Hilbert Modular Forms with Coefficients in Intersection

Hilbert Modular Forms with Coefficients in Intersection Homology and Quadratic Base Change Jayce R. Getz and Mark Goresky 1

1

2

School of Mathematics, Institute for Advanced Study, and Department of Mathematics, Princeton University, Princeton, NJ. 2 School of Mathematics, Institute for Advanced Study, Princeton, NJ.

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Abstract. In their well-known paper [HZ] F. Hirzebruch and D. Zagier defined, for each prime p ≡ 1 (mod 4), a family of 1-dimensional subvarieties Zm (m ∈ Z>0 ) of a toroidal compactification X tor = X0tor (OQ(√p) ) of the Hilbert modular surface Y = Y0 (OQ(√p) ) := SL2 (OQ(√p) )\H2 . They went on to prove that these cycles enjoyed the striking property that the Fourier series ∞ X hβ, [Zm ]iH e2πimz m=0

defines a classical (or elliptic) modular form with nebentypus for any β ∈ H2 (X tor ), where h, iH denotes the intersection product and [Z0 ] is the Poincaré dual of the Kähler class. Motivated by the results of Hirzebruch and Zagier, in this work we consider the following more general setting: Let L be a totally real number field, let c ⊂ OL be an ideal in its ring of integers, and let Y = Y0 (c) be the resulting Hilbert modular Shimura variety of Hecke type. Denote by X = X0 (c) its Baily Borel compactification. It is a projective algebraic variety of dimension n = [L : Q]. Whenever L is a quadratic extension of a subfield E, we isolate a certain subspace IHnE (X) of the intersection homology group IHn (X) that is spanned by Hecke eigenclasses. To describe a special case of our results, assume that the narrow class number of E is 1. For any character χE ∈ Gal(L/E)∧ , using the theory of base change, we construct a family of Hecke correspondences {Tξ }ξ∈E × on X with the following property. For any γ ∈ IHnE (X) and for any β ∈ IHn (X) the following Fourier series constructed from intersection products X hβ, Tξ (γ)iIH q(0,0) (ξx, ξy) ξ∈E × ξ0

is a Hilbert modular cusp form on E with nebentypus χE . Here q(0,0) is a certain Whittaker function which is a product of exponential functions. When the class β can be realized as the fundamental class of a cycle Z ⊂ X (for example, when Z is a Shimura subvariety, possibly one that intersects the cusps of X) we give an alternate expression for the Fourier coefficients of this modular form in terms of period integrals. This provides a general analog of a theorem of Zagier. The appropriate refinements of these results continue to hold when intersection homology is replaced by intersection homology with coefficients in a local system and the class number of E is arbitrary.

To our families

Contents Chapter 1. Introduction 1.1. An observation of Serre 1.2. The setting 1.3. Definition of γχE (m). 1.4. First main theorem 1.5. Second main theorem 1.6. Explicit cycles 1.7. Finding cycles dual to families of automorphic forms 1.8. Comments on related literature 1.9. Comparison with Zagier’s formula 1.10. Outline of the book 1.11. Problematic primes 1.12. Acknowledgements

1 1 3 5 6 7 9 10 12 13 14 15 16

Chapter 2. Review of Chains and Cochains 2.1. Cell complexes and orientations 2.2. Subanalytic sets and stratifications 2.3. Sheaves and the derived category 2.4. The sheaf of chains 2.5. Homology manifolds 2.6. Cellular Borel-Moore Chains

17 17 18 19 20 21 22

Chapter 3. Review of Intersection Homology and Cohomology 3.1. The sheaf of intersection chains 3.2. The sheaf of intersection cochains 3.3. Homological stratifications 3.4. Products in intersection homology and cohomology

24 24 25 26 29

Chapter 4. Integration of Differential Forms 4.1. Differential forms on an orbifold 4.2. Integration 4.3. The decomposition of ξ 4.4. Integration and intersection homology

32 32 33 34 35

Chapter 5. Finite Mappings and Correspondences 5.1. Finite mappings 5.2. Correspondences

37 37 37

Chapter 6.

40

Review of Arithmetic Quotients v

vi

CONTENTS

6.1. Setting 6.2. Baily-Borel compactification 6.3. L2 differential forms 6.4. Invariant differential forms 6.5. Hecke correspondences 6.6. Mappings induced by a Hecke correspondence 6.7. The reductive Borel-Serre compactification 6.8. Saper’s theorem 6.9. Modular cycles

40 41 42 44 45 47 47 49 49

Chapter 7. Generalities on Hilbert Modular Forms and Varieties 7.1. Hilbert modular Shimura varieties 7.2. Weights 7.3. Hilbert modular forms 7.4. Hecke operators 7.5. The Petersson inner product 7.6. Fourier series 7.7. L-functions 7.8. Relationship with Hida’s notation

52 52 55 57 60 61 62 67 74

Chapter 8. The Automorphic Description of Intersection Cohomology 8.1. Coefficient rings 8.2. An orbifold stratification of X0 (c) 8.3. Representations and local systems 8.4. Pairings 8.5. Hecke correspondences 8.6. The action of the component group 8.7. The automorphic description of intersection cohomology 8.8. Pairings and the Petersson inner product

76 76 78 79 81 85 88 88 94

Chapter 9. Hilbert Modular Forms with Coefficients in a Hecke Module 9.1. Notation 9.2. Hecke algebras and base change 9.3. Hilbert modular forms with coefficients in a Hecke module 9.4. The Fourier coefficients of [[Z]χE (m), ΦQχE [Z],χE ]IH∗

97 97 97 101 105

Chapter 10.1. 10.2. 10.3. 10.4. 10.5.

109 109 110 114 116 119

10. Explicit construction of cycles Notation for the quadratic extension L/E The cycles Zθ The full version of Theorem 1.3 An integral representation for hωJ0 (f −ι ), [Zθ ]iK Rankin-Selberg integrals

Chapter 11. 11.1. 11.2. 11.3.

Eisenstein Series with Coefficients in Intersection Homology Eisenstein series Invariant classes revisited Definition of the VχE (m)

127 127 128 128

CONTENTS

11.4.

Statement and proof of Theorem 11.2

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129

Appendix A. Proof of Proposition 2.4 A.1. Cellular cosheaves A.2. Proof of Lemma 2.3 A.3. Proof of Proposition 2.4

131 131 132 133

Appendix B. Recollections on Orbifolds B.1. Effective actions B.2. Definitions B.3. Refinement B.4. Stratification B.5. Sheaves and cohomology B.6. Differential forms B.7. Integration over chains B.8. Groupoids

135 135 138 139 141 142 144 145 146

Appendix C. Fourier expansions of Hilbert modular forms C.1. Characters of L\AL C.2. The different C.3. Characters of GL1 (L)\GL1 (AL ) C.4. Statement of the theorem C.5. Fourier analysis on GL2 (L)\GL2 (AL ) C.6. Whittaker models C.7. Decomposition of Wh C.8. Computing Wφ∞ and Wh0 C.9. Final steps

148 148 149 149 150 151 152 152 154 156

Appendix D. Review of Prime Degree Base Change for GL2 D.1. Automorphic forms and automorphic representations D.2. Hecke operators D.3. Agreement of L-functions D.4. Langlands functoriality D.5. Prime degree base change for GL1 D.6. Conductors of admissible representations of GL2 D.7. The archimedian places D.8. Global base change

158 158 162 165 166 172 173 176 177

Bibliography

181

Index of Notation

189

Index of Terminology

193

CHAPTER 1

Introduction 1.1. An observation of Serre In their seminal paper on the intersection theory of Hilbert modular surfaces, F. Hirzebruch and D. Zagier mentioned that the motivation for their work was to explain an observation of J-P. Serre [HZ]. To describe his observation, let p ≡ 1 (mod 4), let H denote the complex upper half plane and let Y := SL2 (OQ(√p) )\H2 be the (non-compact) Hilbert modular surface attached to the ring of integers of the real √ quadratic field Q( p). Moreover, let · M + := M2+ (Γ0 (p), ( )) p be the “plus space” consisting of those holomorphic elliptic modular forms of (classical)1 weight 2 , level Γ0 (p), and nebentypus ( p· ) whose `-th Fourier coefficient is zero for all primes ` c came up satisfying ( p` ) = 0. In a letter dated December 8th, 1971, Serre observed that b p+19 24 in a computation of the arithmetic genus of a surface related to Y , and that p + 19 + dimC M = . 24 With the benefit of the modern theory of arithmetic quotients and automorphic forms, the work of [HZ], and almost 40 years of hindsight, let us try to work out a possible explanation of Serre’s observation. Let X be the Baily-Borel or minimal Satake compactification of Y . To explain Serre’s observation, we could conjecture that there is a surjective Hecke-equivariant homomorphism (1.1.1)

Φ ∈ Hom(IH2 (X), M + ) = IH2 (X)∨ ⊗ M + ,

where IH• (X) is intersection homology (with respect to middle perversity, see Chapter 3 below). In order to justify the introduction of intersection homology (as opposed to, say usual singular homology) we make two observations. First, the intersection pairing induces a canonical 1We

say “classical weight” because, following Hida, we will later give an alternate normalization of the weight for an isomorphic space of automorphic forms. 1

2

1. INTRODUCTION

isomorphism IH2 (X)∨ = IH2 (X). Therefore Φ defines an element of IH2 (X) ⊗ M + .

(1.1.2)

We call an element of the vector space IH2 (X) ⊗ M + an elliptic modular form with coefficients in IH2 (X). Our second observation is that there is a canonical action of the Hecke algebra T attached to SL2 (OQ(√p) ) on IH2 (X) via correspondences. This is explained in detail in Chapters 5 and 8 below. This gives us a Hecke action on the left factor of (1.1.2). Things are a little more subtle on the right, since the plus space is not preserved by the action of the usual Hecke algebra TQ on spaces of modular forms. However, it is preserved by the image of the base change map b : Tp −→ TQ . Here Tp ≤ T is the subalgebra of Hecke operators whose “components at p” are trivial (see √ (9.2.1)). This map sends T (P0 ) 7→ T (p0 ) if p0 = P0 P0 splits in Q( p)/Q and T (P0 ) 7→ √ T (p02 ) − p2 T (p, p) if p0 is inert in Q( p)/Q (see §9.2.3). The map b may be defined as the homomorphism induced by the usual morphism of L-groups L

GL2/Q → L ResQ(√p)/Q (GL2 )

which defines the quadratic base change lifting sending automorphic representations of GL2/Q to automorphic representations of ResQ(√p)/Q GL2 (see Lemma D.5). It therefore makes sense to require that the homomorphism defined by Φ satisfy (1.1.3)

Φ ◦ T (P0 ) = b(T (P0 )) ◦ Φ

for all prime ideals P0 ⊂ OQ(√p) coprime to p. The existence of a modular form Φ ∈ IH2 (X) ⊗ M + with coefficients in intersection homology satisfying (1.1.3) would give a reasonable explanation of Serre’s observation. From Hirzebruch and Zagier’s work, we might expect that Φ might be defined not only in terms of intersection homology classes, but intersection homology classes of cycles on X. In other words, we might hope that there is a family of cycles {Zn } on X admitting classes {[Zn ]} in intersection homology indexed by the positive integers such that Φ has a Fourier expansion of the form X (1.1.4) Φ := [Zn ]q n n≥0

Here q := e2πiz , so that Φ can be regarded as a function of a complex parameter z ∈ H with values in the intersection homology group IH2 (X). We haven’t specified that [Z0 ] is the class of a cycle, because as Hirzebruch and Zagier it will actually turn out to be the Poincaré dual of a normalized Kähler form instead.

1.2. THE SETTING

3

We hope that the reader is convinced at this point that if a Φ ∈ IH2 (X) ⊗ M + satisfying (1.1.3) and admitting an expansion of the form (1.1.4) exists, then it would be an object that ties together the geometry of X with the theory of quadratic base change. The main result of this work is the existence of such a Φ1 , not only in the classical setting described above, but in √ the analogous setting when Q( p)/Q is replaced by a general quadratic extension of totally real number fields L/E. √ Before we go further, we should point out that the result in the Q( p)/Q case is not new. Indeed, Hirzebruch and Zagier’s work can (roughly) be viewed in the above light, and this was more or less pointed out by Hirzebruch and Zagier themselves. Moreover, since the work of Hirzebruch and Zagier, a host of generalizations of this sort of phenomenon have been discovered, mostly in the context of liftings connected to Shimura varieties of orthogonal type and/or involving arithmetic Chow groups in place of singular homology. We point out a few references with no claim to completeness: [Bo], [BBK], [Co] [G1],[G2], [GrK], [Ku2], [Ku3], [KM], [KM2], [KM3], [KRY],[Mi], [Od1], [Od2], [Od3], [To], [TW1], [TW2]. 1.2. The setting Throughout this paper, with the exception of Appendix D, L/E will be a quadratic extension of totally real number fields of relative discriminant dL/E and relative different DL/E . We use class field theory to identify Gal(L/E)∧ with a pair of Hecke characters {η} trivial at places in Σ(L), the set of infinite places of L. Consider the reductive algebraic Q-rank one group GL := ResL/Q (GL2 ). (See [PR] §2.1.2 for restriction of scalars.) Let K∞ be the normalizer of the standard R-algebraic homomorphism (see 7.1.1) C× ∼ = ResC/R (Gm )(R) −→ GL (R). The associated symmetric space is GL (R)/K∞ ∼ = (C − R)Σ(L) . For each ideal c of the ring of integers OL of L, consider the Hilbert modular variety Y0 (c) := GL (Q)\GL (A)/K∞ K0 (c), where A is the adèles of Q and K0 (c) is the typical Hecke-type compact open subgroup (7.1.9) of the finite adelic points GL (Af ) of GL . We could consider more general compact open subgroups, but fixing the family {K0 (c)}c⊂OL simplifies the Fourier series that will be used. Let X0 (c) be the Baily-Borel Satake compactification of Y0 (c), (see Chapter 7). For an ideal c ⊂ OL , let Tc be the Hecke algebra associated to the compact open subgroup K0 (c) ≤ GL (Af ). It is generated as a Z-algebra by the elements Tc (b) and Tc (b, b) as b ranges over the ideals of OL (see §7.4). Each Tc (b) determines a correspondence and hence an endomorphism Tc (b)∗ of the intersection homology groups IH∗ (X0 (c)) = I m H∗ (X0 (c), C) (see §8.5, §6.5 and §7.2). Let S(0,0) (K0 (c)) := S(0,0) (K0 (c), χtriv )

4

1. INTRODUCTION

be the space of holomorphic Hilbert cusp forms of weight (0, 0) ∈ ZΣ(L) ×√ ZΣ(L) and level K0 (c) (see §7.3). Here χtriv is the trivial character. In the special case L = Q( d) for d ∈ Z>0 , this corresponds to h-tuples of classical weight (2, 2) Hilbert modular forms for certain congruence new subgroups of SL2 (OQ(√d) ) and a certain h depending on c. If f ∈ S(0,0) (K0 (c)) is a newform, then it is a simultaneous eigenform for all Hecke operators, with eigenvalues λf (m), meaning that f |Tc (m) = λf (m)f

(1.2.1) for all ideals m ⊂ OL . Denote by

IH[L:Q] (X0 (c))(f ) the f -isotypical component of IH[L:Q] (X0 (c)) viewed as a Hecke module (see §8.7). Let M E (1.2.2) IH[L:Q] (X0 (c))(f ), IH[L:Q] (X0 (c)) := f new where the sum is taken over those newforms f ∈ S(0,0) (K0 (c)) such that for almost all primes σ P ⊂ OL we have λf (P) = λf (P ) for all σ ∈ Gal(L/E). This subspace admits a decomposition M χE E IH[L:Q] (X0 (c)) = IH[L:Q] (X0 (c)). χE ∈Gal(L/E)∧

Here χE IH[L:Q] (X0 (c)) :=

M

E IH[L:Q] (X0 (c))(f ) ≤ IH[L:Q] (X0 (c)),

g new where the sum is over those g such that g ∈ S(0,0) (K0 (c)) is the base change of a Hilbert 2 modular form of nebentypus χE (see §9) . E In the next section we define, for each γ ∈ IH[L:Q] (X0 (c))χE and χE ∈ Gal(L/E)∧ , a particular family of Hecke translates of γ: χE {γχE (m)}m⊂OE ⊂ IH[L:Q] (X0 (c)).

Theorem 1.1 below, a generalization of the Hirzebruch-Zagier theorem, says that the formal Fourier series constructed from these classes is a Hilbert modular form on E with coefficients E in IH[L:Q] (X0 (c)). This result is a formal consequence of the existence of base change for L/E. Theorem 1.2 below gives a way to explicitly compute the Fourier coefficients of the resulting Hilbert modular forms in terms of certain period integrals. 2In

the introduction, we only consider Hilbert modular forms on L with trivial nebentypus, so we can assume that χE = η for some η ∈ Gal(L/E)∧ . In Chapter 9, we allow Hilbert modular forms on L with more general nebentypus, and hence more general χE .

1.3. DEFINITION OF γχE (m).

5

1.3. Definition of γχE (m). If m ⊂ OE and c ⊂ OL are ideals and χE ∈ Gal(L/E)∧ is a fixed choice of Hecke character, define the Hecke operator (cf. §9.2) Tb(m) := Tbc,χE (m) ∈ Tc ⊗ C as follows. For every prime ideal p ⊂ OE split in L/E, Set  Id     1 T (Pr ) c p Tb(NL/E (P)r ) := 2 r  Tc (P ) + χE (p)NE/Q (p)TχE (p2r−2 )    0

choose a prime ideal Pp lying above p. if r = 0 if NL/E (P) = NL/E (Pp ) = p splits if P = p is inert otherwise.

Here when we say p splits or is inert, we mean p splits or is inert in the extension L/E. If m ⊂ OE is a norm from OL , then the operator Tb(m) is defined multiplicatively, if m ⊂ OE is not a norm from OL , then we set Tb(m) := 0. We note that the linear map (1.3.1)

b: TcE −→ Tc T (m) 7−→ Te(m)

just defined is roughly a section of the base change map b : Tc −→ TcE associated to the usual map of L-groups L GE → L GL used to define the functorial quadratic base change lifting from automorphic representations of GE to automorphic representations of GL . We will return to this point in §9.2 and §D.4 below. An equivalent statement in classical language is given in the following remark: Remark. For each χE ∈ Gal(L/E)∧ and ideal cE ⊂ OE divisible by the conductor of χE let S(0,0) (K0 (cE ), χE ) be the space of cusp forms on K0 (cE ) with nebentypus χE (see §7.3). If new f ∈ S(0,0) (K0 (cE ), χE ) new is a newform on E such that its base change fb to L is an element of S(0,0) (K0 (c)), then

fb|Tb(m) = λf (m)fb if m is a norm from OL coprime to dL/E (c ∩ OE ) (see Proposition 9.2) and where λf (m) is defined by (1.2.1).

6

1. INTRODUCTION

E Given γ ∈ IH[L:Q] (X0 (c))χE , we set notation for the following Hecke translates of QχE γ: ( Tb(m)∗ γ if m + dL/E (c ∩ OE ) = OL (1.3.2) γχE (m) := 0 otherwise.

Let cE ⊂ OE be an ideal divisible by the conductor of χE . For the purposes of the next section, define +,0 S(0,0) (cE , χ) := {f ∈ S(0,0) (cE , χ) : a(m, f ) = 0 if χE (m) = −1 or m + cE 6= OE }.

Here, as above, the notation (0, 0) stands for the element of ZΣ(E) × ZΣ(E) that is zero in every +,0 coordinate, and a(m, f ) is the mth Fourier coefficient of f (see §7.6). The space S(0,0) (cE , χE ) can be viewed as a modification/generalization of the plus-space occurring in [HZ]. 1.4. First main theorem We can now state the first main result, a special case of Theorem 9.4 below. It is essentially a restatement of abelian base change for certain cohomological automorphic representations of GE in a form that can be used as a vehicle for producing analogues of Hirzebruch and Zagier’s theorem: χE Theorem 1.1. If c ⊂ OL is an ideal and γ ∈ IH[L:Q] (X0 (c)), then the formal Fourier series X γχE (ξyDE/Q )q(0,0) (ξx, ξy) Φγ,χE (( y0 x1 )) := |y|AE ξ∈E ξ0

χE +,0 is an element of IH[L:Q] (X0 (c)) ⊗ S(0,0) (N (c), χE ). Moreover, the morphism +,0 h·, Φγ,χE iIH : IH[L:Q] (X0 (c)) −→ S(0,0) (N (c), χE )

defined by Φγ,χE is Hecke equivariant with respect to the base change map b. In other words: ht∗ ξ, Φγ,χE iIH = hξ, Φγ,χE iIH |b(t) for any ξ ∈ IH[L:Q] (X0 (c)) and t ∈ Tc . In Theorem 1.1, DE/Q is the absolute different, the notation ξ 0 means ξ is totally positive, and Y N (c) := m2 bL/E (c ∩ OE ) p, p|c∩OE

where m2 ⊂ OE is an ideal divisible only by dyadic primes that we may take to be OE if c + 2OL = OL and bL/E is an ideal divisible only by those primes ramifying in L/E. Moreover,

1.5. SECOND MAIN THEOREM

7

q(0,0) (ξx, ξy) is a Whittaker function, normalized according to Hida’s conventions in [Hid5] which we recall in §7.3. Finally, we have set X (1.4.1) hξ, Φγ,χE iIH : = |y|AE hξ, γ(ξyDE/Q )iq(0,0) (ξx, ξy) ξ∈E × ξ0

where h , iIH is the Poincaré pairing and ξ ∈ IH[L:Q] (X0 (c)). The “full” version of Theorem 9.4 involves more general local coefficient systems on X0 (c) which in turn necessitates the introduction of higher weight Hilbert modular forms with nebentypus (see Chapter 9). Remarks. (1) There is a well-known dictionary between automorphic forms on GQ := GL2/Q and certain vectors of classical elliptic modular forms that explicitly relates the coefficients of q(0,0) (ξx, ξy) to classical Fourier coefficients (e.g. the coefficient of q in the form encountered in (1.1.4)). (2) As in [KM], the statement that Φγ,χE is an element of +,0 E (N (c), χE ) IH[L:Q] (X0 (c)) ⊗ S(0,0) E (X0 (c)), the Fourier series simply means that for any linear functional hΛ, ·i on IH[L:Q] X |y|AE hΛ, γ(ξyDE/Q )iq(0,0) (ξx, ξy) ξ∈E × ξ0

is an element of +,0 S(0,0) (N (c), χE ),

and similarly for the higher-weight analogues of Φγ,χE considered in Chapter 9. E (X0 (c)) could be an (3) The level of hΛ, Φγ,χE i for a given linear functional hΛ, ·i on IH[L:Q] ideal strictly containing N (c). Theorem 1.1 remains true if we replace IH[L:Q] (X0 (c)) with any Tc -module, as described in the “full” version, Theorem 9.4. This reflects the fact that the proof is a formal consequence of quadratic base change. 1.5. Second main theorem The second main theorem of this work uses more of the structure of the Hecke module IH∗ (X0 (c)) in order to compute the Fourier coefficients of hβ, Φγ,χE i for certain β. In order to state it, let M new,E S(0,0) (c) := Cf, f :λf (Pσ )=λf (P) a.e. σ∈Gal(L/E)

new where the sum is over all newforms f ∈ S(0,0) (K0 (c)) such that for almost all primes P ⊂ OL σ we have λf (P ) = λf (P) for all σ ∈ Gal(L/E). The theory of abelian base change implies

8

1. INTRODUCTION

new,E new (K0 (c)) spanned by forms that are base changes that S(0,0) (c) is precisely the subspace of S(0,0) from E. new,E For any subset J ⊂ Σ(L) and for any f ∈ S(0,0) (c) there is a standard differential form −ι ωJ (f ) on Y0 (c) that is antiholomorphic on (the image of) the copies of GL2 (R) associated to the places in J (under the canonical projection GL (A) → Y0 (c)) and is holomorphic on the image of the copies of GL2 (R) associated to the places in Σ(L) − J (see §8.7). (The −ι arises when translating between the two natural conventions for weights, see §7.3.) new (K0 (c)), Let Wc∗ be the Atkin-Lehner involution (see §8.4), and, for a newform f ∈ S(0,0) let W (f ) be the complex number of norm 1 defined by

Wc∗ f = W (f )fc where fc ∈

new S(0,0) (K0 (c))

is the newform satisfying a(P, f ) = a(P, fc )

for almost all primes P ⊂ OL . Denote by QχE : IH[L:Q] (X0 (c)) → IH χE (X0 (c)) the canonical projection. Finally, set [·, ·]IH∗ : IH[L:Q] (X0 (c)) × IH[L:Q] (X0 (c)) −→ C (a, b) 7−→ ha, Wc∗ biIH∗ (see §8.4). We then have the following special case of Theorem 9.5: Theorem 1.2. Let X0 (c) be the Baily-Borel Satake compactification of the Hilbert modular variety Y0 (c) corresponding to the algebraic group GL and ideal c ⊂ OL . Let Z ⊂ X0 (c) be an oriented subanalytic cycle of dimension [L : Q] (the middle dimension). Suppose its homology class [Z] ∈ H[L:Q] (X0 (c)) lifts to a class [Z] ∈ IH[L:Q] (X0 (c)). Then R (1) The integral Z ωJ (f −ι ) converges for all f ∈ S(0,0) (K0 (c)) and all J ⊂ Σ(E). Let ΦQχE [Z],χE be the formal Fourier series provided by Theorem 1.1. Let [Z](n) be the Hecke translate defined by equation (1.3.2). Suppose (1.5.1)

m + NL/E (c)dL/E = n + NL/E (c)dL/E = OL

where m, n are both norms from OL . Then (2) the mth Fourier coefficient of the product [QχE [Z](n)), ΦQχE [Z],χE ]IH∗ is R R X W Z ωJ (fb−ι ) W Z ωΣ(L)−J (fb−ι ) 1 X |J| c∗ c∗ (−1) λf (n)λf (m) [L:Q] [L:Q]−s b)(fb, fb)P 4 i 2 W ( f f J⊂Σ(E) new where the sum is over the normalized newforms f ∈ S(0,0) (K0 (cE ), χE ) for some cE ⊂ OL new,E b whose base change f to L is an element of S (c). (0,0)

1.6. EXPLICIT CYCLES

9

(3) If equation (1.5.1) does not hold then the mth Fourier coefficient is zero. Here (·, ·)P denotes the Petersson inner product (see §7.5). In the “full” version of Theorem 9.5, we also allow nontrivial local coefficient systems on Y0 (c), and we consider subanalytic cycles Z which define homology classes with local coefficients. This added generality results in a number of technical complications which we postpone addressing until Chapter 9.

1.6. Explicit cycles χE Let QχE : IH[L:Q] (X0 (c)) → IH[L:Q] (X0 (c)) be the canonical projection. Theorem 1.2 raises the question of whether or not one can exhibit cycles admitting classes in intersection homology χE that project nontrivially to IH[L:Q (X0 (c)) under QχE . This can be done, as we now explain. The finite-level Shimura subvarieties associated to the diagonal embedding H = GE ,→ GL determine cycles Z that lift to intersection homology. In the classical Hirzebruch-Zagier case of Hilbert modular surfaces, these cycles are just the image of the composition

H −→ H2 −→ SL2 (OQ(√p) )\H2 of the diagonal map with the canonical projection, and are easily shown to represent classes in intersection homology (although Hirzebruch and Zagier did not express their intersection numbers in this language). For the higher dimensional Hilbert modular varieties, the analogous fact is still true but its verification involves some subtle properties of torus weights in the local cohomology of X at a cusp. The simplest route to this result involves a theorem of L. Saper and M. Stern [SS2] and M. Rapoport [Ra], recently generalized in [S], which is reviewed in §6.9 and applied to the Hilbert modular case in Theorem 6.6. Let Pnew : IH∗ (X0 (c)) −→ IH∗new (X0 (c)) be the projection to the subspace spanned by classes that are orthogonal to the subspace of “old classes,” that is, classes that are pullbacks along any of the usual morphisms X0 (c) → X0 (c0 ) for c0 ⊃ c (see §10.3). The theory of distinguished representations can be used to show that the “new part” Pnew [Z] of the classes of the cycles [Z] mentioned above are elements of the χE subspace IH[L:Q] (X0 (c)) of IH[L:Q] (X0 (c)). Thus Theorem 1.2 is applicable to these cycles. Applying it and the computation of some Rankin Selberg integrals, we have the following theorem, which computes the integrals occuring in Theorem 1.2 in terms of special values of L-functions: Theorem 1.3. If m + NL/E (c)dL/E = n + NL/E (c)dL/E = OE and m, n are both norms from OL , then the mth Fourier coefficient of [Wc∗−1 [Z(n)], Φ[Wc∗−1 Pnew [Z]],χE ]IH∗

10

1. INTRODUCTION

is 2 dL/E c∩OE b L (Ad(f ) ⊗ η, 1)L (As( f ), 1) X dL/E

c0

f

W (fb)L∗ (Ad(fb), 1)

λf (n)λf (m)

where the sum is over the normalized newforms f on E of nebentypus η whose base change fb new (K0 (c), χtriv ). Here c0 is an explicit nonzero scalar (see Theorem to L is an element of S(0,0) 10.5) and λf (n) is the nth Hecke eigenvalue of f . Moreover Y N (c) := m2 dL/E (c ∩ OE ) p2 p|(c∩OE )

where m2 ⊂ OE is an ideal divisible only by dyadic primes, which we may take to be OE if c + 2OL = OL and dL/E is an ideal divisible only by primes ramifying in L/E. In the theorem [·, ·]IH∗ : IH[L:Q] (X0 (c)) × IH[L:Q] (X0 (c)) −→ C denotes a twist of the canonical intersection pairing by the Atkin-Lehner operator Wc (see (8.4.10)). The L-functions L(Ad(f ), s) and L(As(fb), s) will be defined in §7.7 and the base change fb of f will be defined in §7.7.4. We will show how to deduce this theorem from Theorem 1.2 and a Rankin-Selberg integral computation in Chapter 10 below.

1.7. Finding cycles dual to families of automorphic forms In the previous section we mentioned in passing that we used the theory of distinguished χE representations to prove that Pnew [Z] ∈ IH[L:Q] (X0 (c)). In this section we pause to briefly describe the theory of distinguished representations and how it can be used in certain cases to predict the existence of cycles representing classes in subspaces of the intersection homology of locally symmetric varieties. In this work we are interested in cycles on locally symmetric spaces that are defined by subsymmetric spaces. We show that these cycles are nontrivial in intersection cohomology by showing that there are differential forms representing classes in L2 -cohomology that have nonzero integrals against them. Since that L2 -cohomology of locally symmetric spaces can be described in terms of automorphic representations [BW, §XIV.3], it is natural to seek a representationtheoretic analogue of the geometric statement that a differential form on a locally symmetric space has a nonzero integral over a sub-symmetric space. Such a representation-theoretic analogue is provided by the notion of distinction, introduced by Harder, Langlands and Rapoport [HLR].

1.7. FINDING CYCLES DUAL TO FAMILIES OF AUTOMORPHIC FORMS

11

Let G0 ≤ G be a pair of reductive groups over Q. One says that an automorphic representation π of G(AQ ) is G0 -distinguished if Z (1.7.1) φ(g)dg 6= 0 G0 (F )\G0 (AF )∩0 G(R)G(AQ,f )

is convergent and nonzero for some φ in the space of π. One says that (1.7.1) is the integral of φ over G0 . Here dg is induced by a choice of Haar measure and 0 G(R) is defined as in §6.1 below. We must point out that even if π is cohomological (i.e. has nonzero (g, K)-cohomology with coefficients in some representation) just because π is distinguished does not mean that there is a nonzero intersection homology class in the πf -isotypic component of the intersection homology of some locally symmetric space attached to G. One must prove that a “cohomological vector” in the space of π has nonzero period over G0 . To see an example of what one must prove, see [AG]. In particular, in order to understand cycles on locally symmetric spaces, one is forced at some point to work at the level of automorphic forms inside an automorphic representation, or at least its associated (g, K)-module. This is one justification for the introduction of explicit spaces of automorphic forms in Chapter 7 below. If G0 is the fixed points of an involution, then Jacquet has suggested that the automorphic representations of G(AQ ) that are distinguished by G0 are precisely those automorphic representations that are functorial lifts from some other group H with (absolute) root datum determined by G and G0 [JLR]. In other words, there should be an L-map (1.7.2)

L

H −→ L G

such that an L-packet of automorphic representations on G(AQ ) contains an automorphic representation distinguished by G0 if and only if the L-packet is in the image of the putative Langlands transfer attached to (1.7.2) (see §D.4). In this language, the present manuscript is concerned with the case where G = GL and G0 = GE is the fixed points of the involution of GL induced by Gal(L/E). Jacquet’s formalism predicts that H should be a form of GE . In fact, Flicker and Rallis have suggested that it should be a unitary group attached to the extension L/E [Fl]. However, since the derived group of GL has Q-rank one, the set of cuspidal automorphic representations of GL (AQ ) that are a lift from a unitary group attached to L/E is roughly the same as those that are a lift from GE (AQ ), the point being that automorphic representations of GL are all self-dual up to a twist (see Theorem D.11 and [Rog, §11.5]). This is why it is possible for us to relate the cycles we construct to automorphic forms on GE as opposed to some unitary group. Given Jacquet’s conjectural formalism and the work in this manuscript, one is lead to the following rough conjectural generalization of the work in this manuscript: Suppose that G0 ≤ G is the fixed points of an involution, and assume that one can find a group H satisfying Jacquet’s desirata above. Suppose that G0 defines a cycle class Z in some cohomology group H(Γ\X) attached to a locally symmetric space Γ\X defined by G and an arithmetic subgroup Γ ≤ X. If there is a good enough theory of models on H, then there should be an automorphic

12

1. INTRODUCTION

form ΦG,G0 on H(AQ ) with coefficients in H(Γ\X). The automorphic form on H(AQ ) defined by hZ, ΦG,G0 iH should have coefficients given in terms of special values of L-functions. Here by a “good enough theory of models” on H we mean something that can take the place of the Whittaker models which provide the Fourier coefficients we use in this book to make sense out of the notion of the coefficient of an automorphic form. Incidentally, making this notion of coefficients precise in the G = GL case is another justification for introducing explicit spaces of automorphic forms in Chapter 7 below. 1.8. Comments on related literature As a special case of their far-reaching study of of explicit subvarieties of locally symmetric spaces of orthogonal and unitary type, Kudla and Millson (generalizing previous work of Oda) produce a generating series with coefficients in H[L:Q] (X0 (c)) out of cycles whose irreducible components are birational to the components of (possibly non-compact) Shimura curves attached to quaternion algebras over E split by L ([KM]). They then go on to prove that this generating series is a modular form with coefficients in a cohomology group in the sense of part (1) of the comment following Theorem 1.2. There are (at least) three fundamental differences between this special case of the theory in [KM] and our Theorem 9.4. First, as opposed to one Hilbert modular form (with coefficients in cohomology), we obtain a family of modular forms (with coefficients in intersection homology), one for each local system E on X0 (c) and each intersection homology class in IH[L:Q] (X0 (c), E). Second, whereas [KM] consider the intersection product of their generating series with a cycle that is compactly supported in Y0 (c), we consider naturally occuring compact and noncompact cycles (e.g. noncompact subvarieties birational to Hilbert modular varieties associated to E), which are lifted to intersection homology (see Theorem 6.6). Our construction, however, does not require the sophisticated intersection theory [TT] used by Tong in his study [To] of weighted intersection numbers on Hilbert modular surfaces. Finally, the method of proof in [KM] differs from ours, in that we use quadratic base change as a tool to produce our results, whereas the cases of quadratic base change that Kudla and Millson require are incorporated into their arguments using theta liftings. It would be interesting to see if the formal arguments we use to prove Theorem 1.1 in this manuscript could be modified and extended to prove similar theorems in the context of other liftings of automorphic forms, especially when the lifting is not a theta lifting. A relatively simple case of such an extension is provided in the remark after Proposition 9.2, when the automorphic lifting is the GL2 base change associated to a prime-degree Galois extension of totally real fields.

1.9. COMPARISON WITH ZAGIER’S FORMULA

13

In [BL], J. L. Brylinski and J. P. Labesse completely determine the automorphic description of intersection cohomology for Hilbert modular varieties. They compute the associated automorphic L function and show that it coincides with the L function for the Frobenius action on the associated Shimura variety, as predicted by R. Langlands. Although the focus of their article differs from that of ours, there is some overlap. Period integrals on Hilbert modular surfaces are also considered in [Od2]. Hilbert modular varieties are, in a natural way, moduli spaces of Abelian varieties with real multiplication. This point of view, and its relation to Hilbert modular forms, is developed in [Gn]. 1.9. Comparison with Zagier’s formula Theorem 1.2 provides a generalization of a formula for a generating series ΦHZ given by Zagier [Za]. In order to state his formula, denote by fb the Naganuma lift of an f ∈ S2 (Γ0 (p), ( p· )) (here we have used the classical normalization of the weight). Assume for simplicity that the √ narrow class number of Q( p) is 1. Then, denoting by [Zm ]m≥0 the family of classes introduced by Hirzebruch and Zagier in [HZ], Zagier proved that the intersection product h[Zm ], ΦHZ iH is given by:     R 2 b ∞ 0 X X Z1 η(f )    r0  π+ t(m)E2,p (z) − af (m)af (n) q n  . (fb, fb) n=1 Here the prime indicates summation over a basis of normalized newforms (i.e. eigenforms for all the Hecke operators) ∞ X f (z) := af (n)q n n=1

S2 (Γ0 (p), ( p· )),

in π+ is the canonical projection to S2+ (Γ0 (p), ( p· )), the rational number t(m) depends only on m, the nonzero complex number r0 is a certain explicit constant, and E2,p is a weight two Eisenstein series in S2+ (Γ0 (p), ( p· )) (see [Za, (98-99)]). Moreover η(fb) is a certain differential (1, 1)-form attached to fb, and we used the proof of Oda’s period relation for the Naganuma lifting (see [G, p. 154 (7.9)], [Od3] and [Od4]) to modify Zagier’s expression. Let Z 0 be the (open) modular subvariety of Y0 (OQ(√p) ) given by the image of the diagonal embedding: GE (A) ,→ GL (A) −→ Y0 (OQ(√p) ). Here the first map is the diagonal embedding and the second map is the canonical projection. Denote by Z the closure of Z 0 in X0 (OQ(√p) ). Let π : IH2 (X0 (OQ(√p) ) −→ IH2 (X0 (OQ(√p) )) be the projection onto the subspace spanned by the Poincarè duals of differential forms associated to elements of S(0,0) (K0 (OQ(√p) )) (see §8.7) that are base changes of elements of

14

1. INTRODUCTION

× S(0,0) (K0 (pZ), χQ ). Here χQ : A× is the nontrivial character trivial on the image of Q → C √ √ NQ( p)/Q . Choose an embedding σ : Q( p) ,→ R. Then

h[Z]χQ (mZ), Φπ[Z],χQ iIH∗ = |y|AQ

∞ X n=1 (n,p)=1, χQ (n)=1

  R 2 b 0 X Z ω{σ} (f )  a(mZ, f )a(nZ, f ) q(0,0) (nx, ny) r00  (fb, fb)

for some explicit constant r00 ∈ C× , where the sum is over a basis of normalized newforms f ∈ S(0,0) (K0 (pZ), χQ ). (See equation (7.6.5) for the equality between the Hecke eigenvalue λf (nZ) and the Fourier coefficient a(mZ, f ).) In order to obtain this expression, we used the fact that WOL ∗ is the identity for any totally real field L, along with Corollary 4.10 and Proposition 7.9 of [G, Chapter VI]. 1.10. Outline of the book The goal of chapters 2 to 6 is to review the construction and basic properties of the integral of a differential form ω over a cycle ξ. This “standard” material is known to experts, but, to our knowledge, it is not recorded anywhere in the form that we require. In particular, we will need to make use of this formalism in the following context: • The differential form ω is defined on an orbifold Y , rather than on a smooth manifold. • The differential form ω and the chain ξ take values in local coefficient systems E1 and E2 . • The orbifold Y may be the largest stratum of a stratified space X, and the local systems E1 , E2 may fail to extend to all of X. • The cycle ξ represents a class in the intersection homology of X, rather than in ordinary homology. We will also need to know that such an integral can be interpreted using various homologically defined products such as the “Kronecker” pairing between (intersection) homology and (intersection) cohomology, the cup product on cohomology, and the intersection product on intersection homology; and that these various products are compatible with each other whenever there are natural identifications among the different homology and cohomology groups. Although there are no essential difficulties in constructing such an integral, and in describing it homologically, a complete proof of any statement to this effect necessarily involves chain-level operations. These in turn involve geometric properties of the cycle ξ (such as a sub-analytic structure, or triangulability) and analytic properties (such as growth rates) of the differential form. We do not know of any published literature that specifically deals with these (relatively straight forward) issues, so we have included the relevant details in the early chapters: in chapters 2 (chains and cochains), 3 (intersection homology and cohomology), 4 (integration of differential forms), 5 (finite mappings and correspondences), and 6 (arithmetic

1.11. PROBLEMATIC PRIMES

15

quotients). We have also included Appendix B on the definition and basic properties of orbifolds, which fills in some of the technical details that are not easily extracted from the standard references. Then in Chapter 7 we review relevant facts from the theory of Hilbert modular varieties and Hilbert modular forms. This is in preparation for Chapter 8, where we recall the well-known description of the intersection cohomology of the Hilbert modular varieties X0 (c) in terms of Hilbert modular forms. After this preparatory material, we move on to the core of the second half of this paper, namely the proofs of Theorem 1.1, Theorem 1.2, and Theorem 1.3 (see §9.3, 9.4, and 10.3, respectively). Theorem 1.1 is actually implied by the more general Theorem 9.3 where intersection homology is replaced by an arbitrary Hecke module (see §9.3 for details). As indicated above, Theorem 9.3 relies crucially on the theory of quadratic base change for GL2 ; thus we have included a synopsis of prime degree base change for GL2 in Appendix D. Theorem 1.3 relies on Theorem 1.2 and a Rankin-Selberg computation that is contained in Chapter 10. Finally, in Chapter 11, we prove an analogue of Theorem 1.1 and Theorem 1.2 with cuspidal classes replaced by invariant classes (see §8.7 for the definition of a cuspidal and invariant class). In particular, we prove that suitable generating series created out of invariant classes are Eisenstein series with coefficients in intersection homology. As the discussion above indicates, this paper touches on a wide range of topics from the topology and number theory of Q-rank one hermitian symmetric spaces. However, chapters 2 to 6 can be read independently of the rest of this work (though their content and structure reflect the requirements of the later chapters). Similarly, chapters 7 through 11 only use results from chapters 2 through 6 that one can reasonably take to be a “black box.” Appendices B and D are also self-contained. We will close the introduction with a short section giving some comments on the subtle problem of determining the minimal level of a Hilbert modular form whose base change is of a given level.

1.11. Problematic primes The mth Fourier coefficient of the Hilbert modular form with coefficients Φγ,χE that was constructed in Theorem 1.1 is zero if m + dL/E (c ∩ OE ) 6= OE . By placing more assumptions on the level and the character, it is possible to produce an analogue of Φγ,χE (satisfying an analogue of Theorem 1.1) that may have nonzero mth Fourier coefficients for m + dL/E (c ∩ OE ) 6= OE . Proving a theorem along these lines would either require substantial hypotheses on the local admissible representations involved, or a substantial digression on local representation theory. For brevity, we have not attempted to do either in this work. However, we hope that Appendix D might be a useful starting point for further investigations in this direction.

16

1. INTRODUCTION

1.12. Acknowledgements The first author would like to thank Tonghai Yang for answering many questions on automorphic forms and representations and Ken Ono for his constant support. The authors are grateful to the Institute for Advanced Study for its support. The research of Getz was partially supported by the ARO through the NDSEG Fellowship program. The research of Goresky was partially supported by DARPA through grant numbers HR0011-04-1-0031 and HR0011-09-10010. Commutative diagrams were typeset using Paul Taylor’s diagrams package. The body of the book was typeset with amslatex.

CHAPTER 2

Review of Chains and Cochains 2.1. Cell complexes and orientations Recall (e.g. [Hu]) that a closed convex linear cell is the convex hull of finitely many points in Euclidean space. A convex linear cell complex K is a finite collection of closed convex linear cells in some RN such that if σ ∈ K then every face of σ is in K, and if σ, τ ∈ K then the intersection σ ∩ τ is in K. The underlying closed subset of Euclidean space is denoted |K|. Such a complex is a regular cell complex, meaning that each (closed) cell is homeomorphic to a closed ball: no identifications occur on its boundary. If τ ∈ K is a face of σ ∈ K we write τ < σ. A finite simplicial complex is a convex linear cell complex, all of whose cells are simplices. Every cell complex admits a simplicial refinement with no extra vertices. An orientation of a finite dimensional real vector space V is a choice of ordered basis, two being considered equivalent if one can be continuously deformed to the other, through ordered bases. Every (finite dimensional real) vector space has two orientations. An orientation of a convex linear cell is an orientation of the real affine space that it spans. An orientation of a smooth manifold is a continuously varying choice of orientation of each of its tangent spaces. Let K be a convex linear cell complex and let L be a (closed) subcomplex. Let X = |K| and let Y = |K| − |L|. Although Y is not a union of cells, it is a union of interiors of cells. We refer to this decomposition of Y as a pseudo cell decomposition (or a pseudo-triangulation if K is a simplicial complex). Every cell σ ∈ K has two orientations. A choice of orientation for σ determines a unique orientation for each codimension 1 face τ < σ such that the orientation of τ followed by the inward pointing vector − τ→ σ agrees with the orientation of σ. The complex K is purely d dimensional if every cell is the face of some d dimensional cell and there are no cells of dimension greater than d. A oriented cellular pseudomanifold is a convex linear cell complex K, purely of some dimension d, such that every d − 1 dimensional cell is a face of exactly two d dimensional cells; together with a choice of orientation of each d dimensional cell such that the induced orientations cancel on every d − 1 dimensional cell.

17

18

2. REVIEW OF CHAINS AND COCHAINS

2.2. Subanalytic sets and stratifications Let F = semi-algebraic, semi-analytic, or subanalytic. Any finite union, intersection, or difference of F-subsets of RN is again an F-subset of RN . The closure and the interior of any Fsubset of RN is again an F-subset of RN . The image of an F-subset X ⊂ RN by an F-mapping f : RN → Rk is again an F-set for F= semi-algebraic or subanalytic (but this last statement is false for F= semi-analytic). N Let X ⊂ RS be a set of type F. A F-Whitney stratification of X is a locally finite decomposition X = α Xα into disjoint real analytic manifolds or strata, such that • the closure Xα of Xα is an F-subset of RN • if Xα ∩ Xβ 6= φ then Xα ⊂ Xβ . and the pair (Xα , Xβ ) satisfies Whitney’s conditions A and B. A Whitney stratification, if it exists, implies that the local topological type of X is locally constant along each stratum S in the following sense. Assume S is connected (or else relabel the strata by their connected components). R. Thom [Th] and J. Mather [Ma] proved the following: Theorem 2.1. There exists a compact stratified space ` (the link of the stratum S) such that every point x ∈ S has a neighborhood basis in X consisting of neighborhoods Nx homeomorphic to Rs × cone(`) by a stratum-preserving homeomorphism that is smooth on each stratum and takes Rs × {pt} to Nx ∩ S. (Here, s = dim(S) and {pt} denotes the cone point.) Such a neighborhood is called a basic neighborhood. The Thom-Mather theorem implies, in particular, that the local homology Hi (X, X − x; Z) of X is finitely generated at every point x ∈ X. An F-triangulation of a closed set X ⊂ RN of type F is a locally finite simplicial complex K with |K| ⊂ RN together with an F-isomorphism f : RN → RN such that f (|K|) = X and • For each simplex σ ∈ K the restriction f |σ o of f to its interior is a real analytic isomorphism σ o → f (σ o ). • Each f (σ) is a (closed) set of type F in RN . Any two F-triangulations of X have a common refinement. An F-triangulation f of a closed S F-set X ⊂ RN is compatible with an F-Whitney stratification X = α Xα if, for each α the set f −1 (Xα ) is a (closed) subcomplex of K. If X is F-Whitney stratified and F-triangulated by a compatible triangulation, and if x is a point in some stratum S ⊂ X then the link Lx of x (in the sense of P.L. topology) is homeomorphic (2.2.1)

Lx ∼ = Σs `

to the s = dim(S)-fold suspension of the link ` of the stratum S.

2.3. SHEAVES AND THE DERIVED CATEGORY

19

Theorem 2.2. Let Z ⊂ X ⊂ RN be closed subsets of type F. Then X admits an F-Whitney stratification such that Z is a union of strata. Given any F-Whitney stratification of X there exists an F-triangulation that is subordinate to the stratification. The proof of this result has a long history. We list here a few of the important references: [Lo1], [Lo2], [Ha1], [Ha2], [Ha3], [Ha4], [Ha5], [Ha6], [Ha7], [Hir1], [Hir2], [Hir3], [Hir4], [Hir5], [Go], [Jo]. A closed F-subset X ⊂ RN is purely d dimensional if there exists an F-stratification of X such that X is the closure of the union of all of its d-dimensional strata. Then X is an oriented pseudomanifold if there exists a (simplicial) oriented pseudomanifold K of pure dimension d and an F -triangulation f : |K| → X. For Whitney stratified sets, an oriented pseudomanifold structure may be described without reference to a triangulation. Let X be a purely d-dimensional F-set. Then X is a pseudomanifold if it can be Whitney stratified with no strata of dimension d−1. In this case an orientation of X is determined by a choice of orientation of each of the d dimensional strata. 2.3. Sheaves and the derived category Let X be a real or complex algebraic, analytic, semi-analytic or sub-analytic set. Then X is locally compact, Hausdorff, and is homeomorphic to a (locally finite) simplicial complex. Throughout this section we fix a regular, commutative, Noetherian ring R (with unit) of finite cohomological dimension. (A principal ideal domain, for example, is such a ring.) Recall that a complex of sheaves of R-modules S• on X is a collection of sheaves Si and differentials di : Si → Si+1 . The associated cohomology sheaf of degree i is Hi (S• ) = ker di / Im di−1 . If each Si is fine, flabby, soft, or injective, then the cohomology H ∗ (X, S• ) (resp. cohomology with compact supports Hc∗ (X, S• )) is given by the cohomology of the complex of global sections (resp. global sections with compact supports). It is customary to denote by S• [n] the shift of S• by k, that is, (S• [n])k = Sn+k . A morphism S• → T• is a quasi-isomorphism if it induces an isomorphism on the associated cohomology sheaves. In this case, the complex T • is called an injective resolution of S• if each Tj is injective (in the category of sheaves of R modules). A complex of sheaves S• is cohomologically locally constant (CLC) if each of the cohomology sheaves Hi (S• ) is locally constant. The complex S• is cohomologically constructible with respect to a given stratification of X if each of the cohomology sheaves Hi (S• ) is locally constant on each stratum. Let Dcb (X) denote the bounded constructible derived category: its objects consist of complexes of sheaves that are bounded from below and are cohomologically constructible with respect to some Whitney F -stratification of X. In this category, every quasiisomorphism is invertible. See, for example, [I], [GeM], [GM2]. Many functors F defined on the category Sh(X) of sheaves on X pass to derived functors RF. In particular, we shall use the standard notations Rf∗ , Rf! , f ∗ , f ! for the derived push-forward, derived push-forward with proper supports, the pull-back and the extraordinary pull-back on sheaves. If S• , T• are

20


complexes of sheaves on X then RHom• (S• , T• ) denotes the complex of sheaves that is obtained from the double complex of pre-sheaves which associates to any open subset j : U ⊂ X the R module Hom(j ∗ Sp , j ∗ Iq ) where T • → I • is an injective resolution of T• . In this case HomDcb (X) (S• , T• ) = H 0 (X; RHom• (S• , T• ). If S• is cohomologically constructible then it follows from the Thom-Mather theorem (§2.2) that the stalk cohomology (or “local cohomology”) H i (jx∗ S• ) = Hxi (S• ) = Hi (S• )x (of S• at the point x ∈ X) coincides with the cohomology H i (Ux , S• ) of any basic neighborhood Ux of x in X. Here jx : {x} → X denotes the inclusion. Similarly H i (Ux , S• ) ∼ = H i (j ! (S• )) x

c

is the stalk cohomology with compact supports. 2.4. The sheaf of chains Let E be a local coefficient system (= locally constant sheaf) of R modules on a set X ⊂ Rn of type F. There are many quasi-isomorphic versions of the sheaf C• (X, E) of chains on X. We briefly recall the construction of the sheaf of F-chains. Let T be a (locally finite) F-triangulation of X. For each simplex σ of T the restriction of E to σ has a canonical trivialisation, so we may unambiguously refer to the fiber Eσ . An i-dimensional (T -simplicial) Borel-MoorePchain with coefficients in E is a (locally finite) linear combination of oriented simplices ξ = t et σt with et ∈ Eσt whose support |ξ| is closed in X; we identify et σt with −et σt0 where σ 0 is the same simplex as σ but with the opposite orientation. The collection of all Borel-Moore i-chains with respect to the F-triangulation T forms an R module CiBM,T (X, E) and the usual boundary map gives a homomorphism BM,T ∂i : CiBM,T (X, E) → Ci−1 (X, E). If T 0 is a refinement of the triangulation T then the natural 0 homomorphism CiBM,T (X, E) → CiBM,T (X, E) induces an isomorphism BM,T 0 H BM,T (X, E) ∼ (X, E) =H i

i

on homology. Define the complex of (Borel-Moore) F-chains CiBM (X, E) = lim CiBM,T (X, E) T

to be the limit over all F-triangulations of X. The Borel-Moore chains then form a pre-sheaf (with respect to the open subsets of type F). For if U ⊂ V are open F-subsets of X and if T is a triangulation of V then it is possible to find a triangulation T 0 of U such that each simplex of T 0 is contained in a unique simplex of T. This procedure gives a homomorphism CiBM (U, E) → CiBM (V, E). The sheaf of F-chains C• (X, E) on X is the complex of sheaves whose R module of sections over an open set U ⊂ X is Γ U, C−i (X, E) = CiBM (U, E)

2.5. HOMOLOGY MANIFOLDS

21

with d−i = ∂i (for i ≥ 0). (It is placed in negative degrees so that the differentials raise degree.) The sheaf C• (X, E) is soft ([H] §II.5), so the sheaf cohomology over any open set U ⊂ X can be obtained as the cohomology of the complex of sections over U . With this in mind, the Borel-Moore homology is defined by HiBM (U, E) := H −i (U, C• (X, E)). The complex of (compact) F-chains on U ⊂ X is the complex Ci (U, E) = Γc (U, C−i (X, E)) of sections with compact support. The (local) homology sheaf H−i (C• (X, E)) is the cohomology sheaf of the sheaf of chains. It is a topological invariant and its stalk cohomology is the local homology, that is, Hx−i (C• (X, E)) = Hi (X, X − x; E). The cohomology with compact support Hc−i (X, C• (X, E)) is the ordinary homology Hi (X, E). A similar construction [Br] may be made with singular chains, and the resulting complex of sheaves (which is a topological invariant and does not depend on a choice of piecewise linear structure) is canonically quasi-isomorphic to the sheaf of F-chains. We will sometimes refer to “the” sheaf of chains C• (X, E) without reference to a particular PL or analytic structure on X. If R is a field and if E = R is the constant local system then the sheaf of chains on X is called the dualizing sheaf (with coefficients in R) and it is denoted D•X . (For an arbitrary regular Noetherian ring R of finite cohomological dimension, the dualizing sheaf is obtained from the sheaf of chains by P tensoring with an injective resolution of R [Bo2, §7.A].) We remark that if ξ = t at σt ∈ CiBM (U, R) is a chain with constant coefficients, and if s ∈ Γ(|ξ|, of E over the support of ξ then we obtain, in a natural way a chain P E) is a section BM sξ = t at s(σt )σt ∈ Ci (U, E). 2.5. Homology manifolds As in the previous section, we assume the coefficient ring R is a regular Noetherian ring of finite cohomological dimension, and we let F refer to semi-algebraic, semi-analytic, or subanalytic. Let Y be a purely n dimensional set of type F (so Y is contained in some Euclidean space and its closure is also an n-dimensional set of type F ). The set Y is an R-homology manifold if ( 0 if j 6= n Hj (Y, Y − y; R) = R if j = n or, equivalently, if the local homology sheaf H−j (C• (Y, R)) is a local system of rank 1 for j = n and vanishes for j 6= n. Assume Y is an R-homology manifold. The R-orientation sheaf OY = H−n (C• (Y, R)).

22


is the local system whose fiber at each point y ∈ Y is Hn (Y, Y − y; R). If an orientation of Y exists (§2.1) then it determines an isomorphism between OY and the trivial local system R. If Y is also connected (but not necessarily compact) then HnBM (Y, OY ) ∼ = R. A choice of generator [Y ] of this group is called a fundamental class. It can be represented by a (Borel-Moore) chain ξ ∈ CnBM (Y, OY ) whose support is the union of all the n-dimensional simplices in a triangulation of Y, that is, |ξ| = Y. There is a canonical quasi-isomorphism P : OY → C• (Y, R)[−n] which assigns to each sufficiently small open F-ball U ⊂ Y the chain [U ] ∈ Γ (U, C−n (Y, OY )) . It induces a quasi-isomorphism OY ⊗ E → C• (Y, E)[−n]

(2.5.1)

for any finite dimensional local system E of R modules on Y. The resulting isomorphisms of cohomology groups are often referred to as Poincaré duality isomorphisms, BM (Y, E) H i (Y, OY ⊗ E) ∼ = Hn−i i H (Y, OY ⊗ E) ∼ = Hn−i (Y, E). c

The morphism P may also be viewed as a quasi-isomorphism P : E → C• (Y, OY ⊗ E)[−n] with resulting Poincaré duality isomorphisms BM H i (Y, E) ∼ (Y, OY ⊗ E) = Hn−i i Hc (Y, E) ∼ = Hn−i (Y, OX ⊗ E).

2.6. Cellular Borel-Moore Chains In §4.4 of this article we will need to integrate differential forms (defined on the noncompact top stratum Y of a modular variety X) over chains (which are themselves noncompact), with coefficients in local systems on Y that may not extend over its compactification X. Integration of non-compact chains on non-compact manifolds leads to a host of potential pathological difficulties, none of which (fortunately) occur in the setting of modular cycles on modular varieties. The purpose of this section is to provide a few standard but not previously easily referenceable technical tools which will be used to guarantee that the integrals we will eventually consider are well-behaved. The main point is that the manifold Y and the modular cycles in Y are compactifiable. Let X be a set of type F (= semi-algebraic, semi-analytic, or subanalytic) and let f : |K| → X be an F -triangulation of X, where K is a locally finite simplicial complex. Let L ⊂ K be a closed subcomplex such that the open set Y = f (|K| − |L|) is dense in X. The resulting decomposition of Y is a pseudo cell decomposition in the sense of §2.1. Let E be a local coefficient system of R modules on Y. If σ is a cell of K whose interior σ o is contained in Y, then the fibers Ex , Ey of E over any two points x, y ∈ σ ∩ Y are canonically isomorphic. Therefore we may refer unambiguously to the fiber Eσ . If τ < σ and τ o ⊂ Y then there is a canonical isomorphism Φστ : Eσ → Eτ .

2.6. CELLULAR BOREL-MOORE CHAINS

23

An r dimensional elementary Borel-Moore cellular chain (on Y with coefficients in E) is an equivalence class of formal products aσ σ where σ is an oriented r dimensional cell of K such that σ o ⊂ Y and where aσ ∈ Eσ ; modulo the identification aσ σ ∼ (−aσ )σ 0 where σ 0 is the same cell but with the opposite orientation. The boundary ∂aσ σ of an elementary r dimensional chain is defined to be X ∂aσ σ = Φστ (aσ )τ τ

where the sum is taken over those r − 1 dimensional faces τ < σ such that τ o ⊂ Y. brK (Y, E) (with respect to the pseudo-cell The R-module of cellular Borel-Moore chains C decomposition K) is the module of finite formal linear combinations of elementary r chains. P brK (Y, E) be a cellular Borel-Moore chain. Its support |ξ| is the intersection Let ξ = i ai σi ∈ C of Y with the union of those cells σi such that ai 6= 0. If K 0 is a (finite) refinement of K brK (Y, E) → C brK 0 (Y, E) which preserves (and we write K 0 < K) there is a canonical injection C supports. The proof of the following Lemma will appear in Appendix A below. Lemma 2.3. If K 0 is a finite refinement of K then the induced mapping on homology b K (Y, E) → H b K 0 (Y, E) H r

r

is an isomorphism. Now let T be a (piecewise linear) triangulation of Y that is subordinate to K, that is, a triangulation such that every (closed) simplex in T is contained in a (closed) cell of K as a convex linear subset. (If Y is not compact then T will consist of infinitely many simplices.) Then we obtain a canonical injection brK (Y, E) → CrBM (Y, E). (2.6.1) C Proposition 2.4. The mapping (2.6.1) induces an isomorphism on homology, b rK (Y, E) → HrBM,T (Y, E) ∼ H = H BM (Y, E). r

In summary, any pseudo cell decomposition of Y may be used to compute its Borel-Moore homology. The proof, which is standard but surprisingly messy, is in Appendix A below.

CHAPTER 3

Review of Intersection Homology and Cohomology In this chapter we recall the relation between intersection homology, constructed using (p, i)-allowable chains as in [GM1], and intersection cohomology, constructed via sheaf theory.

3.1. The sheaf of intersection chains In this section we suppose X is a (piecewise linear or subanalytic) purely n dimensional stratified pseudomanifold ([GM1, GM2, GM3]), but see also §3.3. As in Chapter 2, we denote by R a regular, commutative, Noetherian ring (with unit) of finite cohomological dimension; for example, any principal ideal domain. Let p : N → N be a perversity, that is, a mapping such that p(0) = p(1) = p(2) = 0 and p(c) ≤ p(c + 1) ≤ p(c) + 1 for all c. Let E be a local coefficient system (of R modules) on the nonsingular part Y ⊂ X of X. Let i ≥ 0 be an integer. The pre-sheaf Ip C−i (E) of (subanalytic or piecewise linear) intersection chains assigns to each open set U ⊂ X the subgroup I p CiBM (U, E) of all (subanalytic or piecewise linear) BorelMoore chains ξ in U (= chains with closed support in U ) with coefficients in E which satisfy the following (p, i)-allowability condition: (3.1.1) (3.1.2)

dim(ξ ∩ S) ≤ i − cod(S) + p(cod(S)), dim(∂ξ ∩ S) ≤ i − 1 − cod(S) + p(cod(S))

for each singular stratum S ⊂ X. The allowability condition guarantees that all the i-dimensional and all the i − 1-dimensional simplices in ξ are contained in the nonsingular part Y , where the local system E is defined. This pre-sheaf is in fact a soft sheaf ([H] §II.5) so for any open set U ⊂ X the sheaf cohomology of U may be obtained as the cohomology of the complex −i of sections over U. It is denoted I p HiBM (U, E) or I p Hclosed (U, E) and it is referred to as the intersection homology with closed supports of U with coefficients in E. The intersection homology with compact supports of U is the homology of the complex I p C∗ (U, E) of sections with compact support, that is, the chains ξ in U with coefficients in E satisfying (3.1.1) and (3.1.2) such that |ξ| is compact. It is denoted I p Hi (U, E). Clearly I p Hi (X, E) = I p HiBM (X, E) if X is compact. 24

3.2. THE SHEAF OF INTERSECTION COCHAINS

25

The stalk cohomology of the sheaf Ip C• (E) at a point x in some stratum S of X is given by (3.1.3)

( 0 if − i ≥ p(c) − n + 1 Hx−i (Ip C• (E)) = I p Hi (X, X − x, E) = p I Hi−s−1 (`x , E) if − i ≤ p(c) − n.

Here, s = dim(S), c = n − s is the codimension of S and `x is the link of the stratum S at the point x. The intersection homology is a topological invariant of (X, E) and it does not depend on a choice of subanalytic or PL structure, or the choice of stratification. 3.2. The sheaf of intersection cochains As in the previous section we suppose X is a (real) n-dimensional (PL or subanalytic, not necessarily compact) pseudomanifold. If E is a local system of R modules on the nonsingular part Y of X then Deligne [GM2] has given an alternate construction of the sheaf of intersection cochains as an element of the constructible derived category Dcb (X) of sheaves on X. Given a stratification of X let Xk denote the closed subset of X consisting of all strata of dimension less than or equal to k. We follow the indexing scheme of [Bo2], starting with the local system E in degree 0 and setting Ip S• (E) = τ≤p(n) Rin∗ · · · τ≤p(3) Ri3∗ τ≤p(2) Ri2∗ (E) where the truncation functor τ≤k kills all stalk cohomology in degrees greater than k and where ik is the inclusion of Uk = X − Xn−k into Uk+1 . The cohomology of Ip S• (E) is the intersection cohomology of X, H i (X, Ip S• (E)) = I p H i (X, E). If S ⊂ X is a stratum of codimension c and if x ∈ S then the stalk cohomology Hxi (Ip S• (E)) at x of this intersection cohomology sheaf is ( 0 if i > p(c) i ∗ p • (3.2.1) H (jx I S (E)) = p i I H (`x , E) if i ≤ p(c) where `x is the link of the stratum S at the point x, cf. §2.2 and where jx : {x} → X denotes the inclusion. The stalk cohomology with compact support at x is ( I p H i−1−n+c (`x , E) if i > p(c) + 1 + n − c (3.2.2) H i jx! Ip S• (E) = 0 if i ≤ p(c) + 1 + n − c. Let T• be a cohomologically constructible complex of sheaves on X. This means that for each k, the (local) cohomology sheaves of the restriction T•k = T• |(Xn−k − Xn−k−1 ) are finite dimensional local systems (i.e., locally constant sheaves). Fix a perversity p. Consider the following possible conditions: (a) H m (jx∗ T• ) = 0 for all m > p(k), all x ∈ Xn−k − Xn−k−1 , and all k. (b) H m (jx! T• ) = 0 for all m < p(k) + n − k, all x ∈ Xn−k − Xn−k−1 , and all k.

26

3. REVIEW OF INTERSECTION HOMOLOGY AND COHOMOLOGY

The following lemma may be proven using the same argument as in [GM2] §3.5, or using §1.3.4 of [BBD]: Lemma 3.1. Let E be a local system on Y. If T• satisfies condition (a) above then any quasiisomorphism T• |Y → E has a unique extension in Dcb (X) to a morphism T• → Ip S• (X, E). If T• satisfies condition (b) above then any quasi-isomorphism E → T• |Y has a unique extension in Dcb (X) to a morphism Ip S• (X, E) → T• . Consequently the quasi-isomorphism of equation (2.5.1) extends (uniquely) to Poincaré duality quasi-isomorphisms (3.2.3) (3.2.4)

P : Ip S• (E ⊗ OY ) → Ip C• (E)[−n] P : Ip S• (E) → Ip C• (OY ⊗ E)[−n]

which determine isomorphisms BM I p H i (X, E ⊗ OY ) ∼ (X, E) = I p Hn−i p i p BM I H (X, E) ∼ = I H (X, E ⊗ OY ) n−i

I

p

Hci (X, E ⊗ OY ) I p Hci (X, E)

∼ = I Hn−i (X, E) ∼ = I p Hn−i (X, E ⊗ OY ). p

3.3. Homological stratifications We will eventually be concerned with the Baily Borel compactification X of a modular variety Y that is obtained as an arithmetic quotient Y = Γ\D of a symmetric domain D by an arithmetic group Γ that is not necessarily torsion free. Such a space has a canonical “stratification” by boundary strata (cf. [BB]), but the strata are not necessarily smooth manifolds; rather they are orbifolds (see Appendix B), and hence they are rational homology manifolds. Thus one is led to consider orbifold stratifications and more generally, homological stratifications, in which each stratum is a homology manifold, along which the space is locally homologically a product. Homological stratifications were defined and shown to exist in [GM2] §4.2 where they were used to prove that intersection homology is a topological invariant. See also [RS]. The requirements on a stratification vary with the application, and there are many variants on the notion of a homological stratification. Let X be an n dimensional pseudomanifold of type F (= semi-algebraic, semi-analytic or subanalytic). Let Xn−2 be a closed subset of dimension ≤ n − 2 such that X − Xn−2 is an R-homology manifold. Let E be a local coefficient system of R modules on X − Xn−2 . By assumption, there exists a stratification of X such that the largest stratum Y ⊂ X is contained in X − Xn−2 . Therefore the intersection complex Ip S• (E) is well-defined on X.

3.3. HOMOLOGICAL STRATIFICATIONS

27

Definition 3.3.1. A homological stratification of X (depending on R, p, and E) is a filtration by closed F -subsets X = Xn ⊃ Xn−2 ⊃ Xn−3 ⊃ · · · ⊃ X−1 = φ

(3.3.1)

such that (1) each Xj − Xj−1 is a j-dimensional R-homology manifold, (2) the intersection complex Ip S• (E) is cohomologically locally constant (CLC) on each stratum Xj − Xj−1 (meaning that each of its cohomology sheaves are locally constant on each stratum), (3) for each j ≥ 2 the complex Rhj∗ Ip S• (E) is cohomologically locally constant on the stratum Xj − Xj−1 . Here, hj : X − Xj−1 → X − Xj denotes the inclusion. Similarly let us say that a filtration (3.3.1) is an R-orbifold stratification if each Xj − Xj−1 is a j-dimensional R-orbifold along which X is locally topologically trivial in the sense of the Thom-Mather theorem (2.2) (where ` is assumed to have an R-orbifold stratification rather than a Whitney stratification). Proposition 3.2. Let X be an F -set with a filtration (3.3.1) by closed F -subsets. If this filtration is a Whitney stratification then it is also a Q-orbifold stratification (with the trivial orbifold structure on each stratum). If it is a R-orbifold stratification then it is also a Rhomological stratification for any local system E of R-modules and any perversity p. If it is an R-homological stratification then there exists a refinement that is a Whitney stratification. In other words, Whitney =⇒ R-orbifold =⇒ R-homological =⇒ Whitney refinement. Proposition 3.3. Let S = {X = Xn ⊃ Xn−2 ⊃ · · · ⊃ X−1 = φ} be a homological stratification (for R, p, and E) as defined above. Let •

IpS S (E) = τ≤p(n) Rin∗ · · · τ≤p(3) Ri3∗ τ≤p(2) Ri2∗ (E) be the complex of sheaves obtained from Deligne’s construction with respect to the homological • stratification S. Let IpS C (E) be the complex of sheaves of (Borel-Moore) chains that satisfy the allowability conditions (3.1.1) with respect to the homological stratification S. Then the identity mapping E → E extends, uniquely in Db (X), to quasi-isomorphisms • (1) Ip S• (E) → IpS S (E) and • p • (2) Ip C (E) → IS C (E). In other words, a homological stratification may be used in place of an honest stratification for either of these constructions of intersection (co)homology. Proof. Statement (1) is proven in [GM2] §4.2. Now let us prove statement (2). Let S 0 be an honest stratification that refines the homological stratification S. Then every stratum

28


A0 ∈ S 0 is contained in a unique stratum A ∈ S. Let p+ be the stratum-dependent perversity that assigns to any such stratum A0 ∈ S 0 the number p+ (A0 ) = p(A) + dim(A) − dim(A0 ). Let p− be the stratum-dependent perversity that assigns to any such stratum A0 ∈ S 0 the number p− (A0 ) = p(A). (Here we have written p(A) rather than p(cod(A)) for simplicity.) The + • sheaves IpS C (E) and IpS 0 C• (E) are identical, because the p+ -allowability restrictions (3.1.1) with respect to strata A0 ∈ S 0 coincide with the p-allowability restrictions with respect to the corresponding strata A ∈ S. The argument of [GM2] §3.6 implies that there are unique quasi-isomorphisms p p+ p+ Ip 0 S• (E ⊗ OX )[n] ∼ = I 0 C• (E) and I 0 S• (E ⊗ OX )[n] ∼ = I 0 C• (E). S

S

S

S

+

(Even though the perversity p is stratum-dependent, the same proof works.) So it remains to + show that IpS 0 S• (E ⊗ OX ) and IpS 0 S• (E ⊗ OX ) are canonically quasi-isomorphic. For notational simplicity we now drop explicit mention of the local system E ⊗ OX . For any stratum A0 ∈ S 0 we have p+ (A0 ) ≥ p(A0 ) ≥ p− (A0 ). Therefore there are canonical morphisms −

+

IpS 0 S• → IpS 0 S• → IpS 0 S• .

(3.3.2)

We claim these are quasi-isomorphisms. Suppose by induction that we have proven that these are quasi-isomorphisms over the open set X − Xn−c that consists of all homological strata of codimension < c. We wish to conclude that these are also quasi-isomorphisms over the homological strata of codimension c. Let A ⊃ A0 be strata of S and S 0 respectively, with codimensions c and c + r respectively. Let x0 ∈ A0 and let `0 denote the link of the stratum A0 at the point x0 . Since the homological stratum A is refined by the stratification S 0 there exists an honest stratum A0 ∈ S which is open in A. Let x ∈ A0 be a point, sufficiently close to x0 , and let ` denote the link of the stratum A0 at the point x. Since the homological stratification satisfies hypothesis (3) there is a canonical isomorphism between the stalk cohomology (3.2.1) of the Ip S• sheaf at the points x and x0 , that is, between ( ( 0 if i > p(c) 0 if i > p(c + r) I p Hxi = and I p Hxi 0 = p i p i 0 I H (`) if i ≤ p(c) I H (` ) if i ≤ p(c + r) Therefore I p H i (`0 ) = 0 for p(c) < i ≤ p(c + r). Moreover, a similar isomorphism holds for the stalk cohomology with compact supports (3.2.2). This is because, by hypothesis (3) again, each term in the following isomorphic triangles of sheaves on X − Xn−c−1 is CLC: -

Ip S•

-

Rh∗ Ip S•

[1 ]

]

τ ≥p(c)+1 Rh∗ Ip S•

[1

Rh∗ Ip S•

τ≤p(c) Rh∗ Ip S•

Rj∗ j ! Ip S•

3.4. PRODUCTS IN INTERSECTION HOMOLOGY AND COHOMOLOGY

29

Here, h : X − Xn−c → X − Xn−c−1 denotes the inclusion and τ ≥p(c)+1 is the functor that kills cohomology in all degrees ≤ p(c), cf. [GM2] §1.14. (The upper left hand corners of the two triangles are isomorphic by statement (1) of Proposition 3.3.) Since the stratum A is also a homology manifold, jx! = i∗x j ! [n − c] where jx denotes the composition of inclusions {x} −−−→ A −−−→ X − Xn−c . ix

j

Hence the cohomology of jx! Ip S• is locally constant as x varies in A. This gives a canonical isomorphism between (cf. equation (3.2.2)), ( I p H i−1−n+c (`) if i > p(c) + 1 + n − c i ! p • H jx I S = 0 if i ≤ p(c) + 1 + n − c and •

H i jx! 0 Ip S

( I p H i−1−n+c+r (`0 ) if i > p(c + r) + 1 + n − c − r = 0 if i ≤ p(c + r) + 1 + n − c − r

Therefore I p H i−1−n+c+r (`0 ) = 0 for p(c + r) + 1 + n − c − r < i ≤ p(c) + 1 + n − c. In summary we conclude that I p H i (`0 ) = 0 for p(c) + 1 ≤ i ≤ p(c) + r.

(3.3.3)

(If S were also an honest stratification, the link `0 would be the r-fold suspension of `, and this statement would follow directly.) On the other hand, the stalk cohomology at the point x0 ∈ A0 of the sheaves (3.3.2) is given in the following table, Stalk cohomology Hxi 0 (IS• ) p p 0 (i > p(c) + r) 0 (i > p(c + r)) p i 0 p+ i 0 I H (` ) (i ≤ p(c + r)) I H (` ) (i ≤ p(c) + r) +

p− 0 (i > p(c)) p− 0 I H(` ) (i ≤ p(c))

− + We have assumed by induction that I p H i (`0 ) ∼ = I p H i (`0 ) ∼ = I p H i (`0 ). Then equation (3.3.3) implies that the morphisms (3.3.2) induce isomorphisms on the stalk cohomology at the point x0 ∈ A0 . Since x0 was arbitrary we conclude that the morphisms (3.3.2) are quasi-isomorphisms. This completes the proof of Proposition 3.3.

3.4. Products in intersection homology and cohomology As in §3.2, we suppose that X is a (not necessarily compact) subanalytic or piecewise linear n dimensional pseudomanifold with singular set Σ and let Y = X − Σ. Let (3.4.1)

E1 × E2 → E3

30


be a bilinear pairing of local systems on Y. Suppose p, q are perversities such that p + q is also a perversity. The pairing (3.4.1) of local systems extends (uniquely in Dcb (X)) to a morphism (3.4.2)

•

Ip S• (E1 ) ⊗ Iq S• (E2 ) → Ip+q S (E3 )

giving a product I p H i (X, E1 ) ⊗ I q H j (X, E2 ) → I p+q H i+j (X, E3 ). By an orientation of X we mean an orientation of its nonsingular part, Y. Suppose X is orientable and oriented. Then the transverse intersection of chains ([GM1]) induces, for any open set U ⊂ X, an intersection pairing BM I p HiBM (U, E1 ) × I q HjBM (U, E2 ) → I p+q Hi+j−n (U, E3 )

using the fact that orientations of transverse chains ξ, η determines an orientation of the intersection |ξ|∩|η| in the presence of an orientation of Y. The orientation of X also determines a Poincaré duality morphism P, cf. (2.5.1) and (3.2.3) , and the following diagram commutes,

(3.4.3)

I p H i (X, E1 ) ⊗ I q H j (X, E2 )   P⊗P y

−−−→ I p+q H i+j (X, E3 )   yP

I p Hn−i (X, E1 ) ⊗ I q Hn−j (X, E2 ) −−−→ I p+q Hn−i−j (X, E3 ) where the bottom row is the intersection pairing. These pairings and mappings are independent of the stratification ([GM2]). For any perversity r let : I r H0 (X, R) → H0 (X, R) → R denote the augmentation. Suppose E1 × E2 → R is a pairing of local systems of rational vector spaces. If p, q are perversities such that p + q is a perversity and if X is oriented, then the above products give rise to pairings on intersection homology and cohomology, which are essentially three different ways to express the same product: (3.4.4) (3.4.5)

h, iIH ∗ : I p H i (X, E1 ) × I q H n−i (X, E2 ) → R h, iIH∗ : I p Hj (X, E1 ) × I q Hn−j (X, E2 ) → R

(3.4.6)

h, iK : I p H k (X, E1 ) × I q Hk (X, E2 ) → R

defined by ha, biIH ∗ = (P(a·b)) in (3.4.4), ha, biIH∗ = (a·b) in (3.4.5), and ha, biK = (P(a)·b) in (3.4.6), where · denotes the intersection product. These pairings will be referred to as the cup product pairing, the intersection pairing, and the Kronecker pairing respectively. They are compatible with the corresponding products in the (ordinary) homology and cohomology of Y.

3.4. PRODUCTS IN INTERSECTION HOMOLOGY AND COHOMOLOGY

31

For example, in the following diagram, Hck (Y ; E1 ) ×HkBM (Y ; E2 ) −−−→ R h,i  x   α1∗ y β1∗ (3.4.7)

I p H k (X, E1 )×I q Hk (X, E2 ) −−−→ R h,iK  x   α2∗ y β2∗ H k (Y, E1 ) × Hk (Y, E2 )

−−−→ R h,i

these “Kronecker products” satisfy (3.4.8)

hα1∗ (x), yiK = hx, β1∗ (y)i and hα2∗ (x0 ), y 0 i = hx0 , β2∗ (y 0 )iK .

Let D•X denote the dualizing complex on X. The Poincaré duality theorem ([GM1, GM2]) states: Theorem 3.4. Assume R is a field. Assume also (1) p and q are complementary perversities (that is, p(c) + q(c) = c − 2 for all c), (2) the pairing E1 × E2 → R is nondegenerate, and (3) X is compact. Then (3.4.2) becomes a dual pairing, that is, a morphism Ip S• (E1 ) ⊗ Iq S• (E2 ) → D•X [−n]

(3.4.9) such that the induced mapping (3.4.10)

Ψ : Ip S• (E1 ) → RHom• (Iq S• (E2 ), D•X [−n])

is a quasi-isomorphism. Therefore the resulting bilinear forms (3.4.4), (3.4.5), and (3.4.6) are nondegenerate.

CHAPTER 4

Integration of Differential Forms 4.1. Differential forms on an orbifold Suppose Y is a (real) analytic manifold, not necessarily compact. Let Z ⊂ Y be a purely d-dimensional oriented semi-analytic or subanalytic closed subset of Y. Let ω ∈ Ωd (Y ; R) be a smooth differential form. Suppose either (a) the set Z is compact or (b) the differential form ω has compact support in Y. Then the integral Z ω C, then y∞ fj γy∞2 ( y0∞

x∞ 1

59

A1 ) ≤ By∞

for all γ ∈ ResL/Q (SL2 )(Q) ≤ G(Q), y ∈ A× L , and x ∈ AL , Σ(L)

where B, C ∈ R>0 , t, A ∈ R>0 , and A, B, C are independent of y and x. A function f : G(A) → C satisfying (7.3) is said to be weakly increasing. Recall the following result [Hid7, §2.3.2] of Hida: Lemma 7.1. The space Mκ (K0 (c), χ) is zero if κ 6∈ X (L). Proof. By condition (2) in the definition of Mκ (K0 (c), χ), we have 0

−k −2m χ(b0 )b∞ =1

for all b ∈ L× (which we consider as an element of A× L viaQthe diagonal embedding). The 0 restriction χ|Ob× is of finite order, say j, so we conclude that σ∈Σ(L) σ()−j(kσ +2mσ ) = 1 for all L ∈ OL× . As in §7.2, it follows from Dirichlet’s unit theorem that k 0 + 2m ∈ Z1. Because of Lemma 7.1 we will henceforth assume that κ ∈ X (L). We now recall another normalization for the weight of a Hilbert modular form which will be useful to us in Chapter 8. Let Sκcoh (K0 (c), χ) (resp. Mκcoh (K0 (c), χ)) be the space of functions f : G(A) → C with f (α∞ α0 ) smooth as a function of α∞ ∈ G(R) for all α0 ∈ G(Af ) satisfying conditions (1) and (4) (resp. (4’)) and the following modification of conditions (2) and (3): −1 (2coh ) If b ∈ A× L and γ ∈ G(Q), we have f (γαb) = χ(b) f (α). P −ι coh 1,0 (3 ) If u = u∞ u0 ∈ K∞ K0 (c), then f (αu) = χ(u0 )f (α)e( σ∈Σ(L) (kσ0 + 2)θσ ) where Y χ (( ac db )) := χv(p) (dv(p) ), p|c

for all ( ac db ) ∈ K0 (c), we set e(z) = exp(2πiz), and cos(2πθσ ) sin(2πθσ ) u∞ = − sin(2πθσ ) cos(2πθσ )

.

σ∈Σ(L)

There is a natural isomorphism (7.3.4)

Mκ (K0 (c), χ) −→ ˜ Mκcoh (K0 (c), χ) f 7−→ f −ι

where f −ι (x) := f (x−ι ). Here ι is the main involution of GL2 , so xι := det(x)x−1 and x−ι := det(x)−1 x. This isomorphism restricts to induce an isomorphism Sκ (K0 (c), χ) ∼ = Sκcoh (K0 (c), χ). Remarks. (1) There is a dictionary between automorphic forms in Mκcoh (K0 (c), χ) as χ varies over a certain set of quasicharacters and h(K0 (c))-tuples of classical (holomorphic) modular forms on Γj0 (c) for 1 ≤ j ≤ h(K0 (c)) (see [Hid5, (3.5)]). We mentioned the elliptic modular case of this fact in the introduction.

60

7. GENERALITIES ON HILBERT MODULAR FORMS AND VARIETIES

(2) The reason for introducing two normalizations for the weight of a modular form is that it is more natural to consider the Fourier expansions of an element of Sκ (K0 (c), χ) (see [Hid1] and §7.6) and it is more natural to associate differential forms and cohomology classes to elements of Sκcoh (K0 (c), χ) (see §8.7). This explains the superscript “coh.” We will comment more on this in §7.8 below. 7.4. Hecke operators We now recall the definition of the Hecke operators associated to the spaces of automorphic forms Mκ (K0 (c), χ) and Mκcoh (K0 (c), χ) considered in the previous section. Let c ⊂ OL be an ideal and let R(c) be the following set of matrices:     Y Y a b × 0 b G(Af ) ∩ x ∈ M2 (Z ⊗ OL ) : x = with c ∈ c and (dv(p) ) ∈ OL,v(p) . c d   p|c

p|c

Here c0 is the ideal associated to c and v(p) is the place associated to the prime p. Then Tc is the algebra of formal Z-linear sums of double cosets K0 (c)xK0 (c) with x ∈ R(c); multiplication is defined as in [Sh1, p. 51]. Any such double coset can be decomposed as a disjoint sum of left cosets X xi K0 (c) i

for suitable xi ∈ G(Af ). The action of Tc on Mκ (K0 (c), χ) is then defined by X (f |K0 (c)xK0 (c)) (g) := χ(xi )−1 f (gxi ). i

and the action of Tc on Mκcoh (K0 (c), χ) is defined by X (f |K0 (c)xK0 (c)) (g) := χ(xi )−1 f (gx−ι i ), i

Q

In either case, χ (( ac db )) := p|c χv(p) (dv(p) ). With these normalizations of the Hecke action, it is clear that the isomorphism Mκ (K0 (c), χ) ∼ = Mκcoh (K0 (c), χ) given by (7.3.4) is Hecke equivariant. Remark. As a space of functions, M(k0 ,m) (K0 (c), χ) is equal to coh −1 M(k ), 0 ,−m−k 0 ) (K0 (c), χ

but this identification of function spaces is not an isomorphism of Hecke modules.

7.5. THE PETERSSON INNER PRODUCT

61

For every ideal n ⊂ OL we set (7.4.1)

X

Tc (n) :=

K0 (c)xK0 (c).

K0 (c)xK0 (c):det(x)=n

Here the sum is over a set of representatives for the set of double cosets K0 (c)xK0 (c) such that x ∈ R(c) and n is the ideal associated to det(x)0 . Further, for a prime p ⊂ OL with p - c, let $p ∈ OL,p be a uniformizer. Considering $p as an element of A× Lf via the canonical inclusion × × OL,p ,→ ALf , set k $ 0 (7.4.2) Tc (pk , pk ) := K0 (c) 0p $k K0 (c) p

k

k

for k ≥ 0. If p|c, we simply set Tc (p , p ) = 0 for k ≥ 0. Define Tc (n, n) in general by multiplicativity. The Hecke operators satisfy the following well-known identity (see [Sh2]): X (7.4.3) Tc (m)Tc (n) = NL/Q (a)Tc (a, a)Tc (a−2 mn). m+n⊆a

For our later convenience, if f ∈ Sκ (K0 (c), χ) is a normalized eigenform for all Hecke operators, we let λf (n) be its nth Hecke-eigenvalue: (7.4.4)

f |Tc (n) =: λf (n)f.

We denote by Sκnew (K0 (c), χ) the new subspace, defined2 as the intersection of the kernels of the natural trace maps Sκ (K0 (c), χ) −→ Sκ (K0 (d), χ) X f 7−→

χ(x)−1 f (·x)

xK0 (c)∈K0 (d)/K0 (c)

where d ) c. It is well-known that Sκnew (K0 (c), χ) has a basis of forms that are simultaneous eigenforms for all Hecke operators. 7.5. The Petersson inner product In this section, we fix a normalization of the Petersson inner product. For κ ∈ X (L), a 0 −2m quasicharacter χ satisfying χ∞ (b∞ ) = b−k and f, g ∈ Mκ (K0 (c), χ), we set ∞ Z (f, g)P := (7.5.1) g(α)f (α)| det(α)|kAσL+2mσ dµc (α), Y0 (c)

2The

equivalence of this definition with the usual one defining the new subspace as the orthogonal complement of the old subspace with respect to the Petersson inner product is well-known (for example, see [La, Theorem 2.2] for the elliptic modular case).

62


and, for f, g ∈ Mκcoh (K0 (c), χ), we set Z (7.5.2) g(α)f (α)| det(α)|A−kLσ −2mσ dµc (α) (f, g)P := Y0 (c)

whenever these integrals are well defined. Here σ is any element of Σ(L) (note kσ + 2mσ is independent of this choice because κ ∈ X (L)). Moreover dµc is the measure on Y0 (c) defined as follows: Let B 0 ≤ G be the algebraic subgroup whose points in a commutative Q-algebra A are given by B 0 (A) := ( a0 1b ) : a ∈ (L ⊗Q A)× and b ∈ L ⊗Q A . We then set notation for the left Haar measure y x × dµB 0 = |y|−1 AL dxd y 0 1 where dx := dx∞ dx0 and d× y := d× y∞ d× y0 are the measures defined as follows: • • • •

additive measure dx = dx∞ is the Lebesgue measure onR RΣ(L) . additive measure dx0 is the measure on ALf such that ObL dx0 = 1. −1 |dy∞ . multiplicative measure d× y∞ is given by |y∞ × multiplicative measure d y0 is the measure on A× Lf such that × b× d y0 = 1. O

The The The The R L

b Identifying (R× )Σ(L) with the center of G(R), we then let dµ∞ be the bL := OL ⊗ Z. Here O Haar measure on the compact group K∞ /(R× )Σ(L) giving it volume 1, and dµ0 be the Haar measure on the compact open group K0 (c) giving it volume 1. In view of the fact that G(A)/(R× )Σ(L) = G(Q)B 0 (A)K∞ K0 (c)/(R× )Σ(L) , (see [Hid4, §4]) the product dµB dµ∞ dµ0 induces a measure dµc on Y0 (c) which is the measure appearing in (7.5.1). Notice that, for f, g ∈ Mκ (K0 (c), χ), we have (f, g)P = (f −ι , g −ι )P . Moreover, it is well-known that the integral defining (f, g)P is always finite and well-defined for f, g ∈ Sκ (K0 (c), χ) or f, g ∈ Sκcoh (K0 (c), χ). 7.6. Fourier series For details on this section, see Appendix C. Let κ = (k 0 , m) ∈ (Q × Q)Σ(L) . For each σ ∈ Σ(L), let Wmσ be the local archimedian Whittaker function Wmσ : R× −→ C y 7−→ |y|−mσ e−2π|y| .

7.6. FOURIER SERIES

63

For x ∈ AL and y ∈ A× L , we then set (7.6.1)

qκ (x, y) = qκ (x, y∞ ) := eL (x)

Y

Wmσ (yσ ),

σ∈Σ(L)

where eL (·) is the additive character of AL /L such that eL (x∞ ) = e2πiTrL/Q (x∞ ) . Let IL denote the set of fractional ideals of OL , and let y ∈ A× eles b ∈ A× L,+ , the set of id´ L with bσ > 0 for all σ ∈ Σ(L). Let DL/Q be the different. Assume that κ ∈ X (L). Theorem 7.2. Let h ∈ Mκ (K0 (c), χ) be a Hilbert modular form. Then h admits a Fourier series,   X y x   (7.6.2) h = |y|AL c(y) + b(ξyf )qκ (ξx, ξy) , 0 1 ξ∈L× ξ0

valid for all x ∈ AL and all y ∈ A× L such that yσ > 0 for all σ ∈ Σ(L). Moreover, each coefficient eL/Q ] ∈ IL determined by ξyf D eL/Q . b(ξyf ) ∈ C depends only on the fractional ideal [ξyf D Addendum. The constant term c(y) vanishes if h is a cusp form or if k 0 ∈ / Z1 and the ideal e [ξyf DL/Q ] is not integral. Otherwise it is a sum, (7.6.3)

0

−m −k −1−m c(y) = c0 (yf )|y∞ | + c1 (yf )|y∞ |

eL/Q ]. If the of two terms. Here, c0 (yf ) and c1 (yf ) only depend on the fractional ideal [yf D Σ(L) functions Fi (z) of (7.3.3) on h (corresponding to h) are holomorphic, then c1 (·) = 0. In what follows we will express these coefficients b, c0 , c1 slightly differently, defining a(·, ·), a0 (·, ·) and a1 (·, ·) (respectively) to be the corresponding functions IL × Mκ (K0 (c), χ) → C and by abuse of notation we write a(ξyDL/Q , h) := b(ξyf ). Consequently, if h ∈ Sκ (K0 (c), χ) is a cusp form, we write X y x h = |y|AL a(ξyDL/Q )qκ (ξx, ξy). 0 1 × ξ∈L ξ0

This well-known theorem is extremely important to us; indeed, it is built into the statement of Theorems 1.1 and 1.2. For a proof that relies on the “classical viewpoint” of Hilbert modular forms as differential forms on HΣ(L) , see [Hid4, Theorem 1.1], and for an adelic proof, see [We], [Hid5, §6]. For the convenience of the reader, we will give a proof relying on the basic theory of Whittaker models in Appendix C below.

64


The relationship between the Fourier coefficients of a cusp form f ∈ Sκ (K0 (c), χ) and one of its Hecke-translates can be written down explicitly: X (7.6.4) NL/Q (b)χ(b)a(mn/b2 , f ) a(m, f |Tc (n)) = b⊇m+n b+c=OL

(see [Hid5, Corollary 6.2]). Here we define χ(b) := χ(b) where b ∈ A× L is an element satisfying the following conditions: • The component bv = 1 if v|∞ or v is the place associated to a prime dividing c. • The ideal associated to the finite part b0 of the idèle b is b. From (7.4.3) and (7.6.4) one can deduce that if f is a simultaneous eigenform for all Hecke operators normalized so that a(OL , f ) = 1, then (7.6.5)

a(m, f ) = λf (m)

(see [Hid5, p. 477]). With this in mind we define X (7.6.6) L(f, s) := a(m, f )NL/Q (m)−1/2−s m⊂OL

for any f ∈ Sκ (K0 (c), χ). This L(f, s) admits an Euler product if and only if f is a simultaneous eigenform for all Hecke operators. Thus, in the sequel, L denotes a totally real number field and, e.g. L(f, s) denotes an L-function. We hope that this does not confuse the reader. Before we recall the formula for these Euler products, we set notation for the twists of f so we can recall their Euler products at the same time. For completeness, we state the following well-known lemma: × Lemma 7.3. Suppose that η : L× \A× is a character satisfying η∞ (α∞ ) = 1. Let L → C b be an ideal divisible by the conductor f of the Dirichlet character associated to η, and let w ∈ Z. Then, for each f ∈ M(k0 ,m) (K0 (c), χ), there is a modular form f ⊗ η| · |w ∈ M(k0 ,m−w1) (K0 (cb2 ), χη 2 | · |2w AL ) whose Fourier series is given by ( η(m)|NL/Q (m)|w a(m, f ) if m + b = OL w a(m, f ⊗ η| · |AL ) : = 0 otherwise. ( η(m)|NL/Q (m)|w a0 (m, f ) if f = OL , a0 (m, f ⊗ η| · |w AL ) : = 0 otherwise.

In the Fourier coefficients given in Lemma 7.3, we define η(n) := η(e n) where n e ∈ A× L is an element satisfying the following conditions: • The component n ev = 1 if v|∞ or v is the place associated to a prime dividing cb. • The fractional ideal associated to the finite part n e0 of the idèle n e is n.

7.6. FOURIER SERIES

65

That is, we define the value of η on ideals as we defined the value of χ on ideals after equation (7.6.4) above. Assuming Lemma 7.3 for the moment, we set notation for the Euler products of the twists of f by characters η trivial at infinity as above (including the trivial twist f ⊗ Id = f ). These are given by Y L(f ⊗ η, s) := (7.6.7) Lp (f ⊗ η, s) prime ideals p⊂OL

where Lp (f ⊗ η, s) is equal to  −1 −1/2−s  + χ(p)η(p)2 NL/Q (p)−2s  1 − λf (p)η(p)NL/Q (p) −1 1 − λf (p)η(p)NL/Q (p)−1/2−s   1

if p - c if p | c and p - b otherwise.

Notice that the definition of the local Euler factor Lp (f ⊗ η, s) depends on b, the modulus of the Dirichlet character we associated to η. We now give the following proof of Lemma 7.3 (which is taken from [Hid4, §7.F]): Proof of Lemma 7.3. Assume first that η is trivial and b = OL . Then it is easy to check that we may set (7.6.8)

f ⊗ | · |w (α) := |NL/Q (DL/Q )|w | det(α)|w AL f (α).

Now assume that w = 0. Then, in view of Lemma 7.4 below, we can and do assume that b is the conductor of the Dirichlet character associated to η. If b = OL , then it is easy to check that we may set f ⊗ η(α) := η(d)η(det(α))f (α), where d ∈ A× ele trivial at the infinite places whose associated ideal is DL/Q . L is an id` Suppose that b 6= OL . Q Let b be anQidèle trivial at the infinite places whose associated ideal is b. Define a subset Υ of p|b Lp × p-b Op by     Y Y Υ = t = (tp ) ∈ Lp × Op : ordp (tp ) ≥ −ordp (b) for all p | b ,   p|b

p-b

e a set of representatives for Υ modulo O bL := Q Op ; this may be viewed as a ring and let Υ p Q e isomorphic to p|b (Op /bp Op ). We denote by ηb : Υ → C the map defined by setting ( e× η(t) if t ∈ Υ ηb (t) = 0 otherwise.

66


e define u(t) ∈ G(A) by For each t ∈ Υ, ( ( 10 t1v ) u(t)v = ( 10 01 )

if v - ∞ and ordv (b) ≥ 1 otherwise.

Finally, set h(α) := η(det(α))

X

ηb (t)f (αu(t)) .

e t∈Υ

It is easy to check that h ∈ Mκ (K0 (cb ), χη ). Moreover, for any y ∈ A× Lf , we have ( × G(η −1 )η(yyb−1 )a(y, h) if yb,v ∈ OL,v for all v such that ordv (b) ≥ 1 a(y, h) = 0 otherwise 2

2

a0 (y, h) = 0. Here G(η −1 ) is the Gauss sum G(η −1 ) :=

X

−1 ηb (d−1 b t)eL (d t)

e t∈Υ

where d ∈ A× ele trivial at the infinite places whose associated ideal is DL/Q , db is the L is an id` e it defines, and yb = (yb,v ) is an idèle with yb,v = yv if the finite place v divides b element of Υ × and yb,v = 1 otherwise. Since η(yyb−1 ) = η(y) if yb,v ∈ OL,v for all v with ordv (b) ≥ 1, it follows that we may set f ⊗ η := G(η −1 )−1 h. The following lemmas will be used in §9.3 and §11.1: Lemma 7.4. Suppose that f ∈ Mκ (K0 (c), χ) and let b ⊂ OL be an ideal. Define ( a(m, f ) if b + m = OL a(m, f b ) = 0 otherwise. Then the Fourier series f b (( y0 x1 )) defined by 



X   −m |+ a(ξyDL/Q , f b )qκ (ξx, ξy) |y|AL a0 (yDL/Q , f )|y∞ ξ∈L× ξ0

is an element of Mκ (K0 (c

Q

p|b

p), χ). r

Proof. It suffices to show that f p ∈ Mκ (K0 (cp), χ) for a prime p ⊂ OL , for then an inductive argument finishes the proof.

7.7. L-FUNCTIONS

67

By considering Fourier series using (7.6.4), we see that −1 r |Tcp (p). f p = f (α) − |NL/Q (p)|−1 f α $0p 01 Here we denote by $p an idèle that is a uniformizer for the maximal ideal of OL,p at the place r associated to p and is 1 at every other place. It is then easy to see that f p ∈ Mκ (K0 (cp), χ). Lemma 7.5. Suppose that f ∈ Mκ (K0 (c), χ) and let a ∈ A× L . Define ( a(ξyDL/Q , f ) if a ∼ ξyDL/Q in T (Q)\T (A)/ det(G(R)0 K0 (c)) a(ξyDL/Q , πa f ) : = 0 otherwise, ( a0 (yDL/Q , f ) if a ∼ yDL/Q in T (Q)\T (A)/ det(G(R)0 K0 (c)) a0 (yDL/Q , πa f ) : = 0 otherwise, (see (7.1.6)). Then the Fourier series πa f (( y0 x1 )) defined by   X   −m |y|AL a0 (yDL/Q , πa f )|y∞ |+ a(ξyDL/Q , πa f )qκ (ξx, ξy) ξ∈L× ξ0

is a Hilbert modular form. Proof. The function πa f : G(A) → C is defined so that ( −1 f (α) if α ∈ det−1 T (Q)aDL/E det(G(R)0 K0 (c)) πa f (α) = 0 otherwise. With this in mind, the lemma follows in a straightforward manner from the definition of Mκ (K0 (c), χ). 7.7. L-functions In this section we set notation for certain L-functions attached to automorphic forms on GL2 . Though this might be a little tedious for the expert, it is important, as the L-functions arising from the periods we will later consider are imprimitive (i.e. they do not coincide with the associated “canonical” Langlands L-functions at all places). We will normalize everything so that the local L-factors are the same as the canonical Langlands L-factors at all unramified places. We warn the reader that the automorphic representation π(f ) attached to a newform f ∈ Sκ (K0 (c), χ) is in general not unitary. Rather the automorphic representation π(f )| · 0 |[k +2m]/2 is unitary, where k 0 + 2m = [k 0 + 2m]1.

68


Throughout, for any L-function or zeta function D(s) = c ⊂ OL , we will write X a(n)NL/Q (n)−s Dc (s) : =

P

n⊂OL

a(n)NL/Q (n)−s and ideal

n⊂OL n+c=OL

X

Dc (s) : = a(OL ) +

a(n)NL/Q (n)−s .

n+c6=OL ∗,c

c

Moreover, D (s) = “Gamma factors” × D (s) will denote the completed partial L-function. To ease notation, as above we write κ = (k 0 , m) ∈ X (L) and write [k 0 + 2m] for the integer such that kσ0 = 2mσ = k 0 + 2m for all σ ∈ Σ(L). Moreover, if χ (resp. π) is an character (resp. automorphic representation) we write f(χ) (resp. f(π)) for its conductor, considered as an ideal of OL . 7.7.1. The standard L-function. Recall that for f ∈ Sκ (K0 (c), χ) we have Y (7.7.1) Lp (f ⊗ η, s) L(f ⊗ η, s) := p

where Lp (f ⊗ η, s) is defined to be  −1 −1/2−s  + χ(p)η(p)2 NL/Q (p)−2s  1 − λf (p)η(p)NL/Q (p) −1 1 − λf (p)η(p)NL/Q (p)−1/2−s   1

if p - c if p | c and p - b otherwise

(see (7.6.7)). For each prime p ⊂ OL we choose, once and for all, α1,p (f ), α2,p (f ) ∈ C such that Lp (f, s) = (1 − α1,p (f )NL/Q (p)−s )(1 − α2,p (f )NL/Q (p)−s ). Let π(f ) be the cuspidal automorphic representation generated by f . At a finite place v(p) associated to a prime p - f(π(f ))f(η), we have that π(f )v(p) ⊗ ηv(p) is in the (unramified) principal series, with Satake parameters α1,p (f ) and α2,p (f ). It follows that Lp (f ⊗ η, s) :=L(π(f )v(p) ⊗ ηv(p) , s) = 1 − η(p)α1,p (f )NL/Q (p)−s

−1

1 − η(p)α2,p (f )NL/Q (p)−s

−1

for p - cf(η) (see Theorem D.4). [k0 +2m]/2 We note that the automorphic representation π(f )|·|AL is unitary. Applying Theorem 5.3 of [JS1], this implies that the partial Euler product Lf(π(f ))f(θ) (f ⊗ η, s)

7.7. L-FUNCTIONS

69

is absolutely convergent for Re(s) ≥ 1 − [k 0 + 2m]/2. Actually, more is true: Lemma 7.6. If f ∈ Sκnew (K0 (c), χ) is a newform, then the Dirichlet series defining L(f ⊗η, s) is absolutely convergent for Re(s) ≥ 1 − [k 0 + 2m]/2. Proof. In view of the previous paragraph, it suffices to show that Lp (f ⊗η, s) is absolutely convergent for p | f(η)f(π(f )) for Re(s) ≥ 1 − [k 0 + 2m]/2. To see this, we apply [SW, Theorem 3.3] (a generalization of some results of Atkin-Lehner-Li theory to the Hilbert modular case) 0 to conclude that |a(f, p)| ≤ NL/Q (p)−1/2−[k +2m]/2+1 . This implies the desired convergence. 7.7.2. Rankin-Selberg L-functions. Let f ∈ Sκ (K0 (c), χ) and g ∈ Sκ (K0 (c0 ), χ0 ) be simultaneous eigenforms for all Hecke operators. The Rankin-Selberg L-function attached to f and g is defined by X 0 L(f × g, s) := Lcc (χχ0 , 2s) a(n, f )a(n, g)NL/Q (n)−1−s . n⊂OL cc0

The partial L-functions L (f × g, s) admit the following Euler product expansion: Y 0 Lcc (f × g, s) = Lp (f × g, s), prime ideals p⊂OL p-cc0

where Lp (f × g, s) =

2 Y 2 Y (1 − αi,p (f )αj,p (g)NL/Q (p)−s )−1 . i=1 j=1

(see [HT, (7.7)]). It follows in particular that Lp (f × g, s) = L(πv (f ) πv (g), s) for finite places v - cc0 . Thus 0 the partial Euler product Lcc (f × g, s) is absolutely convergent for Re(s) > 1 − [k 0 + 2m] 0 [JS1, Theorem 5.3]. Moreover, Lcc (f × g, s) has a pole at s = 1 − [k 0 + 2m] if and only if the 0 automorphic representations π(f ) and π(g) spanned by f and g satisfy π(f ) ∼ = π(g)∨ |·|−[k +2m] , where π(g)∨ is the contragredient of π(g). The pole, if it exists, must be simple (see [JS2, Proposition 3.6]). 7.7.3. Adjoint L-functions. We now recall the adjoint L-functions attached to an eigen× form f ∈ Sκnew (K0 (c), χ). For a Hecke character φ : L× \A× L → C which is trivial at the infinite places, we set Y L(Ad(f ) ⊗ φ, s) := Lp (Ad(f ) ⊗ φ, s), p-f(φ)

where we define the local Euler factor Lp (Ad(f ) ⊗ φ, s)−1 to be (1 − φ(p)NL/Q (p)−s )(1 − φ(p)

α2,p (f ) α1,p (f ) NL/Q (p)−s )(1 − φ(p) NL/Q (p)−s ) α2,p (f ) α1,p (f )

70


if p - c and, in the ramified cases, we set Lp (Ad(f ) ⊗ φ, s)−1 equal to  −s  if p - f(φ) and π(f )v(p) is principal and minimal 1 − φ(p)NL/Q (p) −1−s 1 − φ(p)NL/Q (p) if p - f(φ) and π(f )v(p) is special and minimal  1 otherwise (see [HT, §7] and the corrections in [Gh, §5.1]). Following [HT], we say that an admissible × representation π of GL2 (Lv ) is minimal if f(π) ⊇ f(π ⊗ ξ) for all quasicharacters ξ : L× v → C . We note that if π(f ) = ⊗0v π(f )v is the cuspidal automorphic representation attached to f and p - c, then Lp (Ad(f ), s) = L(Ad(π(f )v(p) ), s) where v(p) is the place associated to p. We then let Lσ (Ad(f ) ⊗ φ, s) be the Γ factor defined to be 0

(2π)−(s+kσ +1) Γ(s + kσ0 + 1)π −(s+1)/2 Γ((s + 1)/2). For any ideal c0 ⊂ OL we let 0

0

L∗,c (Ad(f ) ⊗ φ, s) := Lc (Ad(f ) ⊗ φ, s)

Y

Lσ (Ad(f ) ⊗ φ, s)

σ∈Σ(L)

be the completed L-function. In the case that c0 = OL , we write L∗ (Ad(f ) ⊗ φ, s) = L∗,OL (Ad(f ) ⊗ φ, s). The following theorem, Theorem 7.1 of [HT] (see also [Hid4, (7.2c)]), will be crucial in the proof of Theorem 10.4 below: Theorem 7.7 (Hida-Tilouine). If f ∈ Sκ (K0 (c), χ) is a newform, then [k0 +2m]+3

(f, f )P = dL/Q P , where {k 0 + 21} := σ∈Σ(L) kσ0 + 2.

0

NL/Q (c)2−{k +1} L∗ (Ad(f ), 1)

Here (f, g)P is the Petersson inner product, normalized as in §7.5. × For our later convenience, if φ : L× \A× L → C is a character trivial at the infinite places, let Y Lcf(θ) (Sym2 (f ) ⊗ φ, s) := Lp (Sym2 (f ) ⊗ φ, s), p-cf(θ)

where Lp (Sym2 (f ) ⊗ φ, s) is the local Euler factor given by (1 − φ(p)χ(p)NL/Q (p)−s )(1 − φ(p)α1,p (f )2 NL/Q (p)−s )(1 − φ(p)α2,p (f )2 NL/Q (p)−s ) One checks immediately that (7.7.2)

Lp (Sym2 (f ) ⊗ (χ)−1 , s) = Lp (Ad(f ), s).

7.7. L-FUNCTIONS

71

7.7.4. Asai L-functions. Let L/E be a quadratic extension of totally real number fields and let hςi = Gal(L/E). We use class field theory to identify hηi = Gal(L/E)∧ for a character 0 η : E × \A× E → C trivial at the infinite places. Let κ = (k , m) ∈ X (E) be a weight, and define κ b = (b k0, m b 0 ) ∈ X (L) by declaring that b kσb0 = kσ0 (resp. m b σb = mσ ) if σ b : L ,→ R extends 0 −2m × × × σ : E ,→ R. Let χE : E \AE → C be a quasicharacter satisfying χE∞ (b∞ ) = b−k for ∞ × b ∈ AE , and set χ := χE ◦ NL/E . × We recall that for any f ∈ Sκb (K0 (c), χ) and any quasicharacter φ : L× \A× L → C , the Asai L-function L(As(f ⊗ φ), s) attached to f and φ is defined to be X φ(nOL )a(nOL , f )NE/Q (n)−1−s , ζ (f(φ)c)∩OE ((φ2 χ)|E , 2s) n⊂OE n+f(φ)=OE

where (φ2 χ)|E denotes the restriction of φ2 χ to E (see [R2] for a nice discussion of this Lfunction). The associated partial L-function admits an Euler product Y L(f(φ)c)∩OE (As(f ⊗ φ), s) = Lp (As(f ⊗ φ), s) p-(c∩OE )

where, if p - dL/E ((f(θ)c) ∩ OE ), we have that Lp (As(f ⊗ φ), s)−1 is equal to 2 Y 2 Y

1 − φ(p)αi,P (f )αj,P (f )NE/Q (p)−s

i=1 j=1

if p = PP splits and 1 − φ(p)α1,p (f )NE/Q (p)−s

1 − φ(p)α2,p (f )NE/Q (p)−s × 1 − φ(p)2 χ(p)NE/Q (p)−2s

if p is inert (see [HLR, §2]). Here we have abused notation and set φ(p) := φ(pOL ) and χ(p) := χ(pOL ). Suppose that f ∈ Sκbnew (K0 (c), χ) is a newform. Then there is another newform f ς ∈ Sκbnew (K0 (c0 ), χ) uniquely determined by the fact that for m ⊂ OL coprime to cc0 we have a(f ς , m) = a(f, ς(m)), where c0 ⊂ OL is some other ideal. We see directly from the local Euler factors given above that (7.7.3)

LDL/E c (f × f ς , s) = LdL/E (c∩OE ) (As(f ), s)LdL/E (c∩OE ) (As(f ) ⊗ η, s).

Remark. Ramakrishnan shows in [R2] that there is an isobaric automorphic representation As(π(f )) of ResE/Q (GL4 ) whose L-function is equal to L(As(f ), s) up to finitely many Euler factors. Similarly, in [R1], he shows that there is an isobaric automorphic representation π(f ) π(f ς ) of ResL/Q (GL4 ) whose L-function is equal to L(f × f ς , s) up to finitely many places. Thus (7.7.3) implies that π(f ) π(f ς ) is the base change of As(π(f )) to ResL/Q (GL4 ).

72


As in the introduction, if f ∈ Sκ (K0 (cE ), χE ), we write fb for the unique newform on L generating the automorphic representation of GL2 (AL ) that is the base change to L of the automorphic representation of GL2 (AE ) generated by f . Thus f ∈ Sκb (K0 (c), χ) for some ideal c ⊂ OL . We say that a newform g ∈ Sκb (K0 (c), χ) is a base change from E if g = fb for some newform f on E. We will require the following proposition in the proof of Theorem 10.4 below. × × × × Proposition 7.8. Fix a quasi-character ϑ : L× \A× L → C (resp. θ : L \AL → C ) such 0 0 [k +2m]/2 [k +2m]/2 that ϑ|·|AL (resp. θ|·|AL ) is unitary and its restriction to E × \A× E is χE (resp. χE η). new b If f ∈ Sκb (K0 (c), χ) is a newform that is a base change of a newform f ∈ Sκ (K0 (cE ), χE ) for some ideal cE ⊂ OE , then

LdL/E (As(fb ⊗ ϑ−1 ), s) = LdL/E cf(ϑ)∩OE (η, s)LdL/E f(ϑ)∩OE (Ad(f ), s) and 0 Lb (As(fb ⊗ θ−1 ), s) = ζ dL/E (cf(θ)∩OE ) (s)LdL/E f(θ)∩OE (Ad(f ) ⊗ η, s),

where b0 :=

Q

p|dL/E p-f(θ)∩OE

p.

This proposition is stated, with a typographical error, on p. 4 of [R2]. For a proof that one can always find characters ϑ and θ satisfying the requirements of the proposition, see Lemma 2.1 of [Hid6]. Remark. Let π be the automorphic representation generated by a simultaneous eigenform f ∈ Sκ (K0 (c), χ) for almost all Hecke operators and let Ad(π) denote the automorphic representation with L-function L(Ad(f ), s) [GJ]. Regardless of κ, Ad(π) is unitary. The unramified [k0 +2m]/2 character | · |AE is present in the proposition to make the automorphic representation attached to As(fb ⊗ θ−1 ) (resp. As(fb ⊗ ϑ−1 )) unitary as well. Proof. We will use the theory of quadratic base change for GL2 as developed in [L] freely in this proof (see Appendix D for a synopsis). First, for all primes p ⊂ OE we have Y LP (f, s) = Lp (f, s)Lp (f ⊗ η, s) (7.7.4) P|p

where f ⊗ η is the newform with λf ⊗η (p) = λf ⊗η (p) for almost all primes p ⊂ OE . Moreover, a(P, fb) = a(Pς , fb) for all primes P ⊂ OL . Thus, for i ∈ {1, 2}, without loss of generality we have: (1) αi,P (fb) = αi,Pς (fb), (2) αi,P (fb) = αi,p (f ) if p splits as p = PP in L/E, and (3) αi,p (fb) = αi,p (f )2 if p is inert in L/E.

7.7. L-FUNCTIONS

73

Using these facts, it is easy to deduce the equalities Lp (As(fb ⊗ ϑ−1 ), s) = (1 − η(p)NE/Q (p)−s )−1 Lp (Ad(f ), s) and Lp (As(fb ⊗ θ−1 ), s) = (1 − NE/Q (p)−s )−1 Lp (Ad(f ) ⊗ η, s) for p - dL/E (c ∩ OE ) from the local Euler factors for L(As(f ⊗ φ), s) and L(Ad(f ), s) ((7.7.2) is also helpful). Now assume that p | (c ∩ OE ) but p - dL/E (f(θ) ∩ OE ). In this case we wish to prove the local equalities Lp (As(fb ⊗ ϑ−1 ), s) = Lp (Ad(f ), s) Lp (As(fb ⊗ θ−1 ), s) = Lp (Ad(f ) ⊗ η, s).

(7.7.5)

By Proposition D.9, p | cE . Thus for any prime P|p we have (7.7.6)

f(π(f )v(p) ) 6= OE,v(p) , f(π(fb)v(P) ) 6= OL,v(P) , and f(χE,v(p) ) = OE,v(p) .

It follows from the classification of irreducible admissible representations of GL2 over a local field that L(π(f )v(p) ) = L(π(fb)v(P) ) = 1 which implies that a(pn , f ) = a(pn OL , fb) = 0 for all n > 0 (see, e.g. §D.6 for a table of the L-functions of admissible representations of GL2 , keeping in mind that any supercuspidal representation π of GL2 satisfies L(π, s) = 1). Thus the left hand side of each of the equations in (7.7.5) is 1. On the other hand, (7.7.6) also implies that πv(p) , if it is principal or special, is not minimal. Thus the right hand side of each of the equalities in (7.7.5) is 1 as well. For the rest of the primes, both statements are tautologies. We close this section with one more proposition: Proposition 7.9. If f ∈ Sκnew (K0 (cE ), χE ) is a newform and fb ∈ Sκbnew (K0 (c), χ) is the unique normalized newform that is a base change of f , then Lf(θ)DL/E (Ad(fb), s) equals LdL/E (f(θ)∩OE ) (Ad(f ), s)LdL/E (f(θ)∩OE ) (Ad(f ) ⊗ η, s), where θ is as in Proposition 7.8. Proof. We again use the theory of quadratic base change for GL2 freely. For unexplained facts on minimal and non-minimal representations, see [HT, p. 243]3. We will prove that for p - (f(θ) ∩ OE )dL/E we have Y (7.7.7) LP (Ad(fb), s) = Lp (Ad(f ), s)Lp (Ad(f ) ⊗ η, s). P|p 3Be

forewarned that Hida and Tilouine use notation different from ours for the local representations. Our notation is the same as that in [JL].

74


Notice first that the set of primes dividing cE is contained in dL/E (c ∩ OE ). Thus if p - dL/E (c ∩ OE ), then the equality follows immediately from the definition of the Euler factors above combined with our observations on the relationship of αi,p (f ) and αi,P (fb) from the proof of Proposition 7.8. Assume now that p - dL/E but p | c ∩ OE (still under the assumption that p - f(θ) ∩ OE ). Then χE is unramified at p and π(f )v(p) is ramified at p. If πv(p) were minimal, then we would have that πv(p) is a principal series representation π(χ1 , χ2 ) or a special representation σ(χ1 , χ2 ) where we may assume that χ1 is unramified. This is clearly impossible if χ1 χ2 = χE , so πv(p) is not minimal. The same argument proves that πv(p) ⊗ η is not minimal. It follows that the left hand side of (7.7.7) is identically 1 and f(πv(p) ) 6⊇ p. Since p - dL/E , it follows that the exponent of the conductor of π bv(P) is at least 2 for any P|p. This implies L(b πv(P) ) = 1 for all P|p and thus π bv(p) is not minimal (see [HT, p. 243]). It follows that the right hand side of (7.7.7) is 1 as well. 7.8. Relationship with Hida’s notation As mentioned above, the content of this chapter is drawn from Hida’s papers [Hid1], [Hid4], and [Hid5]. For the convenience of the reader, we now describe how the notation for spaces of Hilbert modular forms in these papers relates to ours. For this purpose, let K11 (c) := {( ac db ) ∈ K0 (c) : ( ac db ) ≡ ( 10 ∗1 )

(mod c)} .

Moreover, let Sκ (K11 (c)) be the space obtained by replacing K0 (c) by K11 (c) in the definition of Sκ (K11 (c)) and replacing (2) by (2’) If γ ∈ G(Q), we have f (γα) = f (α). in §7.3 above. Define Sκcoh (K11 (c)) similarly. We then summarize the notation of [Hid1], [Hid4], and [Hid5] in the following list: • Let k = k 0 + 21 and w = k 0 + m + 1. Then in [Hid1], the space Sκ (K11 (c)) is denoted by ∗ Sk,w (K11 (c)) and the space Sκcoh (K11 (c)) is denoted by Sk,w (K11 (c)). • Let k = k 0 + 21 and w = 1 − m. Then in [Hid4], the space Sκ (K11 (c)) is denoted by Sk,w (K11 (c)). • In [Hid5], the space Sκ (K0 (c), χ) is denoted by S(k0 ,(m,0)) (K0 (c), χ), where the “0” in the weight reflects the fact that we only consider totally real fields in this paper, whereas Hida treats automorphic forms on ResF/Q (GL2 ) for arbitrary number fields F in [Hid5]. We close this chapter by explaining why we introduced two notations, Sκ (K0 (c), χ) and Sκcoh (K0 (c), χ) for the same Hecke module. On the one hand, it is more convenient to relate Fourier expansions to Hecke eigenvalues if we define the Hecke action as it is defined on Sκ (K0 (c), χ). On the other hand, it is easier to relate the action of the Hecke algebra on Sκcoh (K0 (c), χ) to the action of the Hecke algebra on certain sheaves via Hecke correspondences. This is especially the case

7.8. RELATIONSHIP WITH HIDA’S NOTATION

75

if one wants the Hecke correspondences to preserve local systems of A-modules where A is not necessarily a field (see §8.5). The two Hecke modules Sκ (K0 (c), χ) and Sκcoh (K0 (c), χ) are then seen to be isomorphic using the isomorphism (7.3.4) given above.

CHAPTER 8

The Automorphic Description of Intersection Cohomology In this chapter, we first fix an orbifold stratification of X0 (c) and set notation for certain local systems L(κ, χ0 ) on Y0 (c) associated to the standard irreducible representations of G. These local systems are endowed with natural pairings which we recall in §8.4. There is a family of Hecke correspondences on X0 (c) related to the Hecke algebra mentioned in §7.4 above; we recall this family and use the results of §5.2 to lift these correspondences to the L(κ, χ0 ) in §8.5. This may all be viewed as preparatory work for §8.7, in which we recall a description, due to Harder, of the intersection cohomology groups I m H ∗ (X0 (c), L(κ, χ0 )) as Hecke modules using the Hilbert modular forms recalled in §7.3. Our presentation owes a lot to Hida, who used automorphic forms to fix integral normalizations of the isomorphisms occurring in Harder’s work (see [Hid5], [Hid1] and [Gh]). The automorphic description of the intersection cohomology groups I m H ∗ (X0 (c), L(κ, χ0 )) gives us, in particular, a complete picture of these groups as Hecke modules. This will be crucial in the proof of Theorem 9.4, the full version of the “first main theorem” of the introduction, in Chapter 9 below. In the final section §8.8 we show how the pairings on intersection cohomology introduced in §8.4 relate to the Petersson inner product (see Lemma 8.4). This will be a key tool in the proof of Theorem 9.5, the full version of the “second main theorem” of the introduction, in Chapter 9. 8.1. Coefficient rings For the purposes of this chapter, fix an ideal c ⊂ OL , a weight κ ∈ X (L) and a quasichar0 −2m acter χ satisfying χ∞ (b∞ ) = b−k . Moreover, fix an idéle e c ∈ A× cv = 1 for all ∞ L such that e v | ∞ and the fractional ideal associated to the finite part e c0 of e c is c. Using this notation, let (8.1.1)

0 −1 Wc := e c 0

be the Atkin-Lehner matrix. In §8.3 below, we will fix a particular family of local systems of modules on Y0 (c). For the purposes of this paper, it would suffice to take all of these modules to be C-vector spaces, but, in view of future applications, we work with A-modules for a larger class of subalgebras 76

8.1. COEFFICIENT RINGS

77

A ≤ C. In this section, (which may be skipped by those who are primarily interested in the case A = C), we make these assumptions on A precise. Fix a “large” number field M , Galois over Q, such that (1) All Galois conjugates of L are in M . (2) All Galois conjugates of the number field generated by the values of the finite-order 0 character χ| · |Ak L+2m on A× L are in M . Choose, once and for all, an embedding OM ,→ C. We now view C as a OM -algebra via this embedding. In this chapter, A ≤ C will be an OM -subalgebra satisfying the following conditions: (1A) The algebra A contains 1 and is regular, Noetherian, and of finite cohomological dimension. (2A) The algebra A contains the values of χ|Qp|c O× . v(p) (3A) The integers j with 1 ≤ j ≤ k 0 are invertible in A. (4A) We can choose a torsion-free finite-index normal subgroup K fr E K0 (c) normalized by Wc such that every rational integer dividing |K0 (c)/K fr | is invertible in A. (5A) For any 1 ≤ i ≤ h(K0 (c)) and σ ∈ Σ(L), the ideal (det(σ(ti (c))) ∩ M )A is principal and invertible in A. bL ,→ O bM the embedding that In the last condition, we abused notation and denoted by σ : O σ : L ,→ R induces. Notice that all of these conditions are automatically satisfied if A = C or A is a sufficiently large field. In particular, for condition (4A), we may use the K fr given in the first item of the following set of remarks: Remarks. (1) For each rational prime p ∈ Z, let Q(µp )+ be the maximal totally real subfield of the pth cyclotomic field Q(µp ). If p ⊂ OM is a prime ideal outside of the set of prime ideals dividing {p : Q(µp )+ ≤ L}, then using the argument in Lemma 1 of [Gh, §3.1] we have that OM,p satisfies (4A) with a b a b 1 0 fr n b K = K0 (c) ∩ ∈ G(Z) : ≡ (mod q ) c d c d 0 1 for some auxiliary prime q ⊂ OL and sufficiently large n. (2) If A = OM,p for some prime p ⊂ OM , we can always choose the ti (c) so that the last condition is satisfied. This condition is needed for equation (8.4.7).

78

8. THE AUTOMORPHIC DESCRIPTION OF INTERSECTION COHOMOLOGY

8.2. An orbifold stratification of X0 (c) When K0 (c) has torsion, the open Hilbert modular variety Y0 (c) is not in general a manifold, but only an orbifold. Therefore, in order to discuss local systems of A-modules on Y0 (c) (and more generally, sheaves on X0 (c)) in a natural manner, it is useful to fix an A-orbifold stratification of X0 (c). This is the goal of the current section. For generalities on A-orbifolds, we direct the reader to Appendix B. Recall from (7.1.7) that we have an identification h(K0 (c))

Y0 (c) ←→

[ i=1

h(K0 (c))

Y0i (c)

=

[

Γi0 (c)\HΣ(L) .

i=1

fr

Let K E K0 (c) be the torsion-free, finite index normal subgroup normalized by Wc that appeared in the set of assumptions on A above (any torsion-free, finite index normal subgroup will do if A is a field). Set 0 fr −1 Γi,fr E Γi0 (c). 0 := G(Q) ∩ ti (c)G(R) K ti (c)

Let Σ(L) Y0i,fr := Γi,fr 0 \H

and i,fr i Γqu i := Γ0 (c)/Γ0 . i,fr i,fr in a manner such that the quotient Γqu is naturally isomorphic Then Γqu i acts on Y0 i \Y0 i to Y0 (c). This action extends by continuity to an action on the Bailey-Borel compactification X0i,fr of Y0i,fr . We give X0i (c) the orbifold stratification induced by the canonical projection i,fr i π fr : X0i,fr −→ Γqu i \X0 = X0 (c).

Thus the two strata of X0i (c) are the zero-dimensional stratum X0i (c) − Y0i (c) consisting of the cusps of Γi0 (c) and the complex [L : Q]-dimensional strata Y0i (c). For an orbifold atlas of Y0i (c) we fix a good cover U of Y0i (c) and for each open set U ∈ U assign the orbifold chart fr (U, (π fr )−1 (U ), Γqu i , π |(π fr )−1 (U ) )

(see Appendix B). It is easy to see what to take for the orbifold chart of X0i (c) − Y0i (c) (after all, X0i (c) − Y0i (c) is just a finite collection of points). Using the results recalled in §B.5, one can prove the following lemma: Lemma 8.1. If A ≤ C is an algebra satisfying the requirements given in §8.1, then Y0i (c) is an A-homology manifold. is invertible in A by one of our The key point is that the order of any subgroup of Γi,qu 0 assumptions on A. The lemma implies, in particular, that we may use the orbifold stratification given above to compute the intersection homology and cohomology groups with respect to any local system of A-modules (see §3.3).

8.3. REPRESENTATIONS AND LOCAL SYSTEMS

79

Thus we have described a stratification of X0i (c) by orbifolds for all i, and hence a stratification of X0 (c) by orbifolds. For the rest of this paper, whenever we consider differential forms or local systems on Y0 (c) or X0 (c), we will always mean differential forms or local systems with respect to the orbifold structure we have just fixed. Similarly, when constructing intersection homology and cohomology sheaves of A-modules, we will use the A-homology stratification given above (again, see §3.3). Remark. If A is a field and Γ0 E Γi0 (c) is another torsion-free normal arithmetic subgroup of G(Q), then we may give X0i (c) an A-orbifold structure as above using the canonical projection π 0 : Γ0 \HΣ(L) −→ Γi0 (c)\HΣ(L) instead of π fr . The A-homology stratification given by this orbifold structure is equivalent the original A-homology stratification, for they both admit refinements using the A-orbifold structure induced by the projection π 00 : Γ0 ∩ Γi0 (c)fr \HΣ(L) −→ Γi0 (c)\HΣ(L) . 8.3. Representations and local systems In section we fix notation for certain local systems on Y0 (c) arising from certain representations of G. For an OM -module A ≤ C and weight κ = (k 0 , m) ∈ X (L) satisfying the conditions in §8.1, let L(κ, A) := A[{Xσ , Yσ }σ∈Σ(L) ] be the space of polynomials in indeterminants {Xσ , Yσ }σ∈Σ(L) with coefficients in A that are homogeneous of degree kσ0 in Xσ , Yσ . We view L(κ, A) as a representation of G(A) via (8.3.1) γ · P ((Xσ , Yσ )σ∈Σ(L) ) := det(γ)−m P (Xσ , Yσ ) tσ(γ −1 ) σ∈Σ(L) ). We now construct local systems out of these representations. First, recall that in the identification h(K0 (c)) h(K0 (c)) [ [ i Y0 (c) ←→ Y0 (c) = Γi0 (c)\HΣ(L) , i=1

i=1

we had Γi0 (c) := G(Q) ∩ ti (c)G(R)0 K0 (c)tj (c)−1 for certain ti (c) ∈ G(Af ) (see (7.1.7)). Now let (8.3.2)

bM ) ∩ L(κ, M ), Li (κ, OM ) := ti (c) · L(κ, O

bM ) denotes all the polynomials of the form ti (c) · P with P ∈ L(k 0 , O bM ). where ti (c) · L(k 0 , O Q bM := OM ⊗ Here O finite p Zp , and the implied action is the action of G(Af ) induced by (8.3.1). Set Li (κ, A) := Li (κ, OM ) ⊗OM A. Notice that Li (κ, A) depends on the choice of ti (c) ∈ G(Af ) we made back in §7.1. This fact is the motivation for the last assumption on A in §8.1 above (see (8.4.7) below).

80


× Let χ : A× L → C be the quasicharacter we fixed in §8.1. Let χ0 denote the restriction of b× ,→ A× . Denote by Li (κ, χ0 , A) the Γi (c)-module whose underlying abelian group is χ to O 0 L Lf Li (κ, A) with the action of an element γ = ( ac db ) on P ((Xσ , Yσ )σ∈Σ(L) ) ∈ L(κ, A) given by   Y (8.3.3) (γ · P )((Xσ , Yσ )σ∈Σ(L) ) = det(γ)−m  χv(p) (av(p) ) P (Xσ , Yσ ) tσ(γ −1 ) σ∈Σ(L) . p|c

Moreover, let L∨i (κ, χ0Q , A) be the Γi0 (c)-representation defined by replacing Q p|c χv(p) (dv(p) ) with p|c χv(p) (av(p) ) in (8.3.3). Of course, one must check that the defined i action of Γ0 (c) maps L(κ, A) to itself, but this follows from assumption (2A) of §8.1. We now use these representations to create local systems on Y0i (c). This is slightly complicated by the fact that Γi0 (c) may have torsion. To remedy this, recall the torsion free subgroup i,fr i Σ(L) Γi,fr is a manifold, it makes sense to form the local 0 E Γ0 (c) we fixed in §8.2. Since Γ0 \H fr system Li (κ, χ0 , A) of locally constant sections of the projection Σ(L) Σ(L) Γi,fr × Li (κ, χ0 , A)) −→ Y0i,fr = Γi,fr . 0 \(H 0 \H i,fr As in §8.2, let π fr : Y0i,fr → Γqu = Y0i (c) be the canonical projection which gives rise to our i \Y0 i fixed orbifold structure on Y0 (c). Let Li (κ, χ0 , A) be the sheaf in the A-orbifold sense defined by Li (κ, χ0 , A)U := Li (κ, χ0 , A)fr(πfr )−1 U

for every U ∈ U, the good cover of Y0 (c) we fixed in §8.2. Notice that since this is a sheaf in the orbifold sense, a section of Li (κ, χ0 , A)U must be invariant under the natural action of Γqu i on Li (κ, χ0 , A)U (see B.5 for details). If qu fr 0 fr −1 0 fr f : (U, (π fr )−1 (U ), Γqu i , π |(π fr )−1 U ) −→ (U , (π ) (U ), Γi , π |(π fr )−1 U 0 )

is an embedding of charts, then in particular U ⊆ U 0 , and we let ψf : f ∗ (Li (κ, χ0 , A))U −→ Li (κ, χ0 , A)U be the canonical isomorphism (see §B.5 for the definition of a sheaf in the orbifold sense). We apply the same procedure to form the local system L∨i (κ, χ0 , A) on the orbifold Y0i (c). Let L(κ, χ0 , A) (resp. L∨ (κ, χ0 , A)) be the local system on the orbifold Y0 (c) that is Li (κ, χ0 , A) (resp. L∨i (κ, χ0 , A)) on Y0i (c) ⊂ Y0 (c). In the case A = C, we often drop the “A” in our notation. Thus we set Li (κ, χ0 ) := Li (κ, χ0 , C) L(κ, χ0 ) := L(κ, χ0 , C) L∨i (κ, χ0 ) := L∨i (κ, χ0 , C) L∨ (κ, χ0 ) := L∨ (κ, χ0 , C). Remark. The reason that we assumed κ ∈ X (L) is that this is necessary for the local system Li (κ, χ0 , A) on Y0i (c) to have stalks that are isomorphic to Li (κ, χ0 , A) at points that are not elliptic fixed points (see §7.2 for the case A = C).

8.4. PAIRINGS

81

8.4. Pairings In order to discuss intersection numbers of (intersection homology) cycles on X0 (c) with coefficients, it is first necessary to fix a pairing on the relevant local systems. In this section we recall how to do this in our cases of interest. We first give a pairing of local systems over Y0 (c): h·, ·i : L(κ, χ0 , A) × L∨ (κ, χ0 , A) → AY0 (c) and then use the Atkin-Lehner involution to translate this into a pairing I p H∗ (X0 (c), L(κ, χ0 , A ⊗OM M )) × I p H∗ (X0 (c), L(κ, χ0 , A ⊗OM M )) −→ A ⊗OM M. In the case A = C, this pairing will be explicitly related to the Petersson inner product in §8.8 below. We continue to assume that A is an OM -module satisfying the conditions in §8.1 for a × fixed quasicharacter χ : A× L → C and weight κ ∈ X (L). For each 1 ≤ i ≤ h(K0 (c)), define a pairing h·, ·i : Li (κ, χ0 , A) × L∨i (κ, χ0 , A) −→ A

(8.4.1) by 0

* ⊗σ∈Σ(L)

kσ X

0

0 −j kσ

uσ,j Xσ

Yσj , ⊗σ∈Σ(L)

j=1

kσ X

+ 0 −j kσ

wσ,j Xσ

Yσj

j=1 0

[−k0 −2m]

:= | det(ti (c))|AL

kσ Y X (−1)j uσ,j wσ,kσ0 −j 0 , σ∈Σ(L) j=1

kσ j

where [−k 0 − 2m]1 := −k 0 − 2m. Notice that for γ = ( ac db ) ∈ Γi0 (c) we have ab ≡ ε (mod c) 0 × for some ε ∈ OL,+ , the group of totally positive units of OL . Since χ∞ (ε∞ ) = ε−k −2m and χ is trivial on L× by definition, it follows that we have Y 0 χv(p) (ap dp )ε−k −2m = χ(ε) = 1. p|c

With this in mind one can check that the pairing given by (8.4.1) is Γi0 (c)-equivariant and perfect. Thus h·, ·i induces pairings h·, ·iIH∗ , h·, ·iIH ∗ , h·, ·iK as in §3.4 which yield morphisms I p H2[L:Q]−k (X0i (c), L(κ, χ0 , A)) −→ HomA (I q Hk (X0i (c), L(κ, χ0 , A), A) I p H 2[L:Q]−k (X0i (c), L(κ, χ0 , A)) −→ HomA (I q H k (X0i (c), L(κ, χ0 , A), A) which are isomorphisms if A is a field and p, q are complimentary perversities by Theorem 3.4. Remark. Even if we take maximal torsion free quotients, it seems that these morphisms are not isomorphisms unless A contains (at least) the inverses of the order of certain homology groups attached to the links of the singularities of X0i (c). The question of determining for

82


which A these morphisms are isomorphisms is connected with the study of congruence primes for Hilbert modular forms (see [Gh] and the references therein). In order to create a pairing consistent with the Petersson inner product in the sense of Lemma 8.4 below, we must adjust h·, ·i slightly. Unfortunately this will require a substantial digression for the purpose of defining the action of the Atkin-Lehner involution on the local systems given in §8.3. Recall that in §8.1 we fixed an idèle e c ∈ A× cv = 1 for all v | ∞ L with e whose associated fractional ideal is c and set (8.4.2)

0 −1 e c 0

Wc :=

.

We now show that Wc yields an involution of the orbifold Y0 (c). Use strong approximation for A× L to write det(ti (c))−1e c = ri det(tj(i) (c))wi

(8.4.3)

× b× where ri ∈ L× + (the totally positive elements of L ) and wi ∈ OL . The index 1 ≤ j(i) ≤ h(K0 (c)) is uniquely determined by ti (c) and c. Then we have 1 0 0 det(ti (c))−1

Wcι = =

0 ri 0 ri

−1 i 0 tj(i) (c) −w 0 0 −1 −1 0 tj(i) (c)Wc−ι −1 0 0 −wi

Wcι .

Note that (8.4.4)

−1 1 0 G(R)0 0 det(ti (c))−1 −1 1 0 G(R)0 0 det(ti (c))−1

Γi0 (c) = G(Q) ∩

1 0 0 det(ti (c))−1

K0 (c)

Γi,fr 0 = G(Q) ∩

1 0 0 det(ti (c))−1

K fr

It follows from (8.4.4) that 0 −1 −1 i Γ0 (c) ri 0 0 −1 −1 i,fr Γ0 ri 0

j(i)

0 −1 ri 0

= Γ0 (c),

0 −1 ri 0

= Γ0

j(i),fr

,

i and hence for Γi ∈ {Γi,fr 0 , Γ0 (c)} we have maps

Wci : Γi \HΣ(L) −→ Γj(i) \HΣ(L) −1 Γi z 7−→ Γj(i) r0i −1 z. 0

8.4. PAIRINGS

83

These maps fit into the following commutative diagram of topological spaces: Y0i,fr ⊂

-

Y (K fr ) Wc --

--

Y0i (c) ⊂

Wci

-

Y0 (c)

Wci

(8.4.5) ? j(i),fr Y0

? ⊂

-

Y (K fr )

Wc

⊂

--

--

?

j(i) Y0 (c)

?

-

Y0 (c)

(for the definition of Y (K) see (7.1.5)). Here the diagonal maps are all canonical projections, Wc is defined for K ∈ {K fr , K0 (c)} by (8.4.6)

Wc : G(Q)\(D × G(Af )/K) −→ G(Q)\(D × G(Af )/K) G(Q)(z, g)K 7−→ G(Q)(z, gWcι )K,

and the inclusions are the same ones that give rise to the decomposition (7.1.7) of Y (K) into connected components. In particular, for the “back face,” we have made use of the natural inclusion h(K0 (c)) a Y0i,fr ,→ Y (K) i=1

that exists due to the fact that we can factor the determinant map as follows: det - T (Q)\T (A)/ det(G(R)0 K0 (c)) G(A) --

de

t --

T (Q)\T (A)/ det(G(R)0 K fr ) It is easy to see from the commutativity of the left face of (8.4.5) that Wci actually induces a morphism of orbifold-stratified spaces j(i)

Wci : X0i (c) −→ X0 (c).

84


Moreover Wci is finite, proper, surjective, orientation preserving, and subanalytic. Let L0i (κ,χ0 , A) be the module formed by replacing ti (c) in the definition of Li (κ, χ0 , A) by 1 0 0 0 det(ti (c))−1 and let Li (κ, χ0 , A) be the associated orbifold sheaf. Then the natural inclusion 0 Li (κ, χ0 , A) ,→ Li (κ, χ0 , A) induces an inclusion Li (κ, χ0 , A) ,→ L0i (κ, χ0 , A)

(8.4.7)

of orbifold sheaves. By assumption (5A) in §8.1, for any 1 ≤ i ≤ h(K0 (c)) and σ ∈ Σ(L), a generator of the ideal (det(σ(ti (c))) ∩ M )A is invertible in A; thus (8.4.7) is an isomorphism. We have sheaf morphisms (in the sense of orbifolds) given over a connected open set U ⊂ Y0i (c) by (8.4.8)

Li (κ, χ0 , A)U −→L ˜ 0i (κ, χ0 , A)U −→ Wci∗ L∨j(i) (κ, χ0 , A)U −1 P 7−→ µi r0i −1 · P, 0

where the first morphism is the isomorphism of (8.4.7). Here Y k0 +2m µi := χ| · |AσL σ (det(tj(i) (c)))χ(det(ti (c)) χv(p) (−1), p|c

−1

the implied action of r0i 0 is the action induced by (8.3.1), and σ is any element of Σ(L). Using the assumptions of §8.1 again, we see that (8.4.8) is an isomorphism. We may view the pair (Id, Wc ) : X0 (c) ⇒ X0 (c) as a correspondence in the sense of §5.2. Setting Wc∗ := Wc∗ c(I)∗ by abuse of notation, we apply the results of Chapter 5 to obtain isomorphisms (8.4.9)

Wc∗ : I p H ∗ (X0 (c), L(κ, χ0 , A)) −→ ˜ I p H ∗ (X0 (c), L∨ (κ, χ0 , A)) Wc∗ : I p H∗ (X0 (c), L(κ, χ0 , A)) −→ ˜ I p H∗ (X0 (c), L∨ (κ, χ0 , A)).

The inverse of Wc∗ , up to a constant that is a unit in A by the assumptions of §8.1, is just Wc∗ constructed as above with the roles of L∨ (κ, χ0 , A) and L(κ, χ0 , A) reversed. We then define modified pairings [·, ·]IH∗ : I p H2[L:Q]−i (X0 (c), L(κ, χ0 , A)) × I q Hi (X0 (c), L(κ, χ0 , A)) −→ A [·, ·]IH ∗ : I p H 2[L:Q]−i (X0 (c), L(κ, χ0 , A)) × I q H i (X0 (c), L(κ, χ0 , A)) −→ A by (8.4.10)

[a, b]IH∗ : = ha, Wc∗ biIH∗ [a, b]IH ∗ : = ha, Wc∗ biIH ∗ .

Suppose that p, q are complementary perversities and A is a field. In this case h·, ·iIH∗ and h·, ·iIH ∗ are perfect. Since Wc∗ is an isomorphism, this implies [·, ·]IH∗ and [·, ·]IH ∗ are perfect under the same assumptions on A and the perversities p, q.

8.5. HECKE CORRESPONDENCES

85

8.5. Hecke correspondences In this section we fix notation for Hecke correspondences on Y0 (c) for a fixed ideal c ⊂ OL . We assume the notation of §7.4. In particular, for y ∈ R(c) we consider the double coset K0 (c)yK0 (c). As above, let ι be the main involution on GL2 , so y ι := det(y)y −1 and y −ι = det(y)−1 y. The Hecke correspondence associated to K0 (c)yK0 (c) is then (8.5.1)

(c(I), c(y ι )) : Yyι K0 (c)y−ι ∩K0 (c) ⇒ Y0 (c).

Here c(I) is just the natural projection and c(y ι ) is the composition of the map Yyι K0 (c)y−ι ∩K0 (c) → YK0 (c)∩y−ι K0 (c)yι induced by multiplication on the right by y ι : (8.5.2)

D × G(Af ) −→ D × G(Af ) (x, a) 7−→ (x, ay ι )

followed by the natural projection YK0 (c)∩y−ι K0 (c)yι → YK0 (c) . The maps c(I) and c(y ι ) both extend by continuity to the Bailey-Borel and the reductive Borel-Serre compactifications as finite, proper, surjective, orientation-preserving maps. We wish to follow the model of §6.5 and lift this correspondence to the local systems L(κ, χ0 , A) in §8.3. As in the previous section, this is complicated by the fact that, in general, Y0 (c) is only an A-orbifold and not a manifold. Moreover, we cannot simply apply the results of §6.5 as they stand because the representations Li (κ, χ0 , A) of Γi0 (c) do not extend to representations of G unless χ0 is the trivial character. Let us first write the correspondence (c(I), c(y)) in terms of its action on a particular component Y0i (c) as we did for Wc in (8.4.5) above; we follow [Hid1, §7]. Fix a y ∈ R(c) and temporarily write K 0 := K0 (c) ∩ y ι K0 (c)y −ι 0

K fr := K fr ∩ y ι K fr y −ι , where K fr E K0 (c) is the torsion-free finite-index subgroup fixed in §8.1. In analogy with the definition of Γi,fr 0 in §8.2, let 0

0

Γi,fr : = G(Q) ∩ ti (K 0 )K fr ti (K 0 )−1 0 0

0

Σ(L) Y0i,fr : = Γi,fr . 0 \H

By strong approximation, for each 1 ≤ i ≤ h(K 0 ) there is a unique 1 ≤ j(i) ≤ h(y −ι K 0 y ι ) = h(K 0 ) such that1 (8.5.3) 1This

ti (K 0 )y ι = γi tj(i) (y −ι K 0 y ι )y −ι u0 u∞ y ι j(i) also depends on y, and is in general not equal to the j(i) we considered in §8.4.

86


for some γi ∈ G(Q), u0 ∈ K 0 and u∞ ∈ G(R)0 . Then Γj(i) (y −ι K 0 y ι ) =G(Q) ∩ tj(i) (y −ι K 0 y ι )y −ι G(R)0 K 0 y ι tj(i) (y −ι K 0 y ι )−1 =G(Q) ∩ γi−1 ti (K 0 )G(R)0 K 0 ti (K 0 )−1 γi =γi−1 Γi (K 0 )γi , j(i),fr0

0

= γi−1 Γi,fr and similarly Γ0 0 γi . In analogy with (8.4.5), we have the following commutative diagram of topological spaces: 0

Y0i,fr

0

⊂

-

Y (K fr )

c(y ι ) --

--

c γi−1

Yi (K 0 ) ⊂

c γi−1

(8.5.4)

-

?

j(i),fr0 Y0

Y (K 0 )

? ⊂

-

Y (K fr )

⊂

--

--

?

j(i) Y0 (c)

c(y ι )

?

-

Y0 (c)

Here all of the projections are canonical, the injections are those induced by strong approximation as in (7.1.7), c(y ι ) is defined as above, and the c(γi−1 ) are defined to be the maps j(i),fr0

0

Σ(L) Γi,fr −→ Γ0 0 \H 0

\HΣ(L)

j(i),fr0 −1 γi z

Γ0i,fr z 7−→ Γ0 and

Γi (K 0 )\HΣ(L) −→ Γj(i) (c)\HΣ(L) Γi (K 0 )z 7−→ Γj(i) (c)γi−1 z. The commutativity of the left face of (8.5.4) implies that c(γi−1 ) induces an orbifold morphism j(i) c(γi−1 ) : Yi (K 0 ) → Y0 (c), and it is easy to see that the extension of this map to the BaileyBorel compactification Xi (K 0 ) can be thought of as an orbifold morphism when restricted to the 0-dimensional singular stratum. This morphism is proper, surjective, finite, and orientationpreserving.

8.5. HECKE CORRESPONDENCES

87

One can draw a similar (but less complicated) diagram in order to see that the canonical projection c(I) : X(K 0 ) → X0 (c) can be viewed as an A-orbifold morphism on each strata. Again, it is proper, surjective, finite, and orientation-preserving. Thus we may view the pair of morphisms (c(I), c(y ι )) : X(K 0 ) ⇒ X0 (c) as a correspondence of pseudomanifolds in the sense of §5.2 such that the restriction of either map to a nontrivial stratum is an orbifold morphism. We now lift this correspondence to the local systems L(κ, χ0 , A) and L∗ (κ, χ0 , A). By assumption (5A), we can and do assume that det(γi )m ∈ A. Without loss of generality, we may choose the ti (K 0 ) and ti (y −ι K 0 y ι ) so that the idéles det(ti (K 0 )) and det(ti (y −ι K 0 y ι )) are trivial at all places dividing c and ∞. Moreover, by definition of γi , we have γi = ti (K 0 )u−1 y ι tj(i) (y −ι K 0 y ι ) ∈ ti (c)R(c)ι G(R)0 tj(i) (c)−1 . Using these facts and recalling our conventions on the shape of the ti (K) for arbitrary compact open subgroups K from §7.1, we see that the action (8.3.1) of γi−ι gives a map Li (κ, χ0 , A) −→ Lj(i) (κ, χ0 , A)  −1 Y P 7−→  χv(p) (ap ) γi−1 · P p|c

(see [Hid5, p. 470]) where γi = ( ac db ). This yields a morphism of local systems (in the sense of A-orbifolds): (8.5.5)

c(I)∗ Li (κ, χ0 , A)Yi (K) −→ c(y ι )∗ Lj(i) (κ, χ0 , A)Yi (K)

In defining this morphism, we think of P as an element of Li (κ, χtriv , A) and define the action b× → C× is the trivial character. If A is not a field, this of γ −1 accordingly. Here χtriv : O L morphism will not in general be an isomorphism. However, the argument of Chapter 5 still applies to produce a morphism T (y ι ) := c(y ι )∗ c(I)∗ : (8.5.6)

T (y ι ) : I p H∗ (X0 (c), L(κ, χ0 , A)) −→ I p H∗ (X0 (c), L(κ, χ0 , A)) I p H ∗ (X0 (c), L(κ, χ0 , A)) −→ I p H ∗ (X0 (c), L(κ, χ0 , A))

for any perversity p. If m ⊂ OL is an ideal, then define (8.5.7)

Tc (m)∗ :=

X

T (y ι ),

K0 (c)yK0 (c) y∈R(c), det y=m

where the sum is over double cosets K0 (c)yK0 (c) with y ∈ R(c) such that det(y) = m.

88


8.6. The action of the component group Consider the component group π0 (G(R)) of G(R): (8.6.1)

0 π0 (G(R)) = {(γσ )σ∈Σ(L) : γσ = ( ±1 0 1 )}.

For each J ⊂ Σ(L), we have an element ιJ = (γσ )σ∈Σ(L) ∈ π0 (G(R)) such that γσ is the identity if and only if σ 6∈ J. Multiplication on the right by the element ιJ induces a real analytic involution2 (8.6.2) where

ιJ : G(R)/K∞ −→ G(R)/K∞ (zσ )σ∈Σ(L) 7−→ (wσ )σ∈Σ(L) , ( zσ wσ := zσ

if σ ∈ J if σ ∈ 6 J.

Here we used the identification G(R)/K∞ = (C − R)Σ(L) (see (7.1.4)). For any compact open subgroup K ≤ G(Af ), the involution ιJ induces another involution (8.6.3)

ιJ : G(Q)\(D × G(Af ))/K −→ G(Q)\(D × G(Af ))/K G(Q)(x, g)K 7−→ G(Q)(xιJ , g)K

which we denote by the same symbol by abuse of notation. One can check that the ιJ yield involutions of X0 (c) as a pseudomanifold which are A-orbifold morphisms on each nonempty stratum. This requires the argument we gave to prove the analogous statement for Wc after (8.4.5) in §8.4 above. These morphisms are proper, finite and surjective, but they are not in general orientation preserving (in fact they usually change Hodge types). One may prove in the same manner as we did for Wc in §8.4 that ιJ lifts to induce sheaf morphisms from L(κ, χ0 , A) and L∨ (κ, χ0 , A) to themselves. In particular, we obtain involutions ιJ∗ : I p H∗ (X0 (c), L(κ, χ0 , A)) −→ I p H∗ (X0 (c), L(κ, χ0 , A)) ιJ∗ : I p H ∗ (X0 (c), L(κ, χ0 , A)) −→ I p H ∗ (X0 (c), L(κ, χ0 , A)) for any perversity p. For more details, see [Hid1, §7]. Using (8.4.6), (8.5.2) and (8.6.3), one may check that ιJ∗ commutes with Wc and T (y) for all J ⊂ Σ(L) and y ∈ R(c). 8.7. The automorphic description of intersection cohomology In this section, all local systems will be C-vector spaces, and all orbifolds will be Q-orbifolds. × As above, let κ = (k 0 , m) ∈ X (L) and let χ : L× \A× L → C be a quasicharacter satisfying 0 −2m χ∞ (b∞ ) = b−k for b∞ ∈ A× ∞ L∞ . Moreover assume that the conductor of χ divides an ideal c ⊂ OL , which we will fix throughout this section. Given such a weight and character, in §8.3 we set notation for certain explicit local systems L(κ, χ0 ) := L(κ, χ0 , C). We now give a 2The

involution ιJ , which will always have a subscript, should not be confused with the main involution ι.

8.7. THE AUTOMORPHIC DESCRIPTION OF INTERSECTION COHOMOLOGY

89

useful basis of differential forms for the intersection cohomology groups I m H ∗ (X0 (c), L(κ, χ0 )) of middle3 perversity m. We begin with the so-called cuspidal classes. Given an f ∈ Sκcoh (K0 (c), χ) and a subset J ⊂ Σ(L), we now associate a differential form ωJ (f ) ∈ Ω[L:Q] (Y0 (c), L(κ, χ0 )) (in the sense of orbifolds). In order to do this, fix once and for all, an ordering σ1 , . . . , σ[L:Q] of Σ(L) and let dz := dzσ1 ∧ · · · ∧ dzσ[L:Q] . Moreover, for α ∈ G(Q) and z ∈ (C − R)Σ(L) , let Y j(α, z) := (8.7.1) j(σ(α), zσ ) σ∈Σ(L)

be the typical automorphy factor, where j (( ac db ) , z) := (cz + d) for ( ac db ) ∈ GL2 (R)0 and z ∈ C (here GL2 (R)0 is the connected component of the identity in GL2 (R)). S 0 (c)) i Y0 (c) from (7.1.7). As in §7.3, let Recall the decomposition Y0 (c) = h(K i=1 fi (α∞ ) := f (ti (c)α∞ ) : G(R)0 → C.

√ √ Set z0 = ( −1, . . . , −1) ∈ HΣ(L) , and for each z ∈ HΣ(L) , choose a αz ∈ G(R)0 such that αz z0 = z (where αz acts on z0 by Möbius transformation). The invariance properties of f imply that the expression   Y 0 0 0 ω i (f )(z) := j(αz , z0 )k +21 det(αz )−k −m−1 fi (αz )  (−X + zY )kσ  dz σ∈Σ(L)

enjoys the following properties: • It is well-defined independently of the choice of αz . • It satisfies ω i (f ) ◦ γ = γ · ω i (f ) for γ ∈ Γi0 (c) (see [Hid1, §2 and §6] for proofs of these statements). For the second statement, on the left γ denotes the map z −→ γz and on the right the implied action is the action of Γi0 (c) on Li (κ, χ0 ). Hence ω i (f ) defines an element of the sheaf Ω[L:Q] (Y0i (c), Li (κ, χ0 )) of differential forms on the orbifold Y0 (c). Let ω(f ) be the element of Ω[L:Q] (Y0 (c), L(κ, χ0 )) that is ω i (f ) on the ith component for each 1 ≤ i ≤ h(K0 (c)). We then set ωJ (f ) := ι∗J ω(f ), where ιJ is defined as in §8.6. Remark. On the ith component Y0i (c) of Y0 (c) the differential form ωJ (f ) is holomorphic on the image of HΣ(L)−J and antiholomorphic on the image of HJ under the quotient map HΣ(L) → Y0i (c) (see [Hid5, p. 460, §2.4]). 3Recall

that the middle perversity is defined by m(c) := (c − 2)/2.

90


We wish to use the Zucker isomorphism Z of §6.3 to associate intersection cohomology classes to these differential forms. We cannot apply the Zucker isomorphism as it stands, since, as mentioned above, L(κ, χ0 ) does not necessarily extend to a representation of G. We now explain how to overcome this obstacle. In §8.1 we fixed a finite-index torsion-free normal subgroup K fr E K0 (c). The canonical projection π fr : Y0fr = Y (K fr ) → Y0 (c) was used in §8.2 to endow Y0 (c) with its orbifold structure. We assume for this section and the next that K fr is contained in K11 (c) := {( ac db ) ∈ K0 (c) : ( ac db ) ≡ ( 10 ∗1 )

(mod c)} .

Since C is a field, this is harmless for the purposes of this section (see the remark at the bL → C× denotes the trivial character, then Li (κ, χtriv ), viewed as end of §8.2). If χtriv : O a representation of Γi,fr 0 , extends to a representation of G, and hence yields a local system i,fr Li (κ, χtriv ) on Y0 (see §6.5). Over a connected open set U ⊂ Y0i,fr we have orbifold sheaf morphisms (8.7.2)

(π fr )∗ Li (κ, χ0 )U ,→ Li (κ, χtriv )U −→ (π fr )∗ (L(κ, χ0 ))U X P 7−→ γ −1 · P, Γi0 (c)fr γ∈Γi0 (c)fr \Γi0 (c)

where the first map is the canonical inclusion and the implied action is the action of Γi0 (c) on Li (κ, χ0 ). The inclusion yields a morphism (π fr )∗ : Ω• (Y0 (c), L(κ, χ0 )) −→ Ω• (Y0fr , L(κ, χtriv )). Choose a K∞ -invariant Hermitian metric on Li (κ, χtriv ) (viewed as a representation of G). Then the convergence of the Petersson inner product of f with itself, together with the fact [L:Q] that ωJ (f ) is closed, implies that (π fr )∗ ωJ (f ) ∈ Ω(2) (Y0 (c), L(Y0fr , κ, χtriv )) (see, e.g. [Hid5, §2.5]). The second morphism extends to a morphism Ip S(L(κ, χtriv )) −→ Ip S((π fr )∗ L(κ, χ0 )) by the functoriality of Deligne’s construction of the intersection cohomology sheaf (see [GM2, §3.5]), and this morphism induces a homomorphism (π fr )∗ : I p H ∗ (X0 (c), L(κ, χtriv )) −→ I p H ∗ (X0 (c), L(κ, χ0 )) using the arguments of Chapter 5. Thus, if we restrict to the case p = m, for each f ∈ Sκ,J (K0 (c), χ) and J ⊂ Σ(L), then we can and do consider the class (π fr )∗ Z[(π fr )∗ ωJ (f )] ∈ I m H [L:Q] (X0 (c), L(κ, χ0 )). We then have the following lemma.


91

Lemma 8.2. The linear map (8.7.3)

ωJ0 : Sκcoh (K0 (c), χ) −→ I m H [L:Q] (X0 (c), L(κ, χ0 )) 1 f 7−→ (π fr )∗ Z[(π fr )∗ ωJ (f )] [K0 (c) : K fr ]

is Hecke-equivariant for all J ⊂ Σ(L). We will indicate the proof of Lemma 8.2 at the end of this section. The subspace of I m H [L:Q] (X0 (c), L(κ, χ0 )) given by (8.7.4)

[L:Q] (X0 (c), L(κ, χ0 )) := I m Hcusp

M M χ

ωJ0 (Sκcoh (K0 (c), χ))

J⊂Σ(L)

is known as the space of cuspidal cohomology classes in I m H [L:Q] (X0 (c), L(κ, χ0 )). Here the 0 −2m × first sum is over all quasicharacters χ : L× \A× and such that χ∞ (a∞ ) = a−k ∞ L → C χ|Ob× = χ0 . We set L

j I m Hcusp (X0 (c), L(κ, χ0 )) = 0

if j = 6 [L : Q]. We now isolate the subspace of invariant classes (invariant in a slightly more general sense than that of §6.4). These can only occur in the case that κ = (0, m) ∈ X (L) for some m ∈ ZΣ(L) . As explained in §6.4 in the special case that χ is the trivial character, there is a canonical injection M fr i ς: H i (g, K∞ ; Cφ ⊗ L(κ, χtriv ))K ,→ H(2) (Y0fr , L(κ, χtriv )), φ

where the sum is over functions φ : G(Q)\G(A) → C of the form χ◦det for some quasicharacter i χ : L× \A× → C× satisfying χ∞ (b∞ ) = b−m ∞ (see [Har, §3.2]). Here H (g, K∞ ; Cφ ⊗ L(κ, χtriv )) is the relative Lie algebra cohomology of the the Lie algebra g of G(R). We thus have a subspace of invariant cohomology classes ! M m i fr ∗ i K fr (8.7.5) I Hinv (X0 (c), L((0, m), χ0 )) := (π ) Zς H (g, K∞ ; Cφ ⊗ L((0, m), χtriv )) φ

sitting inside I m H i (X0 (c), L(κ, χ0 )). For a more explicit description of these classes, see §11.2 below. We set ∗ I m Hinv (X0 (c), L(κ, χ0 )) = 0 if κ 6= (0, m) for some m ∈ Z1. The following theorem of Harder [Har] (as rephrased by Hida [Hid5, §3.1]) tells us that these two subspaces of the intersection cohomology actually fill the space:

92


Theorem 8.3 (Harder). We have ∗ ∗ I m H ∗ (X0 (c), L(κ, χ0 )) = I m Hinv (X0 (c), L(κ, χ0 )) ⊕ I m Hcusp (X0 (c), L(κ, χ0 )).

Remark. We note that this theorem is not phrased in the language of intersection cohomology in either [Har] or [Hid5, §3.1]. However, it is straightforward to deduce the given statement of the theorem from the statement in [Hid5, §3.1] using the fact that there is a well-known identification I m H [L:Q] (X0 (c), L(κ, χ0 )) = Image Hc[L:Q] (Y0 (c), L(κ, χ0 )) → H [L:Q] (Y0 (c), L(κ, χ0 )) since X0 (c) has isolated singularities. As promised, we now give the proof of Lemma 8.2: Proof of Lemma 8.2. We first show that ω : Sκcoh (K0 (c), χ) −→ Ω[L:Q] (L(κ, χ0 )) f 7−→ ω(f ) is Hecke-equivariant. For this, fix an f ∈ Sκcoh (K0 (c), χ) and a y ∈ R(c). For every compact open subgroup K ≤ G(Af ) and for each each 1 ≤ i ≤ h(K), define a function Fi,K (z) : HΣ(L) → C by choosing an αz ∈ G(R)0 such that αz z0 = z and setting 0

0

Fi,K (z) : = j(αz , z0 )k +21 det(αz )−k −m−1 f (ti (K)αz ) Here j(αz, z0 ) is the typical automorphy factor (8.7.1). Writing  δ(z) := 

 Y

0

(−X + zY )kσ  dz,

σ∈Σ(L)

and K 0 := K0 (c) ∩ y ι K0 (c)y −ι , we have that c(I)∗ ω(f ) is equal to Fi,K 0 (z)δ(z) on the ith component of Y (K 0 ) for 1 ≤ i ≤ h(K 0 ). Now, for each 1 ≤ i ≤ h(K 0 ), use strong approximation to choose a γi ∈ G(Q) satisfying (8.7.6)

ti (K 0 )y ι = γi0 tj(i) (y −ι K 0 y ι )y −ι uy ι


for some u ∈ K 0 and index 1 ≤ j(i) ≤ h(y −ι K 0 y ι ) = h(K 0 ) as in (8.5.3). Setting y = we claim that for every 1 ≤ i ≤ h(K 0 ), we have   Y Fi,K 0 (γi z)  χv(p) (av(p) ) γi−1 · δ(γi z) (8.7.7)

93 a0 b0 c0 d 0

,

p|c

−1

 =

Y

0

χv(p) (d0v(p) )

j(αz , z0 )k +21

p|c

f (tj(i) (y −ι Ky ι )αz y −ι )δ(z) . det(αz )m+k0 +1

Here γi = ( ac db ) acts by a Möbius transformation on HΣ(L) and γi acts on δ considered as an element of Li (κ, χtriv ); this is consistent with the action we used in (8.5.5) to define the lift of the correspondence T (y ι ). Assuming (8.7.7), it is easy to check that ω is Hecke-equivariant. By a computation, proving the equality (8.7.7) is reduced to proving the equality  −1  −1 Y Y (8.7.8)  χv(p) (av(p) ) f (ti (K 0 )γi∞ βz ) =  χv(p) (d0v(p) ) f (tj(i) (y −ι K 0 y ι )βz y −ι ) p|c

p|c

where we fix the choice βz := ( y0 x1 ) of a matrix satisfying βz z0 = z. We compute f (tj(i) (y −ι K 0 y ι )βz y −ι ) = f (γi0 tj(i) (y −ι K 0 y ι )γi∞ βz y −ι )

(by invariance under G(Q))

= f (ti (K 0 )u−1 γi∞ βz )

(by definition of γi0 ).

Since we chose ti (K 0 ) and tj(i) (y −ι K 0 y ι ) to be trivial at the places dividing c (see §8.5) we have that for p | c, −ι u−1 v(p) = γiv(p) yv(p) and it follows that Y

0

0

−1 −1 χv(p) (av(p) dv(p) dv(p) av(p) )=

p|c

Y

χv(p) (det(u−1 v(p) )) = 1.

p|c

Hence f (ti (K 0 )u−1 γi∞ βz ) = χ(uι )f (ti (K 0 )γi∞ βz ) Y 0 −1 = χv(p) (dv(p) av(p) )f (ti (K 0 )γi∞ βz ) p|c

=

Y

0 χv(p) (d0v(p) a−1 v(p) )f (ti (K )γi∞ βz )

p|c

which implies (8.7.8). Thus we have proven that ω is Hecke-equivariant.

94


The rest follows easily from the Hecke-equivariance of the Zucker conjecture isomorphism Z discussed in §6.3. We only point out that the lift of the correspondence we chose in §8.5 differs by a constant from the canonical lift considered in §6.3. 8.8. Pairings and the Petersson inner product As in the previous section, we assume that K fr is contained in K11 (c) := {( ac db ) ∈ K0 (c) : ( ac db ) ≡ ( 10 ∗1 )

(mod c)} .

and all orbifolds we consider will be Q-orbifolds. In this section, we will relate the Petersson inner product (recalled above in §7.5) of two modular forms to a pairing on certain associated differential forms. In order to state our result, first recall that we have a linear isomorphism (8.8.1)

0

Wc∗ : Sκ (K0 (c), χ) −→ Sκ (K0 (c), χ−1 | · |−2k −4m ) −k0 −2mσ

f (α) 7−→ χ−1 (det α)| det α|ALσ

f (αWc ),

where Wc is the Atkin-Lehner matrix given in (8.1.1) and σ is any element of Σ(L) (see [Hid1, §7 p. 354]). Moreover, if f is a newform, then (8.8.2)

0

Wc∗ (f ) = W (f )NL/Q (c)(kσ +2mσ )/2 fc , 0

where W (f ) is a complex number of norm 1 and fc ∈ Sκ (K0 (c), χ−1 | · |−2k −4m ) is the cusp form whose Fourier coefficients satisfy a(P, fc ) = a(P, f ) for almost all primes P ⊂ OL (see [Hid5, (4.10 b)]). We recall in particular that if the level of f is OL , then W (f ) = 1. The main result of this section is the following lemma, which will be of use to us in Chapter 9: Lemma 8.4. Let f, g ∈ Sκnew (K0 (c), χ) be newforms. Then 0 [ωJ0 (f −ι ),ωΣ(L)−J (g −ι )]IH ∗ 0

= (−1)[L:Q]+|J| (2i)[L:Q]+s NL/Q (c)[k +2m]/2 W (g)(f, g)P , where s :=

X

kσ0 .

σ∈Σ(L)

Here, as above, [k 0 + 2m]1 := k 0 + 2m. Before we prove Lemma 8.4, we provide a slight variant of the definition of ωJ (f ) for 0 f ∈ Sκcoh (K0 (c), χ) given in §8.7 above. For f ∈ Sκcoh (K0 (c), χ−1 | · |−2k −4m ), let   Y 0 0 0 ω ∨,i (f )(z) := j(αz , z0 )k +21 det(αz )−k −m−1 f (ti (c)αz )  (−X + zY )kσ  dz σ∈Σ(L)

8.8. PAIRINGS AND THE PETERSSON INNER PRODUCT

95

for αz and j(g, z) as in the beginning of §8.7. One can check that ω ∨,i (f ) defines an element of 0 [L:Q] the orbifold sheaf Ω(2) (Y0i (c), L∨i (κ, χ−1 | · |−2k −4m )). Let ω ∨ (f ) be the element of the orbifold [L:Q]

0

sheaf Ω(2) (Y0 (c), L∨ (κ, χ−1 | · |−2k −2m )) that is ω ∨,i (f ) on Y0i (c) for each 1 ≤ i ≤ h(K0 (c)), and set ωJ∨ (f ) : = ι∗J ωJ (f ) 0

ωJ∨ (f ) : = [K0 (c) : K fr ]−1 (π fr )∗ Z(π fr )∗ ωJ∨ (f ). We now give a proof of Lemma 8.4. Proof of Lemma 8.4. As in §8.4, we let h·, ·iIH ∗ denote the pairing induced by the standard pairing Li (κ, χ0 ) × L∨i (κ, χ0 ) → C given in (8.4.1) and “·” the associated product. We have 0 [K0 (c) : K fr ]2 [ωJ0 (f −ι ), ωΣ(L)−J (g −ι )]IH ∗

= (π fr )∗ Z(π fr )∗ ωJ (f −ι ), Wc∗ (π fr )∗ Z(π fr )∗ ωΣ(L)−J (g −ι ) IH ∗ = (π fr )∗ Z(π fr )∗ ωJ (f −ι ) · Wc∗ (π fr )∗ Z(π fr )∗ ωΣ(L)−J (g −ι ) .

Here denotes the augmentation. Using an argument similar to that given in the proof of Lemma 8.2 with (8.4.3) in place of (8.7.6) one proves that (8.8.3)

0

Wc∗ ωJ0 (g −ι ) = ωJ∨ ((Wc∗ (g))−ι )

where σ is any element of Σ(L). Thus (π fr )∗ Z(π fr )∗ ωJ (f −ι ) · Wc∗ (π fr )∗ Z(π fr )∗ ωΣ(L)−J (g −ι )

∨ = (π fr )∗ Z(π fr )∗ ωJ (f −ι ) · (π fr )∗ Z(π fr )∗ ωΣ(L)−J ((Wc∗ (g))−ι ) ∨ = (π fr )∗ Z(π fr )∗ ωJ (f −ι ) · Z(π fr )∗ ωΣ(L)−J ((Wc∗ (g))−ι )

∨ = [K0 (c) : K fr ] Z(π fr )∗ ωJ (f −ι ) · Z(π fr )∗ ωΣ(L)−J ((Wc∗ (g))−ι ) . By the comparison of the L2 wedge product and the intersection product contained in Lemma 6.3, we have that ∨ Z(π fr )∗ ωJ (f −ι ) · Z(π fr )∗ ωΣ(L)−J ((Wc∗ (g)−ι )) Z ∨ = (π fr )∗ ωJ (f −ι ) ∧ (π fr )∗ ωΣ(L)−J ((Wc∗ (g))−ι ) fr Y0 Z fr ∨ = [K0 (c) : K ] ωJ (f −ι ) ∧ ωΣ(L)−J ((Wc∗ (g))−ι ). Y0 (c)

Using the definitions of ωJ (f ) and (·, ·)P , we compute Z ∨ ωJ (f ) ∧ ωΣ(L)−J ((Wc∗ (g))−ι ) Y0 (c)

96


Z

∨ ι∗J ω∅ (f −ι ) ∧ ωΣ(L) ((Wc∗ (g))−ι )

=

Y0 (c)

=(−1)

|J|

h(K0 (c)) Z Y0i (c)

i=1

∧y

0

X

−k0 −m−1

y −k −m−1 (f −ι )i (( y0 x1 ))

Y

(−X + zY )kσ dz

σ∈Σ(L)

W (g)NL/Q (c)

[k0 +2m]/2

Y

x (gc−ι )i (( −y 0 1 ))

(−X + zY )kσ dz

σ∈Σ(L) |J|

= (−1)

h(K0 (c)) Z

X i=1

Y0i (c)

0

x (f −ι )i (( y0 x1 )) W (g)NL/Q (c)−(k +2m)/2 (gc−ι )i (( −y 0 1 )) [−k0 −2m]

0

× y −2k −2m−21 | det(ti (c))|AL

0

(z − z)k dz ∧ dz

0

= (−1)|J| W (g)NL/Q (c)[k +2m]/2 (2i)s (−2i)[L:Q] (f, g)P , and this completes the proof of the lemma.

CHAPTER 9

Hilbert Modular Forms with Coefficients in a Hecke Module The goal of this chapter is to prove Theorems 9.4 and 9.5, the full versions of Theorems 1.1 and 1.2 given in the introduction. In §9.2, we slightly generalize the Tb(m) of the introduction and prove that they have the “key property” mentioned there. In §9.3, we prove Theorem 9.4 as a corollary of Theorem 9.3, in which I m H∗ (X0 (c), L(κ, χ0 )) is replaced by an arbitrary Hecke module. We then prove Theorem 9.5 in §9.4. 9.1. Notation Let L/E be a quadratic extension of totally real number fields of discriminant dL/E . We × use class field theory to identify the group of Hecke characters η : E × \A× trivial on E → C × × ∧ the image of the norm NL/E : AL → AE with Gal(L/E) . Note that these characters are trivial at the infinite places, so we may use the construction of §7.6 to define twists of Hilbert modular forms by these characters. As above, we let Σ(E) := {σ1 , . . . , σ[E:Q] } denote the set 0 of embeddings E ,→ R and let Σ(L) := {σ10 , . . . , σ[L:Q] } denote the set of embeddings L ,→ R. For κ = (k 0 , m) ∈ X (E), define κ b = (b k 0 , m) b ∈ ZΣ(L) by stipulating that b kσ0 0 = kσ0 and m b σ0 = mσ 0 if σ extends σ. Fix an ideal c ⊂ OL , and let χ be a quasicharacter with conductor dividing c. We assume that χ is of the form χ = χE ◦ NL/E for some quasicharacter χE : E × \A× E → C ∞,× −k0 −2m for b∞ ∈ AE . If n ⊂ OL is coprime to c, we often abuse satisfying χE∞ (b∞ ) = b∞ notation and set χ(n) = χ(n), where n ∈ A× ele trivial at the infinite places and at the places dividing c. We will L is any id´ commit the same abuse of notation in dealing with χE (n0 ) for n0 coprime to dL/E (c ∩ OE ). In this section we will use the theory of prime degree base change for GL2 developed by Langlands [L]. This theory is recalled in Appendix D. 9.2. Hecke algebras and base change Fix a pair of ideals cE ⊂ OE and c ⊂ OL , and let (9.2.1)

TcE dL/E := Z[{K0 (cE )xK0 (cE ) ∈ TcE : xv(p) = ( 10 01 ) for p|cE dL/E }] TcDL/E := Z[{K0 (c)xK0 (c) ∈ Tc : xv(p) = ( 10 01 ) for P|cDL/E }]. 97

98

9. HILBERT MODULAR FORMS WITH COEFFICIENTS IN A HECKE MODULE

Thus TcE dL/E ≤ TcE and TcDL/E ≤ Tc are the subalgebras consisting of operators trivial at the ramified places. There is an algebra homomorphism b : TcDL/E −→ TcE dL/E

(9.2.2) induced by the L-map

b : L GL2E −→ L ResL/E GL2 given on the connected components of the L-groups by the diagonal embedding, as explained in §D.4 below. Explicitly, b is defined by stipulating that for P - cDL/E we have ( ¯ split in L/E TcE (p) for p = PP b(Tc (P)) = (9.2.3) 2 TcE (p ) − NE/Q (p)TcE (p, p) for p = P inert in L/E ( ¯ split in L/E TcE (p, p) for p = PP b(Tc (P, P)) = . TcE (p2 , p2 ) for p = P inert in L/E (see Lemma D.5). Since the Tc (P), Tc (P) for P - cDL/E generate the algebra TcDL/E , this suffices to define b. Let Z[χ] (resp. Z[χE ]) be the subalgebra of C generated by χ(m) (resp. χE (n)) as m (resp. n) ranges over the ideals of OL coprime to cDL/E (resp. OE coprime to cE dL/E ). We note that Z[χ] ≤ Z[χE ]. We then have ideals (9.2.4)

I(χ) : = hTc (m, m) − χ(m)im+cDL/E =OL ⊂ TcDL/E ⊗Z Z[χ] I(χE ) : = hTcE (m, m) − χE (m)in+cE dL/E =OE ⊂ TcE dL/E ⊗Z Z[χE ].

It is easy to see from (9.2.3) that b induces a morphism b : TcDL/E ⊗Z Z[χE ]/I(χ) −→ TcdL/E ⊗Z Z[χE ]/I(χE ). Using the p-adic Cartan decomposition and (7.4.3), one checks that its image is is generated as a Z[χE ]-algebra by Tc (NL/E (P)) as P ranges over the prime ideals of OL coprime to cDL/E , and as a Z[χE ]-module by Tc (NL/E (m)) as m ranges over the ideals of OL coprime to cDL/E (compare [Bu, Proposition 4.6.2 and Proposition 4.6.4]). For every prime p of OE split in L/E choose a prime Pp above p. We define  Id if r = 0,    T (Pr ) if NL/E (P) = p is prime, c p TbcE ,χE (NL/E (P)r ) := r r−1  Tc (P ) + χE (p)NE/Q (p)Tc (P ) if P = p is an inert prime,    0 otherwise. Q Here when we say p splits or is inert we mean p splits or is inert in L/E. If m = P∈I NL/E (PrP ) for some set I of primes of OL and some set of integers rP ≥ 0, we define Y (9.2.5) Tb(m) := TbcE ,χE (m) := TbcE ,χE (NL/E (PrP )). P∈I

9.2. HECKE ALGEBRAS AND BASE CHANGE

99

If m ⊂ OE is not of this form, we then set Tb(m) := Tb(m) = 0. Extending Z[χE ]-linearly, b defines a map (9.2.6)

b : TcE dL/E ⊗Z Z[χE ]/I(χE ) −→ TcDL/E ⊗Z Z[χE ]/I(χ).

We have the following lemma: Lemma 9.1. Let c ⊂ OL be an ideal and χE a quasicharacter as in §9.1. The restriction of the map b : TcE dL/E ⊗Z Z[χE ]/I(χE ) −→ TcDL/E ⊗Z Z[χE ]/I(χ). to b(TcDL/E ) is a Z[χE ]-algebra morphism that is a section of b. Proof. The definition of b is multiplicative in the sense that Tb(m)Tb(n) = (TcE (m)TcE (n))b for m + n = OE . Since the Hecke algebras TcDL/E and TcE dL/E are multiplicative as well, to prove thatbis an algebra morphism it suffices to check that b(TcDL/E ⊗Z Z[χE ]/I(χE )) −→ TcDL/E ⊗Z Z[χE ]/I(χ) is a Z[χE ] algebra homomorphism when restricted to the subalgebra A := hTcE (pr )ir∈Z≥0 ∩ b(TcDL/E ⊗Z Z[χE ]/I(χE )) = hTcE (p2r )ir∈Z≥0 for each prime p ⊂ OE coprime to cE dL/E . Here the h , i denotes the span as a Z[χE ]-module, and we are using the p-adic Cartan decomposition together with (7.4.3) to deduce the equality (compare [Bu, Proposition 4.6.2 and Proposition 4.6.4]). If p is split in L/E this is trivial, so we henceforth assume that p is inert in L/E and write P for the prime above p. In this case, the image of A under b is contained in the Z[χE ]-subalgebra B := hTc (P)r ir∈Z>0 ≤ TcDL/E ⊗Z Z[χE ]. Let T(P) : = hTc (Pr )ir≥0 ≤ TcDL/E ⊗Z Z[χE ] T(p) : = hTc (p2r )ir≥0 ≤ TcE dL/E ⊗Z Z[χE ]. Thus A = T(p) ⊗Z Z[χE ]/I(χE ) and B = T(P) ⊗Z Z[χ]/I(χ). There are injective morphisms of Z[χ]-algebras T(P) −→ H(ResL/E GL2 (Ev )//ResO /O GL2 (OE ) ∼ = C[t±1 , t±1 ]S2 M

T(p) −→ H(GL2 (Ev )//GL2 (OEv ) ∼ =

F

v

1

2

±1 S2 C[t±1 1 , t2 ]

where left hand maps are given as in (D.2.6) (the choice of character is irrelevant) and the right hand maps are the isomorphism given explicitly in the proof of Lemma D.5 below. Here we view C as a Z[χE ]-algebra via the tautological embedding Z[χE ] ≤ C. From the description of

100


these maps and isomorphisms given in §D.2 and the proof of Lemma D.5 it follows in particular T(P) and T(p) are free Z[χE ]-algebras in on two generators: T(P) = Z[χE ][Tc (P), Tc (P, P)] T(p) = Z[χE ][Tc (p2 ), Tc (p2 , p2 )]. This implies that A and B are the free Z[χE ]-algebras on one generator given by (9.2.7)

A = Z[χE ][TcE (p2 )] B = Z[χE ][Tc (P)]

In view of (9.2.7) we have an algebra morphism s : A → B defined by stipulating that s(TcE (p2 )) = Tc (P) + χE (p)NE/Q (p), and it is easy to see that s is a section of b. Thus to prove the lemma it suffices to show that s = b. For this it suffices to show s(Tc (p2r )) = Tb(p2r ). E

We proceed by induction on r. The statement is obviously true for r = 0, 1. Assume it is true for r − 1. Applying (7.4.3), we have that s(TcE (p2r )) is equal to (9.2.8)

s(TcE (p2r−2 )TcE (p2 ) − χE NE/Q (p)TcE (p2r−2 ) − χE NE/Q (p2 )TcE (p2r−4 )) = Tb(p2r−2 )Tb(p2 ) − χE NE/Q (p)Tb(p2r−2 ) − χE NE/Q (p2 )Tb(p2r−4 )

Here we have written (and will continue to write) χE NE/Q (m) = χE (m)NE/Q (m) for ideals m ⊂ OE . Substituting Tb(p2r−2 )Tb(p2 ) = (Tc (Pr−1 ) + χE NE/Q (p)Tc (Pr−2 ))(Tc (P) + χE NE/Q (p)) Tb(p2r−2 ) = Tc (Pr−1 ) + χE NE/Q (p)Tc (Pr−2 ) Tb(p2r−4 ) = Tc (Pr−2 ) + χE NE/Q (p)Tc (Pr−3 ) into (9.2.8), we have that s(TcE (p2r )) is equal to Tc (Pr−1 )Tc (P) + χE NE/Q (p)Tc (Pr−2 )Tc (P) + χE NE/Q (p)Tc (Pr−1 ) + χE NE/Q (p2 )Tc (Pr−2 ) − χE NE/Q (p)(Tc (Pr−1 ) + χE NE/Q (p)Tc (Pr−2 )) − χE NE/Q (p2 )(T (Pr−2 ) + χE NE/Q (p)Tc (Pr−3 )) =Tc (Pr−1 )Tc (P) + χE NE/Q (p)Tc (Pr−2 )Tc (P) − χE NE/Q (p2 )(Tc (Pr−2 ) + χE NE/Q (p)Tc (Pr−3 )) =Tc (Pr ) + χE NE/Q (p2 )Tc (Pr−2 ) + χE NE/Q (p)(Tc (Pr−1 ) + χE NE/Q (p2 )Tc (Pr−3 )) − χE NE/Q (p2 )(Tc (Pr−2 ) + χE NE/Q (p)Tc (Pr−3 )) =Tc (Pr ) + χE NE/Q (p)Tc (Pr−1 ) = Tb(p2r ).

9.3. HILBERT MODULAR FORMS WITH COEFFICIENTS IN A HECKE MODULE

This completes the proof of the lemma.

101

An immediate consequence of the fact that b is section of b over its image is the crucial Proposition 9.2 below. In order to state it, recall that for any newform f ∈ Sκnew (K0 (cE ), χE ), there exists a base change fb ∈ Sκbnew (K0 (c0 ), χ) for some c0 ⊂ OL with c0 ∩ OE equal to cE up to powers of primes dividing dL/E . This fb is uniquely characterized as the newform generating the automorphic representation π b that is the base change of π in the sense of §D.4 below. We have the following proposition: Proposition 9.2. Let m ⊂ OE be an ideal coprime to dL/E cE . If m is a norm from OL , then the Hecke operator Tb(m) := Tbc,χE (m) satisfies fb|Tb(m) = λf (m)fb for all newforms f ∈ Sκnew (K0 (cE ), χE ) with fb ∈ Sκbnew (c, χ). Proof. Let H be an unramified connected reductive group over Ev , where v is a finite place of E, and let KH ≤ H(Fv ) be a hyperspecial subgroup. Recall that an unramified irreducible representation of H(Ev ) has a unique vector fixed by KH [C, p. 152]. It follows from this fact that the Hecke eigenvalues of fb (resp. f ) appearing in the proposition are the same as the Hecke eigenvalues of the automorphic representation π b (resp. π) generated by π b (resp. π) (compare §D.1). Since b is a section of b, the proposition thus follows immediately from the definition of the base change π b of π (see §D.4). Remark. The simple shape of the formula given above for the Tb(m) depends on the fact that L/E is quadratic. The root of this is the “a−2 ” in the Hecke algebra identity X Tc (m)Tc (n) = NL/Q (a)Tc (a, a)Tc (a−2 mn), m+n⊆a

(see (7.4.3)). However, if L/E is an arbitrary prime degree extension of totally real fields and p is a prime of OE that totally splits as p = P1 . . . Pn in OL , then an obvious extension of the argument given above proves that 1 (Tc (P1 ) + · · · + Tc (Pn )) fb = λf (p)fb n if the newform fb ∈ Sκnew (K0 (c), χ) is the base change of a Hilbert modular form f on E and p is coprime to c. 9.3. Hilbert modular forms with coefficients in a Hecke module Let M be a (left) Tc ⊗ A-module, where A ≤ C is a subalgebra as in §8.1 satisfying the additional requirements that

102


(1) The algebra A contains the values (χE )v(p) ($p ) for every prime p ⊂ OE , where $p is a uniformizer for the completion OE,v(p) at the place v(p) associated to p. (2) The algebra A contains the Hecke eigenvalues of all the newforms f ∈ Sκnew (K0 (c), χ) that are base changes from E. For example, if the kσ0 all have the same parity, A could be a sufficiently large number field (see [Sh2, Proposition 2.8]). In a moment, we will prove Theorem 9.3, which uses the Tb(m) of the previous section to produce Hilbert modular forms on E with coefficients in M. In order to state this theorem, we must introduce some notation. If f ∈ Sκbnew (K0 (c), χ) is a newform (and therefore a simultaneous eigenform for all Hecke operators), we let λf : Tc −→ A be the linear functional defined by λf (t)f := f |t as above. Let M(f ) denote the f -isotypical component of M (i.e. the set of B ∈ M such that tB = λf (t)B for all t ∈ Tc ) and set M (9.3.1) ME := M(f ) ≤ M, f :λf (Pσ )=λf (P) a.e. ∀σ∈Gal(L/E)

where the sum is over all newforms f ∈ Sκbnew (K0 (c), χ) such that for almost all primes P ⊂ OL we have λf (P) = λf (Pσ ) for all σ ∈ Gal(L/E). We note that the dual (ME )∨ := HomA-mod (ME , A) is tautologically a Tc ⊗ A-module. We denote the endmorphism of (ME )∨ induced by t ∈ Tc by t∗ : (ME )∨ −→ (ME )∨ . We need to cut down the module ME one more time. Let M (9.3.2) MχE := M(f ) ≤ ME f :f =b g g has nebentypus χE

be the submodule spanned by the f -isotypical components such that f = gb is a base change of a newform g ∈ Sκnew (K0 (cE ), χE ) for some cE ⊂ OE . By the theory of quadratic base change (see Corollary D.12) we have that MχE ⊕MχE η = ME , where η is the generator of Gal(L/E)∧ . We denote by QχE : M −→ MχE the projection. Suppose that γ ∈ MχE . In analogy with (1.3.2), for each ideal m ⊂ OE define ( Tbc,χE (m)γ if m is a norm from OL and m + dL/E (c ∩ OE ) = OE γχE (m) := 0 otherwise.

9.3. HILBERT MODULAR FORMS WITH COEFFICIENTS IN A HECKE MODULE

103

Before we state the main result of this section, we set notation for the following subspace of Sκ (K0 (cE ), χE ): (9.3.3)

Sκ+,0 (cE , χE ) := {f ∈ Sκ (K0 (cE ), χE ) : a(m, f ) = 0 if η(m) = −1 or m + cE 6= OE }.

Note that Sκ+,0 (cE , χE ) is not preserved by TcE . It is, however, preserved by b(TcDL/E ); this is easy to check using the explicit description of b given in (9.2.3) above. We then have the following theorem: Theorem 9.3. If γ ∈ MχE , then the formal automorphic Fourier series X Φγ,χE := |y|AE γχE (ξyDE/Q )qκ (ξx, ξy) ξ∈E × ξ0

is an element of MχE ⊗ Sκ+,0 (N (c), χE ). Moreover, for any Λ ∈ (MχE )∨ and t ∈ TcDL/E ⊗ A we have ht∗ Λ, Φγ,χE i = hΛ, Φγ,χE i|b(t) Here, as in the introduction, N (c) : = m2 bL/E (c ∩ OE )

(9.3.4)

Y

p2 ,

p|(c∩OE )

where m2 ⊂ OE is an ideal divisible only by dyadic primes, which we may take to be OE if c + 2OL = OL , and bL/E is an ideal divisible only by the primes ramifying in L/E. Recall from §8.5 that I m H[L:Q] (X0 (c), L(b κ, χ0 , A)) is a Tc ⊗ A-module. Thus it makes sense to define E I m H[L:Q] (X0 (c), L(b κ, χ0 , A)) : = I m H[L:Q] (X0 (c), L(b κ, χ0 , A))E χE I m H[L:Q] (X0 (c), L(b κ, χ0 , A)) : = I m H[L:Q] (X0 (c), L(b κ, χ0 , A))χE

as in (9.3.1)1. With this notation, Theorem 9.3 immediately implies the following generalization of Theorem 1.1 to nontrivial local coefficient systems: Theorem 9.4. series

E If γ ∈ I m H[L:Q] (X0 (c), L(b κ, χ0 , A))χE , then the formal automorphic Fourier

Φγ,χE (( y0 x1 )) := |y|AE

X

γχE (ξyDE/Q )qκ (ξx, ξy)

ξ∈E ξ0

is an element of χE I m H[L:Q] (X0 (c), L(b κ, χ0 , A)) ⊗ Sκ+,0 (N (c), χE ), 1Notice

that when A = C, each of these groups are Hecke submodules of I m Hcusp (X0 (c), L(b κ, χ0 , C)).

104


χE where N (c) is defined as in (9.3.4). Moreover, if Λ ∈ I m H[L:Q] (X0 (c), L(b κ, χ0 , A))∨ and t ∈ TcDL/E , then ht∗ Λ, Φγ,χE i = hΛ, Φγ,χE i|b(t). χE Remark. Notice that the definition of I m H[L:Q] (X0 (c), L(b κ, χ0 , A)) implicitly depends on a × × × choice of quasicharacter χ : L \AL → C whose restriction is χ0 . There may be more than one choice of such a character, a fact which also played a role in the statement of Theorem 8.3 above.

Before giving the proof of Theorem 9.3, we set the notation (9.3.5)

Sκbnew,E (c, χ) ≤ Sκbnew (K0 (c), χ)

for the subspace spanned by those newforms f such that, for almost all prime ideals P ⊂ OL , we have λf (P) = λf (Pσ ) for all σ ∈ Gal(L/E). We denote by (9.3.6)

Sκbnew,χE (c, χ) ≤ Sκbnew,E (c, χ)

the subspace spanned by those newforms f such that f = gb for some g of nebentypus χE . Using Corollary D.12 we have that Sκbnew,χE (c, χ) ⊕ Sκbnew,χE η (c, χ) = Sκbnew,E (c, χ), where, as above, η is the generator of Gal(L/E)∧ . Proof of Theorem 9.3. We may write X (9.3.7) γ=

γ(b g ),

new,χE g b∈Sκ (c,χ) b

where the gb are newforms and γ(g) ∈ MχE is gb-isotypical under the action of Tc . If m + dL/E (c ∩ OE ) 6= 0 or if m is not a norm from OL , then by definition γχE (m) = 0. Otherwise, we have X (9.3.8) Tbc,χE (m)γ(b g ). γχE (m) = new,χE (c,χ)

g b∈Sκ b

Applying Proposition 9.2, we obtain (9.3.9)

γχE (m) =

1 2

X

λg (m)γ(b g)

Newforms g new,E s.t. g b∈S (c,χ) κ b

Here the 12 factor appears because there are exactly two newforms g contributing the same summand. To see this, note that it suffices by Theorem D.11 to check that π(g) ∼ 6= π(g ⊗ η), where η is the nontrivial element of Gal(L/E)∧ and π(g) (resp. π(g ⊗ η)) is the automorphic d is not be cuspidal representation generated by g (resp. g ⊗ η). If π(g) ∼ = π(g ⊗ η), then π(g) (see [GL, Theorem 2 and Appendix C]) which is a contradiction.

9.4. THE FOURIER COEFFICIENTS OF [[Z]χE (m), ΦQχE [Z],χE ]IH∗

105

It follows that Φγ,χE = γ ⊗

1 2

X

g dL/E (c∩OE )

Newforms g new,E s.t. g b∈S (c,χ) κ b

where g dL/E (c∩OE ) is the cusp form obtained from g by deleting the Fourier coefficients having a prime in common with dL/E (c ∩ OE ) as in Lemma 7.4. By Corollary D.12 in conjunction with Lemma 7.4, for each g in the sum we have that g dL/E (c∩OE ) ∈ Sκ (K0 (cE ), χE ) for some ideal Y cE ⊆ m2 bL/E (c ∩ OE ) p2 p|(c∩OE )

where m2 is an ideal divisible only by dyadic primes which we may take to be OE if c+2OL = OL and bL/E is an ideal divisible only by those primes dividing dL/E . The first statement of the theorem follows. For the second statement, suppose that gb contributes to (9.3.7) and let t ∈ TcDL/E . Note that g and gb are unramified outside of cDL/E and (c ∩ OE )dL/E , respectively. Therefore, the eigenvalue of t on gb is the same as the eigenvalue of t on the automorphic representation generated by gb, and similarly for the eigenvalue of t on gb (compare the proof of Proposition 9.2). Thus the eigenvalue of t on gb is equal to the eigenvalue of b(t) on g dL/E (c∩OE ) by the definition of base change (see §D.4). This implies the second statement. 9.4. The Fourier coefficients of [[Z]χE (m), ΦQχE [Z],χE ]IH∗ E (X0 (c), L(b κ, χ0 )) is a modular form As we have seen, the fact that Φγ,χE for γ ∈ I m H[L:Q] m E with coefficients in I H[L:Q] (X0 (c), L(b κ, χ0 )) is a formal consequence of base change. Let χE QχE : I m H[L:Q] (X0 (c), L(b κ, χ0 )) −→ I m H[L:Q] (X0 (c), L(b κ, χ0 ))

be the canonical projection. In this section we use the fact that we are working in intersection homology, as opposed to some other Hecke module, to compute the values of some linear functionals on ΦQE [Z],χE for certain cycles Z. In particular, we express the Fourier coefficients of [[Z]χE (m), ΦQE [Z],χE ]IH∗ in terms of certain periods. Let Z ⊂ X0 (c) be a subanalytic set of (real) dimension [L : Q]. Let s be a section of L(b κ, χ0L ) over Z 0 = Z ∩ Y0 (c). As in §4.4 and Proposition 4.3 the pair (Z, s) determines a BM Borel-Moore cycle, which we also denote by Z ∈ C[L:Q] (Y0 (c); L(b κ, χ0 )) and homology classes BM [Z] ∈ H[L:Q] (Y0 (c); L(b κ, χ0 )) and [Z] ∈ I t H[L:Q] (X0 (c); L(b κ, χ0 )). BM Theorem 9.5. Let Z ∈ C[L:Q] (Y0 (c); L(b κ, χ0 )) as above. Suppose the homology class [Z] ∈ BM 0 H[L:Q] (Y0 (c), L(b κ, χ0 )) is the image β∗ ([Z ]) of a (compactly supported) class [Z 0 ] in H[L:Q] (Y0 (c), L(b κ, χ0 )). Then the following are true:

106


R (1) The class [Z] lies in I m H[L:Q] (X0 (c), L(b κ, χ0 )) and the integral Z ωJ (f −ι ) converges for all f ∈ Sκb (K0 (c), χ) and all J ⊂ Σ(E) (cf. (8.7.3)). (2) If m + NL/E (c)dL/E = n + NL/E (c)dL/E = OE and m, n are both norms from OL , then the mth Fourier coefficient of [[Z]χE (n), ΦQχE [Z],χE ]IH∗ is R R b−ι ) X ω ( f ω (fb−ι ) 1 X J Wc∗ Z Wc∗ Z Σ(L)−J |J| (−1) λf (n)λf (m), [L:Q]+s W (fb)N (kσ +2mσ )/2 (fb, fb) 4 (2i) (c) P L/Q f J⊂Σ(E)

where σ is any element of Σ(E), the sum is over the normalized newforms f whose P base change fb is an element of Sκbnew,χE (c, χ), and s := σ∈Σ(L) b kσ0 . Otherwise, the mth Fourier coefficient is zero. Here fb−ι (x) := fb(x−ι ) as in (7.3.4). Remark. By Theorem 6.6 the hypotheses on Z hold automatically if Z is the closure of the modular subvariety Z 0 associated to ResE/Q GL2 , or a Hecke translate of such. Suppose that Z 0 is the modular subvariety attached to ResE/Q GL2 and that χ0 is the trivial character χtriv . Then a section of L(b κ, χtriv ) over Z 0 can be constructed using the observation that L(b κ, χtriv )Z 0 ∼ = ι∗ (L(κ, χtriv ) ⊗ L∨ (κ, χtriv ))Z 0 where we view L(κ, χtriv ) ⊗ L∨ (κ, χtriv )) as a sheaf over ResE/Q GL2 (Q)\ResE/Q GL2 (A)/KE,∞ (K0 (c) ∩ ResE/Q GL2 (Af )) and ι : ResE/Q GL2 (Q)\ResE/Q GL2 (A)/KE,∞ (K0 (c) ∩ ResE/Q GL2 (Af )) ,→ Y0 (c) is the canonical inclusion (with image Z 0 ). In particular, in §8.4 we constructed a bilinear form h·, ·i : L(κ, χtriv ) × L∨ (κ, χtriv ) → C invariant under the action of ResE/Q GL2 . This implies the existence of an element in L(κ, χtriv )⊗ L∨ (κ, χtriv ) invariant under the action of ResE/Q GL2 , which in turn yields the desired section. For more details in the classical Hirzebruch-Zagier setting, see [To]. This will be discussed in generality in Chapter 10 below. We note, before giving the proof, that Theorem 1.2 is just Theorem 9.5 in the special case that L(κ, χ0 ) is the trivial representation C.

9.4. THE FOURIER COEFFICIENTS OF [[Z]χE (m), ΦQχE [Z],χE ]IH∗

107

Proof. By definition of QχE [Z] = QχE [Z](OE ), Theorem 8.3, and Corollary D.12 we may write 1 X X QχE [Z]χE (OE ) = (9.4.1) aJ,f PωJ0 (fb) 2 f J⊂Σ(E)

for some aJ,f ∈ C, where the sum is over the same set of f as in the theorem. Here P is the Poincaré duality isomorphism of Chapter 3, and the 12 appears for the same reason explained in the proof of Theorem 9.3. We assume, without loss of generality, that aJ,f = aJ,f ⊗η . Thus, if n ⊂ OE is a norm from OL , then   X X 2QχE [Z]χE (n) = Tb(n)∗  aJ,f PωJ0 (fb) J⊂Σ(E)

=

X X J⊂Σ(E)

= = =

(by Hecke-equivariance of P)

aJ,f PωJ0 (fb|Tb(n))

(by Hecke equivariance of ωj0 )

λf (n)aJ,f PωJ0 (fb)

(Proposition 9.2).

f

X X J⊂Σ(E)

aJ,f P Tb(n)∗ ωJ0 (fb)

f

X X J⊂Σ(E)

aJ,f Tb(n)∗ PωJ0 (fb)

f

X X J⊂Σ(E)

f

f

Here, as usual, we abbreviate Tb(n) := Tbc,χE (n). By the same argument, for m ⊂ OE a norm from OL we obtain   X X 2Wc∗ QχE [Z]χE (m) =  λf (m)aJ,f PWc∗ ωΣ(L)−J (fb) . J⊂Σ(E)

f

and thus [[Z]χE (m), QχE [Z]χE (n)]IH∗ is given by 1 X X 0 (fb)]IH∗ λf (n)λf (m)aJ,f aΣ(L)−J,f [PωJ0 (fb), PWc∗ ωΣ(L)−J 4 f J⊂Σ(E)

=

1 X X 0 λf (n)λf (m)aJ,f aΣ(L)−J,f [ωJ0 (fb), Wc∗ ωΣ(L)−J (fb)]IH∗ 4 f J⊂Σ(E)

108


(see §3.4). We are left with proving that 0 (9.4.2) aJ,f aΣ(L)−J,f [ωJ0 (fb), ωΣ(L)−J (fb−ι )]IH ∗ R R −ι −ι b b ω ( f ) ω ( f ) Wc∗ Z J Wc∗ Z Σ(L)−J = (−1)|J| . (2i)[L:Q]+s W (fb)NL/Q (c)(kσ +2mσ )/2 (fb, fb)P For this, recall that if g ∈ Sκb (K0 (c), χ), then [ωJ (g)] is the image α∗ ([ω0 ]) of a compactly κ, χ0 )). This follows from [Har, §3.1] (the compactly supsupported class [ω0 ] ∈ Hci (Y0 (c), L(b ported cohomology is contained in ker(r) in the notation of [Har, §3.1]) (see [Hid5, Proposition 3.1] as well). So the differential form ωJ (g) determines an intersection cohomology class in two (potentially) different ways: as the image α∗ ([ω0 ]) of a compactly supported class, and as ωJ0 (g) := (π fr )∗ Z[(π fr )∗ ωJ (g)]; a push-forward of the image of a square integrable form. Of course, both procedures determine the same intersection cohomology class (see diagram (6.3.1)). Then Proposition 4.3 yields: Z (9.4.3) ωJ (g) = [ωJ0 (g), [Z]]K < ∞, Wc∗ Z

where [·, ·]K denotes the Kronecker pairing between intersection cohomology and intersection homology induced by [·, ·]IH ∗ . Restricting to the case g = fb−ι and using (3.4.4) (which says that the Kronecker pairing plus Poincaré duality equals the intersection pairing) along with (9.4.1), we have [ω 0 (fb−ι ), [Z]]K = [Pω 0 (fb−ι ), [Z]]IH∗ J

J

0 = aΣ(L)−J,f [PωJ0 (fb−ι ), PωΣ(L)−J (fb−ι )]IH∗ 0 = aΣ(L)−J,f [ωJ0 (fb−ι ), ωΣ(L)−J (fb−ι )]IH ∗ .

Here we also used the commutativity of diagram (3.4.3), and our assumption from above that aJ,f = aJ,f ⊗η . Combining this with Lemma 8.4 and (9.4.3), we obtain Z Z −ι b ωJ (f ) = ωJ0 (fb−ι ) Wc∗ Z

Wc∗ Z

= aΣ(L)−J,f (−1)[L:Q]−|J| (2i)[L:Q]+s NL/Q (c)(kσ +2mσ )/2 W (fb)(fb, fb)P . The last equality follows from Lemma 8.4. The same argument proves Z ωΣ(L)−J (fb−ι ) = aJ,f (−1)−|J| (2i)[L:Q]+s NL/Q (c)(kσ +2mσ )/2 W (fb)(fb, fb)P , Wc∗ Z

This, combined with Lemma 8.4, completes the proof of the theorem.

CHAPTER 10

Explicit construction of cycles In this chapter we construct explicit cycles Zθ on Y0 (c) and use the machinery of this book to prove that they admit canonical classes [Zθ ] ∈ I m H[L:Q] (X0 (c), L(b κ, χ0 , C)) (see Theorem 10.3). We then prove that the “new part” of these cycles lies in χE I m H[L:Q] (X0 (c), L(b κ, χ0 , C)),

thus providing examples of cycles whose class in intersection homology satisfies the hypotheses of theorems 9.4 and 9.5. For a precise statement, see theorem 10.4. As a byproduct of the Rankin-Selberg computations that underly the proof of Theorem 10.4, we explicitly compute the Fourier coefficients appearing in Theorem 9.5 in the case where [Z] = Pnew [Zθ ] (the “new part” of Zθ ). The result of this computation is recorded in Theorem 10.5. 10.1. Notation for the quadratic extension L/E Let L/E be a quadratic extension of totally real number fields, and let hςi = Gal(L/E) and hηi = Gal(L/E)∧ . For every σ ∈ Σ(L), write σ 0 := σ ◦ ς. Borrowing terminology from the theory of CM extensions, we say that a subset J ⊂ Σ(L) is a type for L/E if for every √σ ∈ J, the embedding σ 0 : L ,→ R is not in J. Without loss of generality, we write L = E( ∆) for a totally positive ∆ ∈ E. We√define a distinguished type JE ⊂ Σ(L) for L/E by stipulating that σ ∈ JE if and only if σ( ∆) < 0. We assume that the set Σ(L) is ordered so that for z ∈ HΣ(L) we have 0 dz = dzσ1 ∧ dzσ10 ∧ · · · ∧ dzσ[E:Q] ∧ dzσ[E:Q] 0 }. where JE := {σ10 , . . . , σ[E:Q] × Choose a κ ∈ X (E) and fix a quasicharacter χE : E × \A× E → C satisfying χE∞ (b∞ ) = × −k0 −2m b∞ for all b ∈ AE (for a proof that such a character exists, see [Hid6, Lemma 2.1]). Let × χ = χE ◦ NL/E : L× \A× b = (b k 0 , m) b ∈ X (L) by L → C be its base change to L, and define κ b kσ0 := kσ0 0 and m b σ = mσ0 if the infinite place σ ∈ Σ(L) extends σ0 ∈ Σ(E). For our later convenience, write [k 0 + 2m]1 = k 0 + 2m and similarly [b k 0 + 2m]1 b := b k 0 + 2m. b Fix a character × × × θ : L \AL → C such that

(10.1.1)

0

θu := θ| · |[k +2m]/2 109

110

10. EXPLICIT CONSTRUCTION OF CYCLES

is unitary and its restriction to the image of A× E under the diagonal map is χE η. Let f(?) denote the conductor of ?, where ? is, e.g., a character. Denote by h+ L the narrow class number of OL . Finally, let A ≤ C be an algebra satisfying the conditions of §8.1 together with the following conditions: • The algebra A contains the values of and θu . • The algebra A contains θ(t) for each t ∈ Υ, where Υ is the subset of AL,f defined in (10.2.2) below. 10.2. The cycles Zθ Let c ⊂ OL be an ideal. Let ι : ResE/Q GL2 −→ ResL/Q GL2 be the diagonal embedding and let π : ResL/Q GL2 (A) → Y0 (c0 ) be the canonical projection. Associated to the diagonal embedding is a (non-compact) Shimura subvariety Z0 := π(ι(ResE/Q GL2 (A))). We begin this chapter by constructing a section s : Z0 −→ L(b κ, (χθ−2 )0 , A), where χ and θ are the characters we fixed in §10.1. Write c0 := f(θ)2 c, let K 1 (c0 ) := ( ac db ) ∈ K0 (c0 ) : a ≡ 1

(mod f(θ)2 c) ,

and choose a minimal complete set of right coset representatives K 1 (c0 )g1 , . . . , K 1 (c0 )gn for K 1 (c0 )\K0 (c0 ). Recall that for each ideal c ⊂ OL the number of connected components of + X0 (c) is the narrow class number h+ L (see §7.1). For each 1 ≤ r ≤ n and 1 ≤ i ≤ hL , use strong approximation to choose a γri ∈ G(Q) such that ti (c0 )gr ∈ γri ti (c0 )gr−ι K 1 (c0 )grι = γri ti (c0 )K 1 (c0 ). If we define for each 1 ≤ i ≤ h+ L the subgroup Γ1i (c0 ) := ( ac db ) ∈ Γi0 (c0 ) : a ≡ 1

(mod c) ≤ Γi0 (c0 ),

then we see that γ1i Γ1i (c0 ), . . . , γni Γ1i (c0 ) form a minimal complete set of coset representatives for Γi0 (c0 )/Γ1i (c0 ).

10.2. THE CYCLES Zθ

111

Consider the polynomial Si ({Xσ , Yσ }σ∈Σ(L) ) defined by 0

0

−[k0 +2m]/2

Si := | det(ti (c ))|

kσ Y X −1 k0 σ

j

0

0

k0 −j

(−1)kσ −j Xσkσ −j Yσj Xσj 0 Yσ0σ .

σ∈JE j=1

Here we can and do assume that ti (c0 ) = ti (c) for 1 ≤ i ≤ h+ L . Notice that Z0 ∩ Y0i (c0 ) = πi (ι0 (HΣ(E) )), where πi : HΣ(L) → Y0i (c0 ) is the canonical projection and ι0 : HΣ(E) ,→ HΣ(L) is the diagonal embedding. We can use the Si to construct a section of L∨i (b κ, (χθu−2 )0 , A) over Z0 ∩ Y0i (c0 ). Let κ, (χθu−2 )0 , A) si : ι0 (HΣ(E) ) → L∨i (b P −1 be the function that is identically nr=1 γri · Si , where the action is given by considering Si as −2 ∨ κ, (χθu )0 , A). We have the following lemma: an element of Li (b (10.2.1)

κ, (χθu−2 )0 , A) over Z0 ∩ Y0i (c0 ). Lemma 10.1. The function si descends to a section of L∨i (b of ι0 (HΣ(E) ) in Γi0 (c0 ) under Proof. Consider the action of Γi0 (c0 ) on HΣ(L) . The stabilizer this action is contained in Γi0 (c0 )∩ Z(Γi0 (c0 )) ∪ ι(ResE/Q GL2 (R)) , where Z(Γi0 (c0 )) is the center of Γi0 (c0 ). The function si is obviously invariant under Z(Γi0 (c0 )), as Z(Γi0 (c0 )) acts trivially on κ, (χθu−2 )0 , A). We are left with showing that si is invariant under Γi0 (c0 )∩ι(ResE/Q GL2 (R)). L∨i (b For this, notice first that Si is invariant under Γ1i (c0 ) ∩ ι(ResE/Q GL2 (R)) (this follows from the same calculation proving that (8.4.1) is Γi0 (c)-invariant). The action of Γi0 (c0 ) ∩ ι(ResE/Q GL2 (R)) Γ1i (c0 ) ∩ ι(ResE/Q GL2 (R)) on si simply permutes the summands, so the invariance of si under Γi0 (c0 ) ∩ ι(ResE/Q GL2 (R)) follows. Due to Lemma 10.1, it makes sense to consider the Borel-Moore chain BM Z ∈ C[L:Q] (Y0 (c0 ), L∨ (b κ, (χθu2 )0 , A))

that is given on the component Y0i (c0 ) by πi (ι0 (HΣ(E) )), si . For our purposes, we have to twist the chain Z by θu . LetQ b be an idèle Q trivial at the infinite places whose associated ideal is f(θ). Define a subset Υ of p|f(θ) Lp × p-f(θ) Op by     Y Y (10.2.2) Υ = t = (tp ) ∈ Lp × Op : ordp (tp ) ≥ −ordp (b) for all p | f(θ) ,   p|f(θ)

p-f(θ)

112


e be a set of representatives for Υ modulo O bL := Q Op (this may be viewed as a ring and let Υ p Q −1 e → C the map defined by setting isomorphic to p|f(θ) (Op /bp Op )). We denote by θuf(θ) :Υ ( e× θu−1 (t) if t ∈ Υ −1 θuf(θ) (t) = 0 otherwise. e define u(t) ∈ G(A) by For each t ∈ Υ, ( ( 10 t1v ) u(t)v = ( 10 01 )

if v - ∞ and ordv (b) ≥ 1 otherwise.

e let Zt be the image of ι(ResE/Q GL2 (A))u(t)−ι under the canonical projection For each t ∈ Υ, κ, χ0 , A) over each of the Zt . In ResL/Q GL2 (A) → Y0 (c). We wish to construct a section of L∨ (b order to do this, we will first write down Zt ∩ Y0i (c) for each 1 ≤ i ≤ h+ L . Set a b 0 0 0 0 K11 (c ) := ∈ K0 (c ) : a − 1 ∈ c and d − 1 ∈ c c d and define t1 (K11 (c0 )), . . . , th(K11 (c0 )) (K11 (c0 )) ∈ ResL/Q GL2 (A) defined as in §7.1. To ease notation, write h11 (c0 ) := h(K11 (c0 )). By relabeling if necessary, we can and do assume that det(t(i−1)h11 (c0 )/h+ +1 (K11 (c0 )), . . . , det(tih11 (c0 )/h+ (K11 (c0 )) L

L

are equivalent to det(ti (c0 )) in T (Q)\T (A)/ det(ResL/Q GL2 (R)0 K0 (c0 )). Now, using strong e and each 1 ≤ ν ≤ h11 (c0 ), choose a γν (t) ∈ ResL/Q GL2 (Q) such approximation, for each t ∈ Υ that tν (K11 (c0 ))u(t)ι ∈ γν (t)tν (K11 (c0 ))u(t)−ι K11 (c0 )u(t)ι . A simple matrix calculation implies that u(t)−ι K11 (c0 )u(t)ι = K11 (c0 ). This implies that Zt ∩ Y0i (c0 ) is just the image of γν (t)ι0 (HΣ(E) ) under the canonical projection HΣ(L) → Γi (c0 )\HΣ(L) + 0 for any (i − 1)h11 (c0 )/h+ L + 1 ≤ ν ≤ ih11 (c )/hL . Now that we have a componentwise description of Zt ∩ Y0i (c), we need a section κ, χ0 , A). si,t : Zt ∩ Y0i (c) −→ L∨i (b First notice that Γν (K11 (c0 )) is naturally a subgroup of Γi (c0 ) for the unique i such that (i − + 0 1)h11 (c0 )/h+ L + 1 ≤ ν ≤ ih11 (c )/hL . We then define si,t : ι0 (HΣ(E) ) −→ L∨i (b κ, χ0 , A)

10.2. THE CYCLES Zθ

to be the function that is identically (10.2.3)  ih11 (c0 )/h+ X L  θu−1 (det(ti (c0 ))) ν=(i−1)h11 (c0 )/h+ L +1

X

δ −1 ?1

113

−1 θuf(θ) (t)γν (t) ?2

n X

! −1 γri · Si  ,

r=1

δ∈Γi0 (c)/Γν (K11 (c0 ))

κ, χtriv0 , A) and κ, χ0 , A), L∨i (b where the actions ?1 , ?2 , and · are the actions associated to L∨i (b 0 −2 ∨ κ, (χθu )0 , A), respectively. Here Γν (K11 (c )) is defined as in (7.1.8) and χtriv is the trivial Li (b character. An argument similar to that given in the proof of Lemma 10.1 then implies the following lemma: Lemma 10.2. The function si,t : ι0 (HΣ(E) ) → L∨i (b κ, χ0 , A) descends to define a section si,t : Zt ∩ Y0i (c) −→ L∨i (b κ, χ0 , A). By Lemma 10.2, it makes sense to consider the Borel-Moore chain Zθ ∈ C BM (Y0 (c), L∨ (b κ, χ0 , A)) given on Y0i (c) by X (Zt , si,t ). e t∈Υ

This chain admits a canonical class in intersection homology: Theorem 10.3. The chain Zθ admits a canonical class [Zθ ] ∈ IH[L:Q] (X0 (c), L∨ (b κ, χ0 , A)). Moreover, if A = C, then hωJ0 (f −ι ), Zθ iK

Z =

ωJ (f −ι ).

Zθ

Here h , iK is the Kronecker pairing (see (3.4.4)). The proof is essentially contained in Chapter 6, and requires some background on the Borel-Serre compactification of a locally symmetric space which is given there. b0 (c) denote the reductive Borel-Serre compactification of Y0 (c) (see §6.7). It Proof. Let X b0 (c) → X0 (c) that is the identity on Y0 (c) and is equipped with a continuous surjection µ : X may be stratified so that µ is stratum-preserving. b0 (c) and let n be the upper middle perversity, so n(c) := Let Zbt be the closure of Zt in X c−2 d 2 e. Then, by counting the dimensions of the intersections of Zbt with the singular strata of b0 (c) as in the proof of Theorem 6.6, we see that the chain X X (Zbt , si,t ) e t∈Υ

114


b0 (c), L(b defines a class [Zbθ ] ∈ I n H[L:Q] (X κ, χ0 , A)). We let [Zθ ] be the image of this class under the composite map given by applying the map b0 (c), L(b I n H[L:Q] (X κ, χ0 , A)) −→ I n H[L:Q] (X0 (c), L(b κ, χ0 , A)) induced by µ and the identification I n H[L:Q] (X0 (c), L(b κ, χ0 , A))−→I ˜ m H[L:Q] (X0 (c), L(b κ, χ0 , A)) that exists due to the fact that X0 (c) has no odd-dimensional strata. For the final statement of the theorem, see Proposition 4.3 and the proof of Theorem 9.5. Remark. In the case A = C, the map µ∗ in the proof is an isomorphism due to Saper and Stern’s proof of the Goresky-MacPherson and Rapoport conjecture for Q-rank one hermitian symmetric spaces. The conjecture for general Q-rank was later proven by Saper (see §6.9). 10.3. The full version of Theorem 1.3 In the previous section, we constructed a class [Zθ ] ∈ IH[L:Q] (X0 (c), L(b κ, χ0 , C)). Subsection 10.4 and all of §10.5 are devoted to proving the following theorem: Theorem 10.4. Let f ∈ Sκb (K0 (c), χ) be a simultaneous eigenform for all Hecke operators. Q 0 Write b := p|dL/E p. If J ⊂ Σ(L) is a type for L/E and f is a base change of a Hilbert cusp p-f(θ)

form g on E with nebentypus χE , then hωJ0 (f −ι ), [Zθ ]iK is equal to 0

(−1)t1 c1 LdL/E c ∩OE (Ad(g) ⊗ η, 1)ζb0 (1)−1 Lb0 (As(f ⊗ θ−1 ), 1). Here we set 0

c1 =

0

NE/Q (c0E )ΓE (k 0 + 1)ζ b (1)(−1)t1 i[E:Q]+{k } h11 (c)G(θu )[K0 (c) : K11 (c0 )][K0 (c0 ) : K 1 (c0 )] [k0 +2m]/2

2{k0 }+[E:Q]−1 NE/Q (dL/E )π {k0 }+2[E:Q] h+ L dL/Q P where G(θu ) is the Gauss sum (10.4.1) and t1 := σ∈JE −(J∩JE ) kσ0 . If J ⊂ Σ(L) is not a type for L/E or f is not a base change of a Hilbert cusp form g on E with nebentypus χE , then hωJ0 (f −ι ), [Zθ ]iIH = 0. P Here {k} := σ∈Σ(E) kσ for k ∈ ZΣ(E) and c0E ⊂ OE is the unique ideal satisfying K0 (c0 ) ∩ ResE/Q GL2 (A) = K0 (c0E ), where c0 = f(θ)2 c as above. We now prepare some notation so that we may state the full version of Theorem 1.3. Let new Pnew : I m H[L:Q] (X0 (c), L(b κ, χ0 , C)) −→ I m H[L:Q] (X0 (c), L(b κ, χ0 , C))

10.3. THE FULL VERSION OF THEOREM ??

115

new be the canonical projection, where I m H[L:Q] (X0 (c), L(b κ, χ0 , C)) is the subspace spanned by classes that are f -isotypical under the action of the Hecke algebra for some newform f ∈ Sκbnew (K0 (c), χ). Setting γ = Pnew Wc∗−1 [Zθ ] in our notation from §9.3, we write X ΦPnew Wc∗−1 [Zθ ],χE := |y|AE Pnew Wc∗−1 [Zθ ](ξyDE/Q )qκ (ξx, ξy). ξ∈E × ξ0

Assuming Theorem 10.4 for the moment, we prove the “full” version of Theorem 1.3: Theorem 10.5. We have χE ΦPnew Wc∗−1 [Zθ ],χE ∈ I m H[L:Q] (X0 (c), L(b κ, χ0 , C)) ⊗ Sκ+,0 (N (c), χE ).

Moreover, if m + NL/E (c)dL/E = n + NL/E (c)dL/E = OE and m, n are both norms from OL , then the mth Fourier coefficient of [Wc∗−1 [Zθ ](n), ΦPnew Wc∗−1 [Zθ ],χE ]IH∗ is

2 dL/E f(θ)∩OE −1 b 0 L (Ad(f ) ⊗ η, 1)L (As( f ⊗ θ ), 1) X b

c2

f

(−1)[E:Q] W (fb)ζb0 (1)2 L∗ (Ad(fb), 1)

λf (n)λf (m),

where the sum is over the normalized newforms f on E whose base change fb to L is an element of Sκbnew (K0 (c), χ), and 0

c2 :=

0

c21 (−1){κ } 2{k } [b k0 +2m]+3 b

.

b (2i)[L:Q] NL/Q (c)[bk0 +2m]/2+1 dL/Q

If either m + NL/E (c)dL/E 6= OE or n + NL/E (c)dL/E 6= OE , then the mth Fourier coefficient is zero. Here N (c) := m2 dL/E (c ∩ OE )

Y

p2

p|(c∩OE )

where m2 ⊂ OE is an ideal divisible only by dyadic primes, which we may take to be OE if c + 2OL = OL and dL/E is an ideal divisible only by primes ramifying in L/E. Moreover, W (fb) is the root number (8.8.2). Proof. By Theorem 9.5, the mth Fourier coefficient of [Wc∗−1 [Zθ ](n), Φ[Zθ ] ]IH∗ is zero if m + NL/E (c0 )dL/E 6= OE or n + NL/E (c0 )dL/E 6= OE , and otherwise it is equal to R R b−ι ) X ω ( f ω (fb−ι ) J 1 X Zθ Zθ Σ(L)−J |J| (−1) λf (n)λf (m). b 4 (2i)[L:Q]+{bk0 } W (fb)NL/Q (c)[bk0 +2m]/2 (fb, fb)P J⊂Σ(E)

f

The theorem now follows immediately from Theorem 10.4 and Theorem 7.7.

116


10.4. An integral representation for hωJ0 (f −ι ), [Zθ ]iK Our first step to proving Theorem 10.4 is an integral representation of hωJ0 (f −ι ), [Zθ ]iIH . Before we can state it, we must set some notation. Recall that for f ∈ Sκb (K0 (c), χ), there is a modular form f ⊗ θ−1 ∈ S(bk0 ,−bk0 /2) (K0 (c0 ), χθ−2 ) uniquely determined by the fact that ( θ−1 (m)a(f, m) if m + f(θ) = OL a(f ⊗ θ−1 , m) = 0 otherwise (see Lemma 7.3). Let Z0 be the image of ResE/Q GL2 (A) ,→ ResL/Q GL2 (A) → Y0 (c0 ), where the first arrow is the diagonal embedding ι as in §10.2 and the second arrow is the canonical projection. Finally, let G(θu ) be the Gauss sum X −1 −1 (10.4.1) G(θu ) := θuf(θ) (d−1 b t)eL (d t), e t∈Υ

where d ∈ A× ele trivial at the infinite places whose associated ideal is DL/Q and db is the L is an id` e element of Υ it defines. With this notation in mind, we may state the integral representation mentioned above: Proposition 10.6. If J ⊂ Σ(L) is not a type for L/E, then hωJ0 (f −ι ), [Zθ ]iK = 0. If J ⊂ Σ(L) is a type for L/E, then hωJ0 (f −ι ), [Zθ ]iK is equal to Z t1 (−1) C1 (f ⊗ θ−1 )−ι (ιJ (α))dµc0E (α) Z0

where 0

0

−1 0 0 1 0 −[k +2m]/2 C1 := (−2i)[E:Q]+{k } h11 (c)(h+ L ) G(θu )[K0 (c) : K11 (c )][K0 (c ) : K (c )]|dL/Q | P and t1 := {k 0 + 1} + σ∈JE −(J∩JE ) kσ0 .

In the proposition, c0E ⊂ OE is the unique ideal satisfying K0 (c0E ) = ResE/Q GL2 (A) ∩ K0 (c0 ) and µc0E is the measure used to define the Petersson inner product (·, ·)P (see §7.5). Proof. By Proposition 4.3, Theorem 6.6, and Theorem 8.3, we have that Z 0 −ι ωJ (f −ι ), hωJ (f ), [Zθ ]iK = Zθ

where h·, ·iK is the Kronecker pairing.

10.4. AN INTEGRAL REPRESENTATION FOR hωJ0 (f −ι ), [Zθ ]iK

For z ∈ HΣ(L) , define w ∈ (C − R)Σ(L) by ( zσ wσ := zσ and set

117

if σ 6∈ J otherwise

Im(wσ ) Re(wσ ) αw := ∈ ResL/Q GL2 (R). 0 1 σ∈Σ(L)

Temporarily write θi := θu (det(ti (c))) and write !! Ti,t :=

X X ν

θb−1 (t)γν (t) ?2

δ −1 ?1

X

−1 γri · Si

.

r

δ

+ 0 i 0 The sum is over (i − 1)h11 (c0 )/h+ L < ν ≤ ih11 (c )/hL and δ ∈ Γ0 (c)/Γν (K11 (c )). Using (10.2.3), we have h+ Z L X X b ωJ (f −ι ) = det(ιJ )m Bi,t , Zθ

i=1 t∈Υ e

where Bi,t is defined to be + * Z −ι Y f (ti (c)αw ) 0 (−Xσ + wσ Yσ )kσ , θi−1 Ti,t dw k+m+1 Im(w) Zt ∩Y0i (c) σ∈Σ(L) * !+ Z −ι Y X X f (t (c)α ) 0 i w −1 = C10 (−Xσ + wσ Yσ )kσ , θi−1 θb−1 (t)γν (t) ?2 γri · Si dw, k+m+1 i Im(w) Zt ∩Y0 (c) ν r σ∈Σ(L)

where the ?1 , ?2 , · and the ranges of the sums are defined as in (10.2.3) and h·, ·i is the pairing on L(b κ, χ0 , C) × L∨ (b κ, χ0 , C). Here C10 := [Γi0 (c) : Γν (K11 (c0 ))], 0

11 (c ) for any (i−1)h 0 (resp. σ(α) > 0) for all σ ∈ Σ(E). We have the following lemma:

Lemma 10.7. If f ∈ Sκb (K0 (c), χ) is a simultaneous eigenform for all Hecke operators, then X |dE/Q |1/2 ΓE (k 0 + s1) a(mOL , f ⊗ θ−1 )NE/Q (m)−1−s . |dE/Q NE/Q (dL/E )|s (4π){k0 }+[E:Q]s m⊂O

I(f, s) =

E

Here ΓE (k) :=

Q

σ∈Σ(L)

Γ(kσ ), and, as above, {k} :=

P

σ∈Σ(L)

kσ for k ∈ CΣ(L) .

120


Proof. To ease notation, write g := f ⊗ θ−1 . Notice that the Fourier expansion of g ◦ ιJ is 0 X

a(ξyDL/Q , g)q(bk0 ,−bk0 /2) (ξy, ξx)

ξ∈L×

where the 0 indicates that the sum is over those ξ ∈ L× such that σ(ξ) > 0 for σ 6∈ J 0 and σ(ξ) < 0 for σ ∈ J 0 . Thus Z Z g (ιJE ( y0 x1 )) |y|sAE dxd× y × \A× E+ E,+

E\AE

Z

0 X

Z

= × E+ \A× E,+

b0

E\AE ξ∈L×

a(ξyDL/Q , g)|(ξy)k /2 | exp(−2πΣσ∈Σ(L) |σ(ξ)yσ |)eL (ξx)|y|sAE dxd× y.

Now (p dE/Q eL (ξx)dx = 0 E\AE

Z

Thus Z

0 X

Z

× E+ \A× E,+

b0

E\AE ξ∈L×

Z q = dE/Q

a(ξyDL/Q , g)|(ξy)k /2 | exp(−2πΣσ∈Σ(L) |σ(ξ)yσ |)eL (ξx)|y|sAE dxd× y 0 X

× E+ \A× E,+

if ξ = −ς(ξ) otherwise.

b0

a(ξyDL/Q , g)|(ξy∞ )k /2 | exp(−2πΣσ∈Σ(L) |σ(ξ)yσ |)|y|sAE d× y

ξ∈L× ξ=−ς(ξ)

Note that ξ = −ς(ξ) and σ(ξ) < 0 for σ ∈ JE0 if and only if ξ = ξ 0 ∆−1/2 for some ξ 0 ∈ E+× . Translating the positivity conditions on ξ to positivity conditions on ξ 0 , we obtain 0 X

Z × E+ \A× E,+

ξ∈L× ξ=−ς(ξ)

Z

X

= × E+ \A× E,+

Z = A× E,+

b0

a(ξyDL/Q , g)|(ξy∞ )k /2 | exp(−2πΣσ∈Σ(L) |σ(ξ)yσ |)|y|sAE dxd× y b0

a(ξ 0 ∆−1/2 yDL/Q , g)|(ξ 0 ∆−1/2 y∞ )k /2 | exp(−2πΣσ∈Σ(L) |σ(ξ 0 ∆−1/2 )yσ |)|y|sAE d× y

× ξ 0 ∈E+

b0

a(∆−1/2 yDL/Q , g)|(∆−1/2 y∞ )k /2 | exp(−2πΣσ∈Σ(L) |σ(∆−1/2 )yσ |)|y|sAE d× y

10.5. RANKIN-SELBERG INTEGRALS

121

By writing this global integral as an infinite product of local integrals, we obtain Z b0 a(∆−1/2 yDL/Q , g)|(∆−1/2 y∞ )k /2 | exp(−2πΣσ∈Σ(L) |σ(∆−1/2 )yσ |)|y|sAE d× y (10.5.1) A× E,+



Y Z

= (10.5.2)

×

0

|σ(∆−1/2 )y|kσ exp(−4π|σ(∆−1/2 )y|)|y|s

0

σ∈Σ(E)

YZ



∞

dy  |y|

a(y∆−1/2 DL/Q , g)|y|sp d× yp

Ep

p

Here we are using the fact that the coefficients a(·, g) are multiplicative in the sense that a(mn, g) = a(m, g)a(n, g) if m + n = OL . We evaluate the factor corresponding to the infinite places and each of the factors corresponding to finite places separately. First, the factor corresponding to the infinite places satisfies   Y Z ∞ dy 0  |σ(∆−1/2 )y|kσ exp(−4π|σ(∆−1/2 )y|)|y|s  |y| σ∈Σ(E) 0 Y 0 = |σ(∆−1/2 )|−s (4π)−kσ −s Γ(kσ0 + s). σ∈Σ(E)

By definition of the different, notice that we may write ∆−1/2 DL/Q = D0 OL for an ideal D0 ⊂ OE . Moreover Y |NE/Q (D0 )| = |dE/Q NE/Q (dL/E ) σ(∆−1/2 )|1/2 σ∈JE

(see [Hid6, §6]). With this in mind we see that the factor corresponding to the finite places satisfies YZ a(∆−1/2 yDL/Q , g)|y|sv(p) d× y Ev(p)

p

=

YZ p

(10.5.3)

=

Y p

Ev(p)

s × a(y, g)|(D0 )−1 v(p) y|v(p) d y

|(D0 )v(p) |−s v(p)

∞ X

a(pj , g)NE/Q (p)−js−j

j=1

!s =

|dE/Q NE/Q (dL/E )1/2

Y

σ(∆−1/2 )|

σ∈JE

Combining (10.5.1) and (10.5.3) yields the lemma.

∞ YX p

a(pj , g)NE/Q (p)−js−j .

j=1

122


One can use Rankin’s method to give another expression for I(f, s). In order to state it precisely, we first set some notation. For Q-algebras A, define the algebraic Q-group B 0 by a b 0 × B (A) := : a ∈ (E ⊗Q A) and b ∈ (E ⊗Q A) ≤ ResE/Q GL2 (A). 0 1 Set ResE/Q GL2 (A)+ : = ResE/Q GL2 (R)0 ResE/Q GL2 (Af ) ResE/Q GL2 (Q)+ : = ResE/Q GL2 (Q) ∩ ResE/Q GL2 (A)+ B 0 (Q)+ : = B 0 (Q) ∩ ResE/Q GL2 (A)+ B 0 (A)+ : = B 0 (A) ∩ ResE/Q GL2 (A)+ 0 0 KE,∞ : = KE,∞ ∩ ResE/Q GL2 (R)0

KE (c0 ) : = K0 (c0 ) ∩ ResE/Q GL2 (A). × Moreover let OE,+ := OE× ∩ E+× . Let N : ResE/Q GL2 (A) → C× be defined by ( 0 0 |y|AE if α = ( y0 ∗1 ) bu for y, b ∈ A× E , u ∈ KE,∞ KE (c ) N (α) := 0 otherwise. 1,0 Moreover, let char0 be the characteristic function of B 0 (A)+ KE (c0 )KE,∞ . 0 1 Now let E , E , E be the Eisenstein series defined by X E 0 (α, s) : = char0 (γα)N (γα)s × 0 γ∈(OE B (Q)+ )\ResE/Q GL2 (Q)+ +

E 1 (α, s) : =

hE X

E(det(ti (c0E ))α, s)

i=1

where c0E ⊂ OE is the unique ideal satisfying KE (c0 ) = K0 (c0E ) (note that the primes dividing c0 ∩ OE are precisely the primes dividing c0E ). These Eisenstein series are absolutely convergent if Re(s) > 1 (see [Hid3, §9.1 (8) and (10b)]). As above, let Z0 be the image of ResE/Q GL2 (A) ,→ ResL/Q GL2 (A) → Y0 (c0 ), where the first arrow is the diagonal embedding and the second is the canonical projection. Then we have the following proposition. Proposition 10.8. We have Z I(f, s) = Z0

(f ⊗ θ−1 )(ιJE (α))E 1 (α, s)dµc0E (α).

The proof is essentially the same as the computation in [Hid4, §4] (see also [Hid6, p. 20-21]).


123

Proof. As above, write g = f ⊗ θ−1 to ease notation. Notice that the automorphy properties of f , together with the fact that χθ−2 is trivial when restricted to the image of A× E under the diagonal embedding, implies that for α ∈ ResE/Q GL2 (A), b0

g(ιJE (α))j(ιJE (α∞ ), z0 )k +21 1,0 is invariant under right multiplication by KE,∞ KE (c0 ) on ResE/Q GL2 (A) and left multiplication by B 0 (Q)+ on ResE/Q with the remark after the definition of Sκ (K0 (c), χ) in √ GL2 (A)√(compareΣ(L) . §7.3). Here z0 := ( −1, . . . , −1) ∈ H 1,0 Let char0 be the characteristic function of B 0 (Q)+ KE,∞ KE (c0 ). We have

Z I(f, s) = B 0 (Q)+ \B 0 (A)+

g(ιJE (α))| det(α)|s+1 AE dµB 0 (α)

Z

b0

= B 0 (Q)+ \B 0 (A)+

g(ιJE (α))j(ιJE (α∞ ), z0 )k +21 | det(α)|sAE dµB 0 (α)

Z

b0

= 1,0 1,0 B 0 (Q)+ \B 0 (A)+ KE,∞ KE (c0 )/KE,∞ KE (c0 )

g(ιJE (α))j(ιJE (α∞ ), z0 )k +21

+

×

hE X

char0 (det(ti (c0E ))α)| det(det(ti (c0E ))α)|sAE dµc0E (α)

i=1 Z

X

= 1,0 ResE/Q GL2 (Q)+ \ResE/Q GL2 (A)+ /KE,∞ KE (c0 )

g(ιJE (γα))

× 0 γ∈OE B (Q)+ \ResE/Q GL2 (Q)+

+

×

hE X

b0

j(ιJE (γα∞ ), z0 )k +21 char0 (γ det(ti (c0E ))α)| det(det(ti (c0E ))α)|sAE dµc0E (α).

i=1

Here we used the fact that 1,0 ResE/Q GL2 (A)+ = ResE/Q GL2 (Q)+ B 0 (A)+ KE (c0 )KE,∞

(see [Hid4, §4]) and that +

hE X i=1

char0 (γ det(ti (c0E ))α)| det(det(ti (c0E ))α)|sAE

124


1,0 is supported on B(A)+ KE (c0 )KE,∞ . The last line is equal to Z g(ιJE (α)) 1,0 ResE/Q GL2 (Q)+ \ResE/Q GL2 (A)+ /KE (c0 )KE,∞

+

b k0 +21

X

×

j(ιJE (γα∞ ), z0 )

γ∈B 0 (Q)+ \GE (Q)+

hE X

char0 (γ det(ti (c0E ))α)| det(det(ti (c0E ))α)|sAE dµc0E (α)

i=1

Z = 1,0 GE (Q)+ \GE (A)+ /KE (c0 )KE,∞

g(ιJ 0 (α))E 1 (α, s)dµc0E (α).

We can now prove Theorem 10.4: Proof of Theorem 10.4. By Proposition 10.6, we can and do assume that J ⊂ Σ(L) is a type for L/E. By Lemma 10.7, for sufficiently large s we have 1/2

(10.5.4)

ζ

c0 ∩OE

(2s)I(f, s) =

dE/Q ΓE (k 0 + s1) (dE/Q NE/Q (dL/E ))s (4π){k0 }+[E:Q](s)

L(As(f ⊗ θ−1 ), s).

By Proposition 10.8, we have (10.5.5)

|dE/Q |1/2 ΓE (k 0 + s1) L(As(f ⊗ θ−1 ), s) |dE/Q NE/Q (dL/E )|s (4π){k0 }+[E:Q](s) Z 0 = (f ⊗ θ−1 )(ιJE (α))ζ c ∩OE (2s)E 1 (α, s)dµc0E (α) Z0

Write 0

E(α, s) := ζ c ∩OE (2s)NE/Q (c0E )

q |dE/Q |E 1 (α, s),

It follows from the analytic properties of the Asai L-function (see §7.7.4) and the absolute convergence of E(α, s), for s > 1 that both sides of this equality are holomorphic functions of s in the half-plane Re(s) > 1. Moreover, the fact that E(α, s) can be meromorphically continued to whole complex plane with a simple pole at s = 1 (see [Hid4, §6] or [Hid3, §9.1 Theorem 1]) implies the same is true of the right hand side (we could have also applied the results of [R2] to come to this conclusion). By Theorem 6.1 of [Hid4] (see also [HT, p. 245]), we have that 0

Ress=1 E(α, s) = 2−1 π [E:Q] Ress=1 ζ c ∩OE (s).


125

Taking residues at s = 1 on both sides of (10.5.5) we obtain NE/Q (c0E )|dE/Q |ΓE (k 0 + 1) Ress=1 L(As(f ⊗ θ−1 ), s) |dE/Q NE/Q (dL/E )|(4π){k0 }+[E:Q] Z −1 [E:Q] c0 ∩OE =2 π Ress=1 ζ (s) (f ⊗ θ−1 )(ιJE (α))dµc0E (α) ZZ0 0 = 2−1 π [E:Q] Ress=1 ζ c ∩OE (s) (f ⊗ θ−1 )(ιJE (α−ι ))dµc0E (α) Z0

−1 [E:Q]

=2 π

Ress=1 ζ

c0 ∩OE

(s)(−1)[E:Q]+

P

σ∈Σ(E)

0 kσ

C1−1 hωJE (f −ι ), [Zθ ]iK ,

where the last equality is by Proposition 10.6. Replacing JE by the type J, we obtain NE/Q (c0E )ΓE (k 0 + 1) −1 (10.5.6) ), s) 0 }+[E:Q] Ress=1 L(As(f ⊗ θ {k |NE/Q (dL/E )|(4π) 0

= 2−1 π [E:Q] Ress=1 ζ c ∩OE (s)(−1)[E:Q]+

P

σ∈Σ(E)

0 + kσ

P

σ∈JE −(J∩JE )

0 kσ

C1−1 hωJ (f −ι ), [Zθ ]iK .

We claim that if hωJ (f −ι ), Zθ iK 6= 0 then f is a base change of a modular form on E. In order to see this, note first that hωJ (f −ι ), Zθ iK 6= 0 implies that Ress=0 L(As((f ⊗ θ−1 )), s) 6= 0. Write g = f ⊗ θ−1 to ease notation. As recalled in §7.7.4, (10.5.7)

0

0

0

LDL/E c (g × g ς , s) = LdL/E (c ∩OE ) (As(g) ⊗ η, s)LdL/E (c ∩OE ) (As(g), s).

Let π(g), π(f ), etc. be the (unitary) cuspidal automorphic representations attached to g, 0 f , etc. With our normalization, LDL/E c (g × (g ς ), s) has a pole at s = 1 if and only if the contragredient π(g)∨ satisfies π(g)∨ ∼ = π(g)ς [JS2, Proposition 3.6]. This implies that π(f ς ⊗ (θς )−1 ) ∼ = π(f ⊗ θ−1 )ς ∼ = π(f ⊗ θ−1 )∨ ∼ = π(f ) ⊗ χ−1 θ. Thus π(f ς ) ⊗ (θς θ)−1 χ ∼ = π(f ). It is easy to check that (θς θ)−1 χ = χtriv , so we conclude that π(f ς ) ∼ = π(f ). By the theory of base change [L], this implies that f is a base change of a Hilbert modular form g on E. Thus our claim is proven if we show that L(As(g), s) has a pole 0 at s = 1 if and only if LdL/E (c ∩OE ) (As(g), s) does. To prove this, we show that any pole s0 of Lp (As(g), s) satisfies Re(s0 ) < 1. Suppose that p - f(π(g)) ∩ OE . Then for any prime P above p we have the well-known estimate NL/Q (P)−1/5 < |ai,P (g)| < NL/Q (P)1/5 (see [S, (4.1.3)]). We note that one actually knows |ai,P (g)| = 1 by the Ramanujan conjecture (see [B]) but we won’t need this. The absolute convergence then follows from an elementary bounding argument. If p | f(π(g)) ∩ OE , then π(g) is ramified, and in this case we have strong

126


bounds on the P-power Fourier coefficients of g which easily imply the desired holomorphicity (see [SW, Theorem 3.3]). We now show that, under the assumption that hωJ (f −ι ), Zθ iK 6= 0, the Hilbert modular form g on E whose base change to L is f has nebentypus χE . Its nebentypus is either χE or χE η. Suppose that the nebentypus of g is χE η. By an analogue of the proof of Proposition 7.8, we obtain (10.5.8)

LdL/E (c∩OE ) (As(f ⊗ θ−1 ), s) = LdL/E (c∩OE ) (Ad(g), s)LdL/E (c∩OE ) (η, s).

As proven above, the left hand side has a pole at s = 1. On the other hand, LdL/E (c∩OE ) (Ad(g), s)ζ dL/E (c∩OE ) (s) = LdL/E (c∩OE ) (g × (g ⊗ (χE η)−1 ), s) for Re(s) > 1. Since the pole of the right hand side at s = 1 is simple [JS2, Proposition 3.6], it follows that LdL/E (c∩OE ) (Ad(g), s) has no pole at s = 1, contradicting (10.5.8). Now assume that f is the base change of a Hilbert modular form on E with nebentypus χE . In this case (10.5.6) and Proposition 7.8 imply that NE/Q (c0E )ΓE (k 0 + 1) 0 dL/E c0 ∩OE (Ad(f ⊗ η), 1)Lb (As(fb ⊗ θ−1 ), 1) 0 }+[E:Q] L {k |NE/Q (dL/E )|(4π) 0

P

0

P

= 2−1 π [E:Q] ζ b (1)(−1)[E:Q]+ σ∈Σ(E) kσ + Q p. The theorem follows. where b0 = p|dL/E p-f(θ)∩OE

σ∈JE −(J∩JE )

0 kσ

C1−1 hωJ (f −ι ), [Zθ ]iK

CHAPTER 11

Eisenstein Series with Coefficients in Intersection Homology [L:Q]

Thus far we have ignored classes in I m Hinv (X0 (c), L(b κ, χ0 )) and their Poincaré duals in intersection homology. We now take up the study of these classes. We will assume throughout this chapter that the following condition holds: • We have that κ b = (0, m) b ∈ X (L) for some m = [m]1 ∈ Z1. [L:Q]

κ, χ0 )) is zero unless κ = (0, m) for some m ∈ Z1 As stated in §8.7, the group I m Hinv (X0 (c), L(b so this is no loss of generality. We also continue to assume the notational conventions of §9.1. 11.1. Eisenstein series It turns out that the formal Fourier series associated to certain elements of [L:Q]

I m Hinv (X0 (c), L(b κ, χ0 )) are Eisenstein series with coefficients in intersection homology. In order to make this precise, we first set notation for certain Eisenstein series. For each cE ⊂ OE and character ϑ : E × \A× E → × C that is trivial at the infinite places and whose conductor divides cE , let Ec0E ,ϑ (( y0 x1 )) be the Fourier series given by |y|AE

X

σc0 E ,ϑ (ξyDE/Q )qκ (ξx, ξy).

ξ∈E × ξ0

Here for each ideal cE ⊂ OE , the function σc0 E ,ϑ is defined on the set of fractional ideals of E by setting (P ϑ(a/b)NE/Q (b)1−[m] if a is integral b⊆a 0 0 a/b+cE =OE σcE ,ϑ (a) := σcE ,ϑ,[m] (a) : = 0 otherwise. In the definition of σc0 E ,ϑ , the sum is over integral ideals. We have also committed a standard abuse of notation, in that whenever we wrote ϑ(b) for some ideal b ⊂ OE we should have written ϑ(b) for some idèle trivial at the infinite places whose associated ideal is b. This is well-defined by the assumption that the conductor of ϑ divides cE . The following proposition is well-known: 127

128

11. EISENSTEIN SERIES WITH COEFFICIENTS IN INTERSECTION HOMOLOGY

Proposition 11.1 (Shimura). Let cE ⊂ OE . If cE 6= OE , then Ec0E ,ϑ ∈ M(0,m) (K0 (cE ), ϑ). One can obtain this proposition, for example, from the results contained in [Hid4, §6]) after twisting by an appropriate power of | · |AE (see also [Sh2, Proposition 3.4] and [Hid3, Theorem 1, §9.1]). 11.2. Invariant classes revisited In (8.7.5) we defined b χ0 )) I m Hiinv (X0 (cE ), L((0, m), to be

! M

(π fr )∗ Zς

K fr

H i (g, K∞ ; Cφ ⊗ L((0, m), b χtriv ))

,

φ

where the sum is over φ = χ ◦ det : G(Q)\G(A) → C× such that the quasicharacter χ satisfies m b χ∞ (b∞ ) = b− ∞ . We now make this space more explicit. For each subset J ⊂ Σ(L), consider the differential form on HΣ(L) given by υJ := ∧σ∈J yσ−2 dxσ ∧ dyσ . Then, by [Har, §3.2], we have H 2i+1 (g, K∞ ; Cφ ⊗ L((0, m), b χtriv )) = 0 and

D E H 2i (g, K∞ ; Cφ ⊗ L((0, m), b χtriv )) = υJ ⊗ φ : |J| = i . 11.3. Definition of the VχE (m)

We now make the following simplifying assumption that: × 0 × × χE = χ02 E for some quasicharacter χE : E \AE → C .

Under this assumption, for any J ⊂ Σ(L) of order [L : Q]/2 set (11.3.1) κ, χ0 )). VJ = (π fr )∗ Zς υ J ⊗ (χ0E ◦ NL/E ◦ det) ∈ I m H[L:Q] (X0 (cE ), L(b We wish to define classes VJ,χE (m) in analogy with the definition in (9.3.3). Let m ⊂ OE . If m = NL/E (m0 ) for some m0 ⊂ OL , m + dL/E (cE ∩ OE ) = OE , define VJ,χE (m) := TbcE ,χE (m)∗ VJ , otherwise, define VJ,χE (m) = 0. Here TbcE ,χE (m) is the Hecke operator introduced in §9.2.

11.4. STATEMENT AND PROOF OF THEOREM 10.2

129

11.4. Statement and proof of Theorem 11.2 × Recall that we are assuming that χE : E × \A× is a quasicharacter of the form E → C × 02 0 × × χE = χE for a quasicharacter χE : E \AE → C . With these assumptions in mind, we may state the main result of this section:

Theorem 11.2. Suppose κ = (0, m) ∈ X (L) and cE ⊂ OL . Choose an auxilliary ideal b0 ⊂ OE such that b0 6= OE if (cE ∩ OE )dL/E = OE and set cE := b0 (c ∩ OE )dL/E . If J ⊂ Σ(L) has order [L : Q]/2, then the formal Fourier series X ΦVJ ,χE ,b0 (( y0 x1 )) := |y|AE VJ,χE (ξyDL/E )qκ (ξx, ξy) ξ∈E × , ξ0 ξy0 DL/E +b0 =OE

is an element of inv I m H[L:Q] (X0 (c), L(b κ, χ0 )) ⊗ Mκ (K0 (N 0 (c)), χE ).

In particular, ΦVJ ,χE ,b = VJ ⊗

1 0 EcE ,1 ⊗ χ0E η + Ec0E ,1 ⊗ χ0E . 2

Here N 0 (c) := dL/E (c ∩ OE )3 b0 , and the twisting operator that we applied to the Eisenstein series Ec0E ,1 was defined in Lemma 7.3. Remarks. (1) In analogy with Theorem 9.3, Theorem 11.2 admits a generalization where I m H[L:Q] (X0 (c), L(b κ, χ)) is replaced by an arbitrary Hecke module. We omit the (straight forward) details. (2) The auxilliary ideal b0 was introduced for simplicity so that we always work with an Eisenstein series that does not have a constant term at ∞. The notion of the degree of a Hecke operator will be useful in the proof of Theorem 11.2. For any Z-algebra A ≤ C we recall the homomorphism deg : Tc ⊗Z A −→ A defined by letting deg(K0 (c)γK0 (c)) be the number of summands in a decomposition deg(K0 (c)γK0 (c))

K0 (c)γK0 (c) =

X i=1

γi K0 (c)

130

11. EISENSTEIN SERIES WITH COEFFICIENTS IN INTERSECTION HOMOLOGY

for some γi ∈ R(c) and extending A-linearly. See [Sh1] for a proof that deg is well-defined and is an algebra homomorphism. One can check1 that 0 0 deg(Tc (m)) = σc,1 (m) := σc,1,[m] (m),

(11.4.1)

where by the subscript 1 we mean the trivial character and we define the function σ for ideals of OL as we did for ideals of OE in §11.1 above. inv (X0 (c), L(κ, χ0 )), it suffices to prove Proof of Theorem 11.2. By definition of I m H[L:Q] the second assertion of the theorem. For this, it suffices to show that χ0 (n) VJ,χE (n) = E (11.4.2) η(n)σc0 E ,1 (n) + σc0 E ,1 (n) VJ 2 for ideals n ⊂ OL such that n + cE = OE . This is obvious if n is not a norm from OL , so we assume that n is a norm from OL for the remainder of the proof. We defined Tbc,χE multiplicatively, so it suffices to prove (11.4.2) for prime powers. Suppose that p ⊂ OE is a prime coprime to dL/E (c ∩ OE )b splitting as p = PP in OL . Then we have VJ,χ (pr ) : = Tbc (pr )∗ VJ E

E

= = =

χ0E (NL/E (Pr )) deg(Tc (Pr )))VJ 0 χ0E (pr )σc,1 (Pr )VJ χ0E (pr )σc0 E ,1 (pr )VJ

It is easy to see that (11.4.2) in the case n = pr follows from this. Now assume that p ⊂ OE is a prime coprime to dL/E (c ∩ OE )b that is inert in L/E (we denote by P the integral closure of p in OL ). We have VJ,χ (p2r ) := Tbc (p2r )∗ VJ E

E

= Tc (Pr )∗ VJ + χE (p)NE/Q (p)Tc (Pr−1 )∗ VJ = χ0E (NL/E (Pr )) deg(Tc (Pr ))VJ + χE (p)NE/Q (p)χ0E (NL/E (Pr−1 )) deg(Tc (Pr−1 ))VJ 0 0 2r−2 0 = χ0E (p2r )σc,1 (Pr ) + (χ02 )σc,1 (Pr−1 ) VJ E )(p)NE/Q (p)χE (p = χ0E (p2r )σc0 E ,1 (p2r )VJ . This implies (11.4.2) in the case n = p2r for inert p.

1See

[Bu, (6.4) p. 494] for the case of Tc (p) when p - c.

APPENDIX A

Proof of Proposition 2.4 A.1. Cellular cosheaves Let K be a finite regular cell complex. A cellular cosheaf1 is a gadget E that assigns to each cell σ ∈ K an R module Eσ and to each face τ < σ a homomorphism Φστ : Eσ → Eτ such that whenever τ < ω < σ are faces, we have Φστ = Φωτ ◦ Φσω commutes. Thus, E is a contravariant functor from the category (which we also denote by K) whose objects are the cells of K and whose morphisms τ → σ are inclusions of faces τ < σ. A simplicial local system is a cellular cosheaf such that all the homomorphisms Φστ are isomorphisms. An elementary r chain with coefficients in a cellular cosheaf E is an equivalence class of formal products aσ where σ ∈ K is an r dimensional cell and a ∈ Eσ , modulo the identification aσ ∼ (−a)σ 0 where σ 0 is the same cell but with the opposite orientation. The chain module Cr (K, E) is the collection of all finite formal linear combinations of elementary cellular r chains. The boundary of an elementary r chain aσ is X ∂(aσ) = Φστ (a)τ τ

where the sum is taken over those τ < σ of dimension r − 1. Let Hr (K, E) be the homology of the complex C∗ (K, E). It is called the homology of K with coefficients in the cosheaf E. If K 0 is a (finite) refinement of K then the cosheaf E on K determines a cosheaf E0 on K 0 by declaring E0 (σ) = E(τ ) where σ ∈ K 0 , and τ ∈ K is the unique cell containing the interior σ o . Then refinement determines a natural injection Cr (K, E) → Cr (K 0 , E0 ). If σ is a cell in a convex linear cell complex K and if σ b ∈ σ o is a point in its interior, define the stellar subdivision of K with respect to σ b to be the convex linear cell complex in which the cell σ has been divided into the collection of cones σ b ∗ τ (where τ varies over the cells in ∂σ), together with the zero dimensional cell σ b. This definition differs slightly from that in some other texts such as [Hu]. Even if K is a simplicial complex, the resulting subdivision K 0 is only a cell complex. The first barycentric subdivision of K may be obtained by starring (i.e. taking the stellar subdivision) with respect to all the simplices of K in order of increasing dimension. 1Recall

that every simplex σ has a canonical open neighborhood, namely the open star Sto (σ) of σ. The reason for the terminology “cellular cosheaf” is that if τ < σ then the open stars satisfy the reverse containment: Sto (σ) ⊂ Sto (τ ). 131

132

A. PROOF OF PROPOSITION 2.4

Proposition A.1. Let E be a cellular cosheaf on a convex linear cell complex K and let K 0 be a refinement of K with corresponding cosheaf E0 . Then the refinement mapping induces a canonical isomorphism Hr (K, E) ∼ = Hr (K 0 , E0 ) for all r. Proof. (Most of the classical proofs of the invariance of homology under subdivision are either very complicated, or else they do not work in this setting.) Let us consider the case that K 0 is a stellar subdivision of K with respect to a single barycenter σ b ∈ σ o of a cell of dimension r. In this case, we need to show that the homology of the quotient complex D∗ = C∗ (K 0 , E0 )/C∗ (K, E) vanishes. Let L denote the subcomplex of K corresponding to ∂σ. For j 6= 0, r the quotient Dj = Cj (K 0 , E0 )/Cj (K, E) may be identified with the group Cj−1 (L, Eσ ) (with constant coefficients), while D0 = Eσ and Dr = Cr−1 (L, Eσ )/Eσ , this last being the quotient under the diagonal embedding Eσ → Cr−1 (L, Eσ ). In other words, the complex D∗ is the chain complex (with constant coefficients) for the r − 1 dimensional sphere, augmented in degrees 0 and r, so Hr (D∗ ) = 0 for all r. The stellar subdivisions of K form a cofinal system in the directed set of all finite refinements of K, so we conclude that the homology H∗ (K, E) is invariant under refinement. We remark for completeness that a simplicial sheaf is a covariant functor from the category K to the category of R-modules. The cohomology of a simplicial sheaf is defined in a way that is dual to the homology of a cosheaf.

A.2. Proof of Lemma 2.3 Let K be a regular cell complex, L a closed subcomplex, set X = |K| and Y = |K| − |L| with i : Y → X the inclusion mapping. Let E be a local coefficient system on Y. Then we obtain a cellular cosheaf i! E on X which assigns to any cell σ ∈ K the group ( Eσ if σ o ⊂ Y i! E(σ) = . 0 otherwise It follows immediately from the definitions that the chain groups brK (Y, E) Cr (K, i! E) = C b rK (Y, E). Proposition A.1 then implies that the cellular are identical, hence Hr (K, i! E) ∼ = H K Borel-Moore homology groups Hr (Y, E) are invariant under (finite) refinements, which proves Lemma 2.3.

A.3. PROOF OF PROPOSITION 2.4

133

A.3. Proof of Proposition 2.4 In this section, K is a finite convex linear cell complex, L is a closed subcomplex, X = |K|, Y = |K| − |L|, and E is a local coefficient system (of R modules) on Y. Set A = |L| ⊂ X. We have a triangulation T of Y (with infinitely many simplices) that refines K and we wish to brK (Y, E) → CrBM,T (Y, E) → consider the map on homology that is induced from the injection C CrBM (Y, E). First let us consider the case that E extends as a local system over all of X. Then the chain bK (Y, E) is canonically isomorphic to the quotient group Ci (K, E)/Ci (L, E), since both group C i complexes have bases that are indexed by i-dimensional simplices σ ∈ K such that σ 6∈ L, in other words, such that σ o ⊂ Y. It is also easy to see that the boundary homomorphisms are b rK (Y, E) ∼ compatible with this isomorphism, hence H = Hr (X, A; E). On the other hand, the BM mapping Cr (K, E)/Cr (L, E) → Cr (Y, E) also induces an isomorphism on homology, which may be checked by building up X from A, one simplex at a time. Now consider the general case, when E does not necessarily extend as a local system over X. By taking the barycentric subdivision if necessary, we may assume that K is a simplicial complex and that L is a full subcomplex, meaning that if σ is a simplex of K and all its vertices are in L then σ ∈ L. Let M be the subcomplex of K consisting of simplices that are disjoint from A = |L|. Then M is also a full subcomplex of K (for if σ is a simplex of K whose vertices are in M, then the simplex σ is disjoint from A so it lies in M ). Let us now consider the simplices of K that are neither in M or L. The vertices of such a simplex σ fall into two classes: those in L and those in M. Thus we have isolated two faces τL ∈ L and τM ∈ M of σ such that σ = τL ∗ τM is the join of these faces. In other words, every point x ∈ σ lies on a unique line segment Ix with one endpoint in τL and the other endpoint in τM . In this way, the mapping |L| → {0} and |M | → {1} extends uniquely, linearly, and continuously to a simplicial mapping f : X → [0, 1]. (See, for example, [Mu] Lemma 70.1, p. 414.) Let us denote by XS = f −1 (S) for any subset S ⊂ [0, 1]. Then Y = X[0,1) . The simplicial mapping f is necessarily a product mapping over the open interval (0, 1). We obtain in this way a “collaring” X(0,1) ∼ = X{ 21 } × (0, 1) of a neighborhood of infinity. It follows that there is a piecewise linear homeomorphism Y → X[0, 1 ) , which then induces 2 canonical isomorphisms (A.3.1)

HrBM (Y, E) → HrBM (X[0, 1 ) , E) ∼ = Hr (X[0, 1 ] , X{ 1 } ; E) 2

2

2

b K (Y, E). since E extends across X{ 1 } . We now wish to make a similar identification of H r 2 1 Let us refine the interval [0, 1] by adding a vertex { 2 }, and then refine K into a regular cell complex such that f −1 ( 12 ) is a subcomplex: each simplex σ = τL ∗ τM is decomposed into two cells, σ ∩ f −1 ([0, 12 ]) and σ ∩ f −1 ([ 21 , 1]) whose intersection is the cell σ ∩ f −1 ( 12 ). Let K 0 bK 0 (Y, E) be the cellular Borel-Moore chains denote this (cellular) refinement of K and let C r on Y with respect to this pseudo-cellulation K 0 of Y. The chain group decomposes into three

134

A. PROOF OF PROPOSITION 2.4

M z}|{

L z}|{

τL σ τM f ? 1 2

0

1

Figure 1. Decomposing simplices of K 0

brK (Y, E) = Ar ⊕ Br ⊕ Cr , each of which consists of formal linear combinations of subgroups C r dimensional cells σ ∈ K 0 with σ o ⊂ X[0, 1 ) in case Ar , with σ o ⊂ X{ 1 } in case Br , and with 2 2 σ o ⊂ X( 1 ,1) in case Cr . The boundary homomorphism decomposes as follows. 2

Ar+1 ?

⊕

Br+1 -

⊕

Cr+1

?

?

Ar ⊕ Br ⊕ Cr Each cell in Cr is the product of a cell in Br−1 with the open interval ( 12 , 1) so the subcomplex B ⊕ C is acyclic. Hence the left arrow in the following diagram induces an isomorphism on homology groups. The right arrow in this diagram is an isomorphism of chain complexes. A ⊕ B ⊕ C −−−→ (A ⊕ B ⊕ C)/(B ⊕ C) ←−−− (A ⊕ B)/B Altogether, we obtain a canonical isomorphism b K 0 (Y, E) ∼ H = Hr (A ⊕ B/B) = Hr (X r

[0, 21 ] , X 12 ; E).

Combining this with the isomorphisms of Lemma 2.3 and equation (A.3.1) gives the desired isomorphism b rK (Y, E) ∼ b rK 0 (Y, E) ∼ H =H = HrBM (Y, E).

APPENDIX B

Recollections on Orbifolds If a compact group acts with finite isotropy on a smooth manifold, then the quotient space is an orbifold. The singularities in such a space are “mild” in that it is possible to develop a theory of differential forms on such a space, which supports the theorems of Stokes and de Rham. In this section we review the definition and basic properties of orbifolds, or V-manifolds, as described in [Sa], [CR], and elegantly reformulated by [MP] (see also [Ka], [Ba], and [Mo]). Although none of this material is new, we have filled in some technical details, particularly with respect to refinement of an orbifold atlas. As a consequence, several hypotheses appearing in [Sa], [Ka], and [CR] may be omitted from the definition of an orbifold. We note that there are slight differences among the various definitions and terminologies. The orbifolds considered here are reduced in the sense of [CR]. B.1. Effective actions Let G be a finite group of diffeomorphisms of an n dimensional connected manifold M. Assume the action is effective: every element g ∈ G acts non-trivially except for the identity element 1 ∈ G. Write g · m for the action of g ∈ G on m ∈ M and let dg(m) : Tm M → Tg·m M be the differential of this mapping. For each m ∈ M let Gm ⊂ G denote the isotropy group of G, and for each g ∈ G let M g denote the set of points in M that are fixed by g. A choice of G-invariant Riemannian metric on M determines a system of geodesics on M such that the exponential mapping is G-equivariant in the sense that the following diagram commutes over some neighborhood Wm of the origin in Tm M :

(B.1.1)

Tm M −−−→ Tg·m M dg(m)    exp expy y M

−−−→ g

M

See [KN] §VI Prop. 1.1 and §IV Prop. 2.5. This has several immediate consequences. (A) For any g ∈ G the fixed point set M g is a union of smoothly embedded closed submanifolds of M. For if g · m = m then there exists a neighborhood Um of m such that M g ∩ Um = exp(V ) where V = {v ∈ Tm M | dg(m)(v) = v} ∩ Wm is an open set in a vector space. 135

136

B. RECOLLECTIONS ON ORBIFOLDS

(B) Let g ∈ G. If there exists a point m ∈ M g such that dg(m) : Tx M → Tx M is the identity mapping, then g = 1. For, if dg(m) = I then g fixes a whole neighborhood of m. Therefore the set of points y ∈ M such that dg(y) = I is both open and closed, so g acts trivially on M. Since the action is effective, g = 1. (C) The set M 0 of points on which G acts freely is open and dense in M. In fact M 0 is the complement of the finitely many closed submanifolds M g (for g 6= 1), each of which has codimension ≥ 1. Lemma B.1. ([MP]) Let M 0 be another connected n dimensional manifold with an effective action by a finite group G0 . Let i : M/G → M 0 /G0 be an embedding. Suppose f : M → M 0 is a smooth embedding which covers the mapping i. Then (1) There exists a unique mapping λ : G → G0 such that f is equivariant with respect to λ (meaning that λ(g) · f (m) = f (g · m)). (2) The mapping λ is an injective group homomorphism. Its image is Im(λ) = {g 0 ∈ G0 | g 0 · f (M ) = f (M )} . For each m ∈ M the mapping λ induces an isomorphism of isotropy groups Gm ∼ = 0 Gf (m) . (3) If there exists g 0 ∈ G0 such that f (M ) ∩ g 0 · f (M ) 6= φ then f (M ) = g 0 · f (M ) and g 0 is in the image of λ. (4) If h : M → M 0 is another smooth embedding that covers the same embedding i then there exists a unique g 0 ∈ G0 such that h(m) = g 0 · f (m) for all m. Let MS be the category of n-manifolds with finite symmetry. An object in MS is a pair (M, G) where M is a smooth connected n-manifold and G is a finite group acting effectively on M. A morphism (M, G) → (M 0 , G0 ) is a smooth equivariant open embedding f : M → M 0 which induces an open embedding i : M/G → M 0 /G0 . In this case we say that the embedding f covers the embedding i. Proof of Lemma B.1. We will prove (4) first. Assume we are given mappings f, h : M → M 0 which cover the mapping i. If M 0 and (M 0 )0 denote the subsets on which G and G0 act freely then the set W = M 0 ∩f −1 ((M 0 )0 )∩h−1 ((M 0 )0 ) is open and dense in M ; the projection π : W → W/G is a smooth unramified covering; and the restriction i : W/G → (M 0 )0 /G0 is a smooth embedding of smooth manifolds. Fix a point m0 ∈ W. Then f (m0 ) and h(m0 ) lie in the same fiber (π 0 )−1 (iπ(m0 )) where π 0 : M 0 → M 0 /G0 is the projection. Therefore there is a unique g00 ∈ G0 such that h(m0 ) = g00 · f (m0 ). We will show that h(m) = g00 · f (m) for all m ∈ M. ¿From covering space theory we know that h(m) = g00 · f (m) for all m in the connected component of W that contains the point m0 , and this equality also holds in the closure of this connected component. The same remarks apply to every connected component of W. Let W0

B.1. EFFECTIVE ACTIONS

137

denote the union of those connected components of W such that h(m) = g00 f (m) for all m in the closure W 0 of W0 . Recall that W is the complement of a finite collection of closed submanifolds of M having dimension ≤ n − 1. Choose a point m1 ∈ M on one of these submanifolds that separates the region W0 from some other region, say W1 , for which the corresponding group element g10 6= g00 . Choose a G0 -invariant Riemannian metric b(·, ·) on M 0 and consider its pullback f ∗ (b) to M. This is a smooth metric on M. However its restriction to W is the pullback f ∗ (b) = π ∗ i∗ (¯b) of a smooth metric ¯b on M 0 /G0 . Therefore f ∗ (b) is G-invariant on W so by continuity it is G invariant on all of M. Similarly, the metric h∗ (b) coincides with f ∗ (b) on W so they coincide everywhere, and h is also an isometry. By [KN] §VI Proposition 1.1 we again have a diagram which commutes over some neighborhood of the origin in Tm1 M : Tm1 M −−−0−−−→ Tf (m1 ) M 0 d(g0 f )(m)    exp expy y (B.1.2)

M   πy

−−0−→

M0   0 yπ

M/G

−−−→

M 0 /G0

g0 ·f

i

and there is a similar diagram for h. On the one hand h(m) = g00 · f (m) for m in the region W 0 (hence dh(m1 ) = dg00 ◦ df (m1 )) but on the other hand h(m) = g10 · f (m) for m in the region W1 (hence dh(m1 ) = dg10 ◦ df (m1 )). Therefore the group element (g10 )−1 g00 fixes the point f (m1 ) and acts on Tm1 M 0 by the identity mapping, so by (B) above, g10 = g00 . In summary, the set of points m where h(m) = g00 · f (m) is both open and closed, hence h = g00 · f. This completes the proof of part (4) of the lemma. To prove part (1), let us return to the original basepoint m0 ∈ M. Since G acts freely on m0 and since G0 acts freely on f (m0 ), for any g ∈ G there is a unique element, call it λ(g) ∈ G0 such that f (g ·m0 ) = λ(g)·f (m0 ). The mappings m 7→ f (g ·m) and m 7→ λ(g)·f (m) both cover the mapping i and they agree at the point m0 so by the preceding paragraph, they coincide everywhere. Moreover this implies that f (g1 g2 ·m) = λ(g1 )·f (g2 ·m) = λ(g1 )·λ(g2 )·f (m) which proves that λ is a group homomorphism. It is clearly injective: if λ(g) = 1 then f (gx0 ) = f (x0 ). But f is an embedding so gx0 = x0 hence g = 1. This proves part (2). For part (3), suppose f (M ) ∩ g 0 · f (M ) 6= φ. Then the set f −1 (g 0 · f (M )) is open and non-empty. Since M 0 is dense in M there exist m1 , m2 ∈ M 0 such that f (m1 ) = g 0 · f (m2 ). It follows that π(m1 ) = π(m2 ) so there exists a unique g ∈ G such that m2 = g · m1 . The embeddings m 7→ f (g · m) and m 7→ g 0 · f (m) agree at the point m1 so they coincide. Therefore g 0 = λ(g) and hence g 0 · f (M ) = f (g · M ) = f (M ).

138


This lemma allows us to remove several hypotheses in [Sa] and [Ka] concerning the definition of an orbifold.

B.2. Definitions Throughout the remainder of this appendix we fix a locally compact Hausdorff space X and a regular, commutative Noetherian ring R (with unit) of finite cohomological dimension (e.g. a principal ideal domain). An R-orbifold chart (also called a local uniformization) on X is a collection C = (U, M, G, φ) where U ⊂ X is a connected open subset, (M, G) is an an object in MS such that every rational integer dividing |G| is invertible in R, and φ : M → X is a continuous G-invariant mapping which induces a homeomorphism φ¯ : M/G → U ⊂ X onto an open subset U of X. Suppose C = (U, M, G, φ) and C 0 = (U 0 , M 0 , G0 , φ0 ) are charts such that U ⊂ U 0 . We say these charts are compatible, and we write C → C 0 , if there exists a morphism f : (M, G) → (M 0 , G0 ) in MS that covers the inclusion i : U → U 0 . In this case we also write f : (U, M, G, φ) → (U 0 , M 0 , G0 , φ0 ) and we refer to f as an embedding of charts or as a morphism of charts. By Lemma B.1 such a morphism f : M → M 0 , if one exists, is uniquely determined up to the action by elements of G0 . Let us say that an open covering U of X is good if each U ∈ U is connected and if U is closed under pairwise intersections: if U1 , U2 ∈ U then U1 ∩ U2 ∈ U. Definition B.2.1. An R-orbifold atlas U on X consists of a good open cover U and an assignment, for each U ⊂ U of an R-orbifold chart (U, M, G, φ) over U, such that: if U ⊂ U 0 are elements of U then the charts (U, M, G, φ) and (U 0 , M 0 , G0 , φ0 ) are compatible. We say that U is an orbifold atlas over the cover U. We use this somewhat restrictive notion of a “good” open cover, which requires all pairwise intersections U ∩ U 0 to be connected (or empty), in order to facilitate the proof of Proposition B.3 below. However this condition may be weakened to the more standard, but equivalent condition • If U, U 0 ∈ U and if x ∈ U ∩ U 0 then there exists V ∈ U such that x ∈ V ⊂ U ∩ U 0 . In fact, any open cover of an orbifold X admits a “good” refinement [Mo2]. Remark. If X admits a R-orbifold atlas then X is a R-homology manifold. One annoying aspect of these definitions is the fact that, given an R-orbifold atlas and an arrangement of open sets sets in U, U1 −−−→   y

U2   y

U3 −−−→ U4

B.3. REFINEMENT

139

it may be impossible to choose the morphisms so that the corresponding diagram commutes: (M1 , G1 ) −−−→ (M2 , G2 )     y y (M3 , G3 ) −−−→ (M4 , G4 ) Consequently some constructions become quite difficult when using the definition of orbifold as given in [Ka]. On the other hand, suppose U1 ⊂ U2 ⊂ U3 are open sets in X and suppose R-orbifold charts Ci = (Ui , Mi , Gi , φi ) are given over each of these (with 1 ≤ i ≤ 3). It is easy to see that if C1 → C2 (meaning that these charts are compatible) and if C2 → C3 then C1 → C3 . Moreover we have the following: Lemma B.2. Let C1 , C2 , C3 be charts with U1 ⊂ U2 ⊂ U3 as above. Suppose C1 → C3 and C2 → C3 . Then C1 → C2 . Proof. Let f1 : (M1 , G1 ) → (M3 , G3 ) and f2 : (M2 , G2 ) → (M3 , G3 ) be morphisms that cover the inclusions U1 ⊂ U3 and U2 ⊂ U3 respectively. Let M10 be the set of points m such that G3 acts freely on f1 (m), cf. Lemma B.1 part (2). Then there exists g3 ∈ G3 such that g3 · f1 (m) ∈ f2 (M2 ). We claim that g3 · f1 (M1 ) ⊂ f2 (M2 ), from which it will follow that f2−1 ◦ (g3 · f1 ) : M1 → M2 is a morphism covering the inclusion U1 ⊂ U2 . ¿From the theory of covering spaces we know that f2 (M2 ) contains the image g3 · f1 (M1 ) of the connected component M1m of M10 that contains the point m. So, just as in the proof of Lemma B.1 we must consider the behavior of the mapping f3 at a point m1 ∈ M1 that separates several regions of M10 . A choice of G3 -invariant Riemannian metric on M3 determines Gj -invariant Riemannian metrics on Mj (for j = 1, 2) and we have a diagram which commutes in some neighborhood of the origins, d(g3 f1 )(m1 )

df2 (m2 )

Tm1 M1 −−−−−−−→ Tm3 M3 ←−−−− Tm2 M2      exp expy expy y M1

−−−→ g3 ·f1

M3

←−−− f2

M2

where m3 = g3 · f1 (m1 ) and m2 = f2−1 (m3 ). Since df2 is an isomorphism, g3 · f1 takes a whole neighborhood of x1 into the image, f2 (M2 ). Therefore the set of points in M1 that are taken, by g3 · f1 into f2 (M2 ) is both open and closed in M1 . This proves the claim, and hence completes the proof of the lemma. B.3. Refinement A good open covering U of X is said to refine a good open covering U 0 if every U ∈ U is contained in some U 0 ∈ U 0 . An R-orbifold atlas U over a good open cover U is said to refine

140


an R-orbifold atlas U0 over a good open cover U 0 if U refines U 0 and if the chart (U, M, G, φ) is compatible with the chart (U 0 , M 0 , G0 , φ0 ) whenever U ∈ U is contained in U 0 ∈ U 0 . Proposition B.3. Let U, U 0 be good open covers such that U refines U 0 . Let U0 be an Rorbifold atlas over U 0 . Then there exists an R-orbifold atlas over U that refines U0 . Proof. For each U ∈ U choose a chart (U 0 , M 0 , G0 , φ0 ) in U0 such that U ⊂ U 0 . Choose a connected component M of the fiber product F P = U ×U 0 M 0 : F P −−−→   φy

M0  φ0 y

U −−−→ U 0 The group G0 acts on F P so we may define G = {g 0 ∈ G0 | g 0 (M ) = M } . Since this is a subgroup of G0 , we have that every rational integer dividing |G| is invertible in R. We claim that these choices {(U, M, G, φ)| U ∈ U} form an atlas U that refines the atlas U0 . First we must show that the charts C1 = (U1 , M1 , G1 , φ1 ) and C2 = (U2 , M2 , G2 , φ2 ) are compatible whenever U1 ⊂ U2 are elements of U. Let C10 , C20 be the charts in U0 that were 0 associated to U1 and U2 respectively, and let C12 be the chart corresponding to U10 ∩ U20 ∈ U 0 . Consider the diagram of inclusions and chart compatibilities, U10

U10 ∩ U20

-

U20

C10

0 C12

6

6

U1

-

U10

C20

6

6

C1

U2

U20

-

U10

-

C2

0 . C12

C10

Applying the same lemma to → gives ⊂ ∩ Applying Lemma B.2 to U1 ⊂ 0 U1 ⊂ U2 ⊂ U2 gives C1 → C2 as needed. Therefore U is an atlas. To show that it is a refinement of U0 we need to prove that the chart C = (U, M, G, φ) in U is compatible with chart C10 = (U10 , M10 , G01 , φ01 ) in U0 whenever U ⊂ U10 . For this purpose let C20 = (U20 , M20 , G02 , φ02 ) be the chart in U0 that was associated to U during the construction of the atlas U. Then there 0 0 is a chart C12 = (U10 ∩ U20 , M12 , G012 , φ012 ) in U0 that lies over U10 ∩ U20 so we have a diagram of inclusions and compatibilities, C20

6

6

?

U

0 C12

-

U10 ∩ U20 -

U20

-

U10

?

C

C10

0 Lemma B.2 applied to U ⊂ U10 ∩ U20 ⊂ U20 implies that C → C12 → C10 as claimed.

B.4. STRATIFICATION

141

Definition B.3.1. An R-orbifold structure on X is an equivalence class of R-orbifold atlases, two being equivalent if they have a common refinement. The orbifold is orientable (resp. complex) if, for each chart (U, M, G, φ) the manifold M is orientable (resp. complex) and the action of G on M is orientation preserving (resp. holomorphic). The orbifold is subanalytic if X is a subanalytic set and for each chart (U, M, G, φ) the manifold M is subanalytic, the group G acts subanalytically on M , and the mapping φ : M/G → X is subanalytic. For brevity, if the ring R is understood, we often call R-orbifold simply an orbifold. Notice that any R-orbifold is automatically a Q-orbifold, though not conversely. B.4. Stratification Let (U, M, G, φ) be a chart on X. Decompose M into strata according to the isomorphism type of the isotropy groups, that is, if H is a finite group, set MH = {m ∈ M | Gm ∼ = H} . It follows from (A) in §B.1 that MH is a disjoint union of smoothly embedded submanifolds of M. (Its connected components may have varying dimensions.) The group G preserves MH so if m ∈ MH then φ−1 (φ(m)) ⊂ MH . The projection MH → MH /G is a local diffeomorphism so the quotient MH /G is also a union of smooth manifolds. The open dense subset on which G acts freely is M 0 = M{1} . If f : (U, M, G, φ) → (U 0 , M 0 , G0 , φ0 ) is an embedding of charts then G0f (m) = λ(Gm ) for all m ∈ M (where λ : G → G0 is the injection from Lemma B.1). Hence f (MH ) ⊂ MH0 and φ(MH ) ⊂ φ0 (MH0 ) so we may define [ XH = φ(MH ) C

where the union is taken over all charts C in an atlas. Then each connected component of XH is a smooth manifold, topologically embedded in X; the sets XH and XH 0 are disjoint if S H 6= H 0 , and the subset X 0 = C φ(M 0 ) is open and dense in X. Proposition B.4. There exists an embedding of X into Euclidean space so that the decomposition of X into connected components of the various XH forms a (locally finite) Whitney stratification of X. The topological space X can be triangulated so that the closure of each stratum becomes a closed subcomplex. Proof. The quotient of a finite dimensional real vector space V under the action of a finite group can be embedded as a semi-analytic subset of Euclidean space so that the decomposition into strata VH satisfies the Whitney conditions. Every Whitney stratified subset of a manifold can be triangulated so that the closure of each stratum is a subcomplex. These two facts can be used to prove the Proposition. Details for the embedding results may be found in [Bi], [SL, §6], [Sc] [PS], [Ma2], [CuS]. Details for the triangulation results may be found in [Mo2, §1.2] §[Ya], [Go].

142


B.5. Sheaves and cohomology In the next few paragraphs we recall the definition of the cohomology of an R-orbifold, and [Sa] the complex of differential forms that may be used to compute it if R = R or R = C. In general, the cohomology (in the orbifold sense) of an orbifold X differs from the (singular) cohomology of the underlying topological space X. But if the coefficient ring is the rational or real numbers, then these coincide. Suppose a finite group G acts on a smooth manifold M with orbit space π : M → M/G. An G-equivariant sheaf, or G-sheaf (of R-modules) F on M is a sheaf together with an isomorphism φg : g ∗ (F) → F for each g ∈ G, such that φg ◦ g ∗ (φh ) = φhg for all g, h ∈ G. The category of G-sheaves is abelian and it has enough injectives. An equivariant section s : M → F is a section such that φg (s(g · m)) = s(m) for all m ∈ M and g ∈ G. Let ΓG (M, F) denote the abelian group of equivariant sections. The equivariant cohomology is the right derived functor, HGi = Ri ΓG . Thus, if F is a G-sheaf on M then HGi (M, F) is obtained as the cohomology of the complex of global invariant sections of any resolution F → I0 → I1 → · · · by injective G-sheaves. Let EG be a universal space for G with corresponding classifying space BG = EG/G. An equivariant sheaf F on M pulls up to an equivariant sheaf on EG × M and it passes to a sheaf, b on the Borel construction EG ×G M. There is a natural isomorphism which we denote by F, b HGi (M, F) ∼ = H i (EG ×G M, F). Suppose A is a module over a ring in which |G| is invertible and suppose G acts linearly on A. This action determines a locally constant sheaf (or local system) A = EG ×G A on BG. The resulting cohomology is the group cohomology ([W] Prop. 6.1.10), ( AG for i = 0 i i i (B.5.1) H (G, A) = HG ({point}, A) ∼ = H (BG, A) = 0 otherwise where AG denotes the submodule of invariants in A. For any G-sheaf F on M the group G acts on the push-forward π∗ (F) and we let F = π∗ (F)G be the sheaf of invariants. It is the sheafification of the presheaf whose sections over an open set U ⊂ M/G are ΓG (π −1 (U ), F). If y ∈ M and if Gy denotes the isotropy group at y then G there is a natural identification, Fy y ∼ = Fπ(y) between the Gy -invariants in the stalk Fy and the stalk of F at π(y) ∈ M/G. If G acts freely on M then F ∼ = H i (X/G, F). But if G acts with = π ∗ (F) and HGi (X, F) ∼ nontrivial isotropy on X then HGi (X, F) is usually non-zero (but torsion) for infinitely many values of i. However if |G| is invertible in the coefficient ring then we regain an isomorphism between the two cohomology groups:

B.5. SHEAVES AND COHOMOLOGY

143

Proposition B.5. Suppose a finite group G acts on a smooth manifold M. Let F be a sheaf on M of modules over a ring in which |G| is invertible. Then there is a natural isomorphism H i (M/G, F) ∼ = HGi (M, F). Proof. Let y ∈ M. The mapping q : EG ×G M → M/G has for its fiber over the point π(y) the classifying space EG/Gy = BGy of the stabilizer group Gy . The restriction of the b to this fiber is the locally trivial sheaf corresponding to the representation of Gy on the sheaf F b may therefore be identified with H b (Gy , Fy ). So the Leray-Serre stalk Fy . The stalk of Rb q∗ (F) spectral sequence for the map q has, as its E2 page, b =⇒ H a+b (M, F). E2a,b = H a (M/G, Rb q∗ (F)) G By equation (B.5.1) this cohomology sheaf vanishes for b 6= 0, while H 0 (Gy , Fy ) ∼ = Fy is the vector space of invariants, Fy Gy . So the natural morphism F → R0 q∗ (F) is an isomorphism of sheaves. Let X be a locally compact topological space with an R-orbifold atlas U. Recall from [MP] for example, that a sheaf F on the R-orbifold X is a choice, for each chart (U, M, G, φ), of a G-sheaf FU of R-modules on M, and an isomorphism ψf : f ∗ (FU0 ) → FU ) of G-sheaves of R-modules whenever f : (U, M, G, φ) → (U 0 , M 0 , G0 , φ0 ) is an embedding of charts. The morphisms ψf are required to be compatible: ψf 0 f = ψf 0 ψf if f, f 0 are composable morphisms. A section s of the sheaf F is a choice of invariant section sU ∈ ΓG (φ−1 (U ), FU ) in each chart (U, M, G, φ) which are compatible: ψf ◦ f ∗ (s0 ) = s for each morphism f : (U, M, G, φ) → (U 0 , M 0 , G0 , φ0 ) and section s0 ∈ ΓG0 (φ−1 (U 0 ), FU0 ), s ∈ ΓG (φ−1 (U ), FU ). In other words, the sections Γ(X, F) (in this R-orbifold sense) are precisely the sections Γ(X, F) (in the topological sense) of the sheaf F which is obtained from the presheaf of invariant sections in each chart. The category of sheaves (of abelian groups) on the R-orbifold X is abelian and it has enough injectives. The cohomology H i (X, F) is defined to be the right derived functor Ri Γ(X, F). It may be nonzero for infinitely many values of i. If we start with a sheaf F0 of R-modules on X (in the topological sense) then it pulls up to a G-equivariant sheaf in each chart (U, M, G, φ) so it gives a sheaf F in the R-orbifold sense, on which each stabilizer group Gy acts trivially. Therefore the resulting sheaf F on X coincides with the original sheaf F0 . Therefore we obtain, Proposition B.6. Let F0 be a sheaf of R-modules on the topological space X. Let F denote the resulting sheaf in the R-orbifold sense. Then the isomorphism F ∼ = F0 induces an isomorphism ∗ H ∗ (X, F0 ) ∼ H (X, F) between the (singular) cohomology of (X, F0 ) and the cohomology in = the sense of R-orbifolds of (X, F).

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B.6. Differential forms For the remainder of this chapter, we take R = Q and we drop it from our notation. A local system E of real vector spaces on an orbifold X is a sheaf of real vector spaces such that each EU is a local system. Let E be a local system of real vector spaces on the orbifold X. The sheaf of differential p-forms on the orbifold X is the sheaf which, in each chart (U, M, G, φ) is given by the sheaf ΩpG (M, E). It is easy to check that this collection satisfies the required compatibility conditions for a sheaf on an orbifold. Thus, a differential p-form with coefficients in E on X is a choice, for each chart (U, M, G, φ), of a smooth differential form ωU ∈ Ωp (M, EU ) which is G-invariant, such that f ∗ (ωU 0 ) = ωU whenever f : (U, M, G, φ) → (U 0 , M 0 , G0 , φ0 ) is a morphism of charts. If ω, η are G-invariant differential forms then so is dω and ω ∧ η so we obtain exterior differentials and products on this complex of differential forms. The differential p forms Ωp (X, E) in fact form a fine sheaf (in the orbifold sense), Ωp (X, E) on X. It follows that the de Rham theorem holds: Proposition B.7. Let E be a local system of real vector spaces on the orbifold X. Then the cohomology of the complex of smooth differential forms Ω• (X, E) is canonically isomorphic to the cohomology H i (X, E). According to the remarks in §B.5 the differential p forms on the orbifold X are precisely the global sections (in the usual sense) of the (topological) sheaf (B.6.1)

p

Ω (X, E)

of invariant sections of Ωp (X, E). Let us examine this sheaf in more detail. Let X 0 denote the part of X over which the isotropy groups are trivial. Then the restriction p Ω (X, E)|X 0 is canonically isomorphic to the usual sheaf of differential forms Ωp (X 0 , E). Thus, a differential form ω on the orbifold X is simply a smooth differential form on X 0 which, near the singular points of X, satisfies an equivariance condition: (EQ) if (U, M, G, φ) is a chart, then the pullback of ω to M 0 extends (uniquely) to a smooth, G-invariant differential form on M. Since the sheaf (B.6.1) is fine, we conclude, in analogy with Proposition B.5 that the cohomology of the complex of differential forms is naturally isomorphic to H ∗ (X, E). In summary, we have: Proposition B.8 ([Sa]). Let E be a local system (in the topological sense) of real vector spaces on a locally compact Hausdorff space X. Assume that X is endowed with an orbifold structure. Let Ω• (X, E) be the complex of smooth differential forms on X 0 (with coefficients in E) which satisfy the above equivariance condition (EQ) near the singularities of X. Denote • ∗ by HdR (X, E) the cohomology of this complex. Then the inclusion of sheaves E → Ω (X, E) is ∗ a fine resolution of E and it induces an isomorphism HdR (X, E) ∼ = H ∗ (X, E) between the de ∗ Rham cohomology and the singular cohomology, and an isomorphism HdR,c (X, E) ∼ = Hc∗ (X, E)

B.7. INTEGRATION OVER CHAINS

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between the compactly supported de Rham cohomology and the singular cohomology with compact supports.

B.7. Integration over chains It is possible to define the simplicial or singular homology of an orbifold. Just as in the preceding sections, this may differ from the singular homology of the underlying topological space. But if the coefficient ring is the rational numbers or the real numbers, then these homology groups coincide. In this section we remark that differential forms on an orbifold can be integrated over subanalytic chains. Proposition B.9. Suppose X is a subanalytic set with a subanalytic orbifold structure. Let ω ∈ Ωp (X, R) be a smooth differential form (in the sense of the preceding section). Let Z ⊂ X be a subanalytic (Borel-Moore) p-dimensional chain on X. Suppose either (a) Z is compact or (b) the differential form ω has compact support. Then the integral Z ω