0 ) ⊗ G = πU0 ∗ HA (U0 /S) ⊗ G.

By Proposition 5.5.8.9 we deduce k 0 k HA (U0 /S, F) = HA (U0 /S) ⊗ HA (U0 /S, G).

Let t denote the number of j such that ij = 0. By applying the above argument for all such j’s, we get k 0 k HA (U0 /S, F) = HA (U0 /S)⊗t ⊗ HA (U0 /S, Log (n−t) ).

This, together with [9] Lemma 1.3.2 and Corollary 1.3.3 imply that k k HA (U0 /S, F)(1) = 0. As tra acts on HAi (S, HA (U0 /S, F)) by acting on the k k (U0 /S, F))(1) = 0, and (U0 /S, F) alone, it implies that HAi (S, HA coefficient HA

by 6.5.14 we deduce that HAj (U0 , F)(1) = 0 for all j ≥ 0. This implies that so do the corresponding de Rham and rigid cohomologies. j Hence, Proposition 4.0.58 implies that Hsyn (U0 , F)(1) = 0 for j ≥ 0 and the

result follows also in this case, so the proof is complete.

Proof of Lemma 6.5.15. If for some j, we have ij ≥ 2 then F = 0 as shown above. Thus, we may assume that 0 F = HA (V /U0 )

⊗k

⊗ Log (1)

⊗n−k

,

215

0 for some 0 ≤ k ≤ n. Since we showed that HA (V /U0 )(r) = 0 for r < 0, it is

enough to show that the same holds for Log (1) . It suffices to show it for the de Rham part, which follows by [9] Lemma 6.14, and the result follows. 5. We deduce by the previous item: · n · (V n /U0 )(∗))(1) = Hsyn (U0 , Log ⊗(n) (∗ − nd))(1) , (U0 , HA Hsyn k · (V n /U0 ))(1) = 0, (U0 , HA Hsyn

for k 6= n. 6. · · Hsyn (U0 , Log (n) (∗)) = Hsyn (U0 , Log ⊗(n) (∗))sgnn . · The action of sgnn on Habs (U0 , Log ⊗(n) (∗)) is via its action on Log (n) . We first sgnn = Log (n) . The involution (i, j) of E n (i.e., the map show that Log ⊗(n)

(u1 , . . . , ui , . . . , uj , . . . , un ) → (u1 , . . . , uj , . . . , ui , . . . , un )) induces the endomorphism l1 ⊗ · · · ⊗ li ⊗ · · · ⊗ lj ⊗ · · · ⊗ ln ∼ = r1∗ (l1 ) ∪ · · · ∪ ri∗ (li ) ∪ · · · ∪ rj∗ (lj ) ∪ · · · ∪ rn∗ (ln ) → r1∗ (l1 ) ∪ · · · ∪ rj∗ (li ) ∪ · · · ∪ ri∗ (lj ) ∪ · · · ∪ rn∗ (ln ) =

(6.5.16)

−r1∗ (l1 ) ∪ · · · ∪ ri∗ (lj ) ∪ · · · ∪ rj∗ (li ) ∪ · · · ∪ rn∗ (ln ) ∼ = −l1 ⊗ · · · ⊗ lj ⊗ · · · ⊗ li ⊗ · · · ⊗ ln . We used the anti-commutativity of the cup product. This implies that σ(l1 ⊗ P j ⊗(n) sgnn · · · ⊗ ln ) = sgnn (σ)lσ(1) ⊗ · · · ⊗ lσ(n) . Thus, if l := j ⊗i li ∈ Log

216

implies l =

1 n!

P

P σ∈ n

P

j

j ⊗i lσ(i) , which is easily seen to belong to Log (n) . This

implies that Log ⊗(n)

sgnn

= Log (n) .

Since these are Q-coefficients, and the action of

P

n

(6.5.17) on the cohomology is given

by acting on the coefficients alone, the result follows. 7. The Leray spectral sequence Theorem 5.9.1 for pn : V n → U0 together with items 5 and 6 implies · ·−n (n) Hsyn (V n , (∗))(1) (∗ − n))(1) . sgnn = Hsyn (U0 , Log

We should remark that the morphism pn admits a good compactification. The proof of this uses toric geometry, and reduces to a simple combinatorial question, but we omit the details here. 8. We base change πU0 : U0 → S along the morphism πE : E → S, to obtain p

E ×S U0 −−−→ U0    πU0  pr1 y y E

(6.5.18)

π

E −−− → S

We apply this base change to Equation 7. We obtain (n)

n+2 2 (1) Hsyn (E ×S V n , n + 1)(1) sgnn = Hsyn (E ×S U0 , LogE×S U0 (1)) .

(6.5.19)

9. We investigate the right hand side of Equation 6.5.19. We claim that: (n) (n) 1 1 2 Hsyn (U0 , πU∗ 0 (HA (E/S)) ⊗ LogE (∗))(1,1) ∼ (E × U0 , LogE×S U0 (∗))(1,1), = Hsyn

where the (1, 1) superscript stands for the intersection of the (a1 )-eigenspace for the action of a × id and the (a1 )-eigenspace for the action of id × a, both acting on E ×S V n . the first map multiplies the first coordinate, E, and the second

217

map multiplies the ”second” coordinate V n . If one forgets to take multiplication by a for the first coordinate, the following would not hold! Also note that this is the same as the (1,1)-eigenspace appearing in [9] and [81]. Proof. Let p : E ×S U0 → U0 be the projection on the second coordinate. Since E ×S U0 is connected and dimU0 = 1, it follows that p is flat. By flat base (n)

(n)

change we deduce that LogE×U0 = p∗ LogU0 . Hence, (n)

(n)

k k HA (E ×S U0 /U0 , LogE×U0 ) = HA (E ×S U0 /U0 , p∗ LogU0 ) =

(n) (n) k k HA (E ×S U0 /U0 ) ⊗ LogU0 ∼ (E/S) ⊗ LogU0 . = πU∗ 0 HA

By the Leray spectral sequence Theorem 5.9.1 for p : E ×S U0 → U0 (clearly this morphism admits a good compactification) we deduce that: (n)

(n)

j i i+j Hsyn (U0 , πU∗ 0 (HA (E/S)) ⊗ LogE (∗))(1,1) ⇒ Hsyn (E ×S U0 , LogE×S U0 (∗))(1,1) .

Notice that the (1, 1) eigenspace appears in the spectral sequence similarly to [9] equation 6.9. We now examine the index in question, that is i + j = 2. Now, we recall that multiplication by a induces multiplication by ai on the i’th homology (!) group, and therefore multiplication by a2−i on the i’th absolute geometric cohomology group. As we choose the (1)-eigenspace for the action of tra on the first cohomology group, the only non-trivial part is j = 1, implying that i = 1, and the claim follows. 10. (n) (n) 1 1 1 (E/S))⊗LogE (∗))(1,1) ∼ (E/S)∨ ), LogU0 (∗))(1,1) Hsyn (U0 , π ∗ (HA = Ext1Ssyn (U0 ) (π ∗ (HA

Combining items 8, 9, and 10, the result follows.

218

The fact that the residue maps, which appear in corollary 6.5.5, induce isomorphisms Ext1Ssyn (E\e(S)) (H, Log (n) (1))(1,1) → Ext1Ssyn (E\e(S)) (H, Log (n) (1))(1,1) induced by Log (n) (1) → Log (n−1) (1) is clear by the isomorphism proved in this theorem together with corollary 6.5.5. Remark 6.5.20. Notice that Kings has no (1)-eigenspace, definition 2.3.2 of Kings. Corollary 6.5.21 (compare [56] corollary 2.3.3). The syntomic regulator (1,1)

(1,1)

n+2 n+2 rsyn : HM (E × V n , n + 1)signn → Hsyn (E × V n , n + 1)signn (n)

(n)

maps PolM,E (cf. definition 6.5.8) to polE (cf. definition 6.4.10). Proof. We use here the compatibility of the syntomic regulator with the localization sequence. This is going to be developed by Chiarellotto and his coauthors (the precise identity of which is not yet clear). (0)

2 The case n = 0: we have rsyn (PolM,E ) is the image of rsyn (Λ(E \ e(S))) in Hsyn (E ×S (0) (1) U0 , K(1))(1,1) . This is known to be the class of LogE ∼ = polE (cf. Lemma 1.1.2.10,

which has a syntomic analog). The isomorphisms established by the second state(n)

ment of Theorem 6.5.9 imply that rsyn (PolM,E ) is the unique element that maps to rsyn (∆(E \ e(S))). Again, using these isomorphisms it is clear that the same is true (n)

(0)

for polE (they are the unique elements which map to polE , and we already proved the claim for n = 0).

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p-adic elliptic polylogarithms and arithmetic applications

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