Hecke Operators on Jacobi Forms of Lattice Index and the Relation to

Hecke Operators on Jacobi Forms of Lattice Index and the Relation to Elliptic Modular Forms

DISSERTATION zur Erlangung des Grades eines Doktors der Naturwissenschaften vorgelegt von Ali Ajouz

eingereicht bei der Naturwissenschaftlich-Technischen Fakultät der Universität Siegen Siegen 2015

This page is intentionally left blank.

Author’s declaration

Ich erkläre hiermit an Eides statt, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer, nicht angegebener Hilfs mittel angefertigt habe. Die aus anderen Quellen direkt oder indirekt übernommenen Daten und Konzepte sind unter Angabe der Quelle gekennzeichnet. Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde vorgelegt. Es wurden keine Dienste eines Promotionsvermittlung sinstituts oder einer ähnlichen Organisation in Anspruch genommen.

SIGNED: ................................... DATE: ....................................

Erstgutachter: Prof. Dr. Nils-Peter Skoruppa Zweigutachter: Prof. Dr. Lynne Walling Tag der mündlichen Prüfung: 10-07-2015 i


Abstract Jacobi forms of lattice index, whose theory can be viewed as extension of the theory of classical Jacobi forms, play an important role in various theories, like the theory of orthogonal modular forms or the theory of vertex operator algebras. Every Jacobi form of lattice index has a theta expansion which implies, for index of odd rank, a connection to half integral weight modular forms and then via Shimura lifting to modular forms of integral weight, and implies a direct connection to modular forms of integral weight if the rank is even. The aim of this thesis is to develop a Hecke theory for Jacobi forms of lattice index extending the Hecke theory for the classical Jacobi forms, and to study how the indicated relations to elliptic modular forms behave under Hecke operators. After defining Hecke operators as double coset operators, we determine their action on the Fourier coefficients of Jacobi forms, and we determine the multiplicative relations satisfied by the Hecke operators, i.e. we study the structural constants of the algebra generated by the Hecke operators. As a consequence we show that the vector space of Jacobi forms of lattice index has a basis consisting of simultaneous eigenforms for our Hecke operators, and we discover the precise relation between our Hecke algebras and the Hecke algebras for modular forms of integral weight. The latter supports the expectation that there exist equivariant isomorphisms between spaces of Jacobi forms of lattice index and spaces of integral weight modular forms. We make this precise and prove the existence of such liftings in certain cases. Moreover, we give further evidence for the existence of such liftings in general by studying numerical examples.

iii


Zusammenfassung Jacobiformen von Gitterindex, deren Theorie als Erweiterung der Theorie klassischer Jacobiformen betrachtet werden kann, spielen in unterschiedlichen Theorien, wie der Theorie der Modulformen auf orthogonalen Gruppen oder der Theorie der Vertexoperatoralgebren, eine bedeutende Rolle. Jede Jacobiform mit Gitterindex hat eine Theta-Entwicklung, die bei ungeradem Index eine Verbindung zu Modulformen von halbganzen Gewichten herstellt und daher via Shimura-Liftung eine Beziehung zu Modulformen ganzen Gewichts impliziert, und bei geradem Index eine direkte Verbindung zu Moldulformen ganzzahligen Gewicht suggeriert. Das Ziel dieser Dissertation ist, eine Hecke-Theorie für Jacobiformen mit Gitterindex zu entwickeln, indem die Hecke-Theorie für die klassischen Jacobiformen erweitert wird, und zu untersuchen, wie sich die angedeuteten Beziehungen zu elliptischen Moldulformen unter Hecke-Operatoren verhalten. Nachdem die Hecke-Operatoren als Doppelnebenklassen-Operatoren definiert werden, wird deren Wirkung auf die Fourier-Koeffizienten der Jacobiformen und die multiplikativen Relationen, welche von Hecke-Operatoren erfüllt werden, untersucht, indem z.B. die Struktur-Konstanten der Algebra berechnet werden, die von den Hecke-Operatoren erzeugt werden. Daraufhin wird gezeigt, dass der Vektorraum der Jacobiformen mit Gitterindex über eine Basis von simultanen Eigenformen für die HeckeOperatoren verfügt und es wird die präzise Beziehung zwischen der HeckeAlgebra und der Hecke-Algebra für Moldulformen ganzzahligen Gewichts aufgezeigt. Letztgenannte Beziehung stützt die Erwartung, dass äquivariante Isomorphismen zwischen den Räumen der Jacobiformen mit Gitterindex und Räumen Moldulformen ganzzahligen Gewichts existieren. Es erfolgt eine Präzisierung und der Beweis, dass solche Liftungen in bestimmten Fällen v

vi existieren. Zudem geben wir weitere Argumente für die Existenz solcher Liftungen im allgemeunen Fall durch das Studium numerischer Beispiele.

Acknowledgments I would like to thank my supervisor, Prof. Dr. Nils-Peter Skoruppa, for his continued guidance and support throughout the duration of my Ph.D. He taught me a lot with his mathematical knowledge and skills, and gave me many valuable suggestions as well as kind help in the elucidation of difficulties. I also want to thank Prof. Dr. Lynne Walling, Prof. Dr. Volker Michel, and Prof. Dr. Gerd Mockenhaupt for agreeing to be in the jury.

vii


To my wife, Karam.

ix


Table of Contents Page Introduction and Statement of Results 1 Preliminaries

xiii 1

1.1 Elliptic Modular Forms . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2 Lattices and Finite Quadratic Modules . . . . . . . . . . . . . .

10

2 Hecke Theory of Jacobi Forms of Lattice Index

15

2.1 Some Lemmas on Lattices and Gauss Sums . . . . . . . . . . .

15

2.2 Jacobi Groups of Integral Lattice . . . . . . . . . . . . . . . . . .

27

2.3 Theta Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.4 Jacobi Forms of Lattice Index . . . . . . . . . . . . . . . . . . . .

34

2.5 Hecke Operators on the Space of Jacobi Forms . . . . . . . . .

38

2.6 The Action of Hecke Operators on Fourier Coefficients . . . . .

42

2.7 Hecke Operators and Euler Products . . . . . . . . . . . . . . . .

52

3 Basis of Simultaneous Eigenforms

65

3.1 The Action of the Orthogonal Group . . . . . . . . . . . . . . . .

65

3.2 Jacobi Cusp Forms and Hecke Operators . . . . . . . . . . . . .

67

3.3 Jacobi-Eisenstein Series and Hecke Operators . . . . . . . . . .

73

4 Lifting to Elliptic Modular Forms

83

4.1 Lifting via Shimura Correspondence for Half Integral Weight

83

4.2 Lifting via Stable Isomorphisms between Lattices . . . . . . .

87

5 Lifting from Elliptic Modular Forms xi

91

xii

Table of Contents 5.1 Jacobi Forms of Prime Discriminant and Hecke Operators . . 92 5.2 Operators on the Vector-Valued Components . . . . . . . . . . . 100

6 Appendix 105 6.1 Examples Using the Method of Theta Blocks . . . . . . . . . . . 105 List of Notations

115

Bibliography

119

Introduction and Statement of Results Jacobi forms of scalar index are a mixture of modular forms and elliptic functions. They have an arithmetic theory very analogous to the usual theory of modular forms; this began with Maass’s proof of the Saito-Kurokawa conjecture and was developed systematically by Martin Eichler and Don Zagier in the monograph “The Theory of Jacobi Forms” [EZ85]. Because they have two variables, Jacobi forms have associated to them two characteristic integers: the weight, which describes the transformation properties of the form with respect to the modular group SL2 (Z), and the index, which describes the transformation properties in the elliptic variable. We shall use Jk,m for the space of Jacobi forms on SL2 (Z) of weight k and index m. The basic features of the theory of Jacobi forms of scalar index are: • Jk,m is finite-dimensional. • There exists a Hecke theory for Jacobi forms of scalar index, i.e., for each positive number ` relative prime to m, there exists a natural Hecke operator T(`) on Jk,m , and the space Jk,m has a basis consisting of simultaneous eigenforms with respect to all T(`). • There exists a natural notion of Jacobi Eisenstein series and Jacobi cusp forms. • There exists a Petersson scalar product < φ, ψ > on the space of Jacobi cusp forms. xiii

xiv

Introduction and Statement of Results

• A perfect correspondence with elliptic modular forms of integer weight: the result of the paper of Skoruppa and Zagier [SZ88] is a relation between Jacobi forms of weight k and index m on the one hand and ordinary elliptic modular forms of weight 2k − 2 and level m. Indeed, they proved that Jk,m is isomorphic as module over the Hecke algebra to a natural subspace of the space M2k−2 (m) of modular forms of weight 2k − 2 on Γ0 (m). More precisely, the lifting from Jk,m to M2k−2 (m) is constructed as follows: Let φ ∈ Jk,m , then φ has a Fourier expansion of the form φ(τ, z) =

X

C φ (d, r)q

r2 −d 4m

ζr .

d ≤0,r d ≡ r 2 mod 4m

Let D < 0 be a fundamental discriminant and r be an integer with D ≡ r mod 4m, then S D,r (τ) : φ →

∞ ‡ X `=0

coefficient of q

r 2 −D 4m

·

ζ in φ | T(`) e2π i`τ r

(1)

maps Jk,m to a certain subspace of M2k−2 (m), and the maps S D,r commute with the action of the Hecke operators. (This is not quite true, since one would need here to define T(`) for gcd(`, m) > 1 which is not a part of the common theory.) Jacobi forms of lattice index, which can be viewed as a generalization of the classical ones if one consider 2m as a Gram matrix of the lattice (Z, (x, y) → 2mx y), have been studied in [Zie89], [BK93], [CG11], [Sko07], [Boy11], [Bri04], and other authors. Recall that an integral lattice over Z is a pair L = (L, β), where L is a free Z-module of a finite rank, and β : L × L → Z is a non-degenerate symmetric and Z-bilinear. We shall use L# , lev(L), rk(L) and det(L) to denote the dual, the level, the rank, and the determinant of the lattice, respectively. Also, for each r ∈ L# , we shall use Nr for the smallest positive integer such that Nr · r ∈ L. Suppose that L is even and positive definite, i.e., such that β(x, x) is even and strictly positive unless x = 0. A Jacobi form φ(τ, z) of weight k ∈ N and index L = (L, β) is a holomorphic function of a variable τ in the complex upper

xv half plane H and a variable z ∈ L ⊗Z C that satisfies certain transformation laws (see Definition 2.4.1) and has a Fourier expansion of the form: φ(τ, z) =

X

¡ ¢ C φ (D, r) e (β(r) − D)τ + β(r, z) ,

D ≤0,r ∈L# D ≡β(r) mod Z

The starting point for the current thesis is the expectation that there should be lifting for Jacobi forms of lattice index to elliptic modular forms. However, there is currently not even a Hecke theory for these forms. In this thesis, I shall focus on Hecke operators acting on the vector space of Jacobi forms of lattice index. I shall use these operators and the arithmetic properties of Jacobi forms to give examples of correspondences with elliptic modular forms. More precisely, we will discuss correspondences supporting the mentioned expectation of the following sense: Jacobi forms of weight k correspondence Elliptic modular forms of weight ←−−−−−−−−−−→ k 1 := 2k − 1 − rk(L) and index L if rk(L)≡1 mod 2 Jacobi forms of weight k correspondence ←−−−−−−−−−−→ and index L if rk(L)≡0 mod 2

Elliptic modular forms of weight k 2 := k −

rk(L) 2

The main results of this thesis can be subdivided as follows: 1. A Hecke theory for Jacobi forms of lattice index (chapter 2 and chapter 3): In this part we develop a systematic Hecke theory for Jacobi forms of lattice index along the lines of Hecke’s theory of modular forms. 1.a. Explicit description of the action of Hecke operators (section 2.5, section 2.6 and section 2.7: Let L = (L, β) be a positive definite even lattice over Z. We set 8 rk(L) 0

2. If rk(L) is even, we set k2 := k −

rk(L) 2

X

T(`) :=

d,s>0 sd 2 |`, s square-free

d k1 −2 T0 ( d`2 ).

and χL (s)(sd 2 )k2 −2 T0 ( sd` 2 ).

The operators T0 (`) and T(`) are well-defined and map Jacobi forms of weight k and index L to Jacobi forms of the same weight and same rank. For rk(L) = 1 the operator T(`) equals the Hecke operator for classical Jacobi forms as e.g. in [SZ88] and [EZ85]. Next, we describe the action of these operators on Jacobi forms in terms of Fourier coefficients and give their commutation relations. Theorem (see Theorem 2.6.1). Let L = (L, β) be a positive definite even lattice over Z of odd rank. Let φ be a Jacobi form of weight k and index L = (L, β) with Fourier expansion X

φ(τ, z) =

¡ ¢ C φ (D, r) e (β(r) − D)τ + β(r, z) .

D ≤0,r ∈L# D ≡β(r) mod Z

Let ` ∈ N with gcd(`, lev(L)) = 1, and let ¡

¢ T(`)φ (τ, z) =

X D ≤0,r ∈L# D ≡β(r) mod Z

¡ ¢ C T(`)φ (D, r) e (β(r) − D)τ + β(r, z) .

xvii Then one has C T(`)φ (D, r) =

X a

a k1 /2−1 %(D, a)C φ

¡ `2

a2

¢ D, à0 r .

The sum is over those a | `2 such that a2 | `2 lev(L)D , and a0 is an integer such that aa0 ≡ 1 (mod lev(L)). Moreover, %(D, a) equals f · χL (D/ f 2 , a/ f 2 ) if gcd(lev(L)D, a) = f 2 with f ∈ N, and it equals 0 if gcd(lev(L)D, a) is not a perfect square. Theorem (see Theorem 2.6.3). In the same notations as in the preceding theorem assume the rank of L is even. Then one has X

C T(`)φ (D, r) =

a k2 −1 χL (a)C φ

a|`2 ,lev(L)D

¡ `2

a2

¢ D, à0 r .

Using this explicit description of the action of the operator T(`) on Jacobi forms of lattice index in terms of Fourier expansion, we can determine their multiplicative properties. Theorem (see Theorem 2.7.11 and Theorem 2.7.4). Let `1 , `2 ∈ N such that `1 , `2 coprime to lev(L). We have the following multiplicative relation: 8 X · ‡ `1 ` 2 k 1 −1 > > d T > d2 < d |`1 ,`2 ‡ · T(`1 ) · T(`2 ) = X `1 ` 2 k 2 −1 > > χL (d)d T d > : 2 2

If rk(L) is odd, If rk(L) is even.

d |`1 ,`2

As an easy consequence of these theorems we obtain an important insight into the arithmetic proprieties of Jacobi eigenforms of lattice index. More precisely, let L = (L, β) again be an even positive definite lattice over Z, and consider a Jacobi form φ of weight k and index L = (L, β) with Fourier coefficients Cφ (D, r), which is an eigenfunction of all T(`) with gcd(`, lev(L)) = 1, say, T(`)φ = λ(`)φ. Theorem (see Theorem 2.7.17 and Theorem 2.7.9). Let r ∈ L# and lev(r) is the smallest positive integer such that lev(r)β(r) ∈ Z, and D ≤ 0 such that D ≡ β(r) mod Z and lev(r)D is a square-free integer. Then

xviii


1. if rk(L) is odd, one has Y ‡

1 − χL (D, p)p k1 /2−1−s

·−1

X `∈N gcd(`,lev(L))=1

p-lev(L)

= C φ (D, r)

C φ (`2 D, ` r)`−s

·−1 Y ‡ 1 − λ(p)p−s + p k1 −1−2s , p-lev(L)

2. if rk(L) is even, one has ‡

X

χL (`)`k2 −1−s

·

X `≥1 gcd(`,lev(L))=1

`|lev(L)D gcd(`,lev(L))=1

= C φ (D, r)

¡ ¢ C φ `2 D, ` r `−s

1 + χL (p)p k2 −1−s ¢ . k 2 −1 χ (p) p− s + p2(k 2 −1− s) L p-lev(L) 1 − λ(p) − p Y

¡

The products are over all primes p not dividing lev(L). Note that the right-hand sides of these identities are nothing else than P C φ (D, r) L(s, φ), where we set L(s, φ) = ` λ(`)`−s (all sums are taken over ` coprime to lev(L)). If the φ in the first identity lifts to an elliptic modular form f of weight k1 = 2k − 1 − rk(L) then L(s, φ) should be (up to a finite number of Euler factors) the L-series of f . We observe that the right-hand side of the first identity has indeed the right shape. The right-hand of the second identity looks, at the first glance, slightly more complicated. However, if we think of an elliptic modular form of weight rk(L) k 2 = k − 2 with nebentypus, say, χL ξ, and with Hecke eigenvalues γ(`), then P 2 −s (again taken over all ` coprime to lev(L)) equals L(s, φ) if we ` ξ(`)γ(` )` replace λ(p) with ξ(p)γ(p2 ). This suggests, for each ξ and suitable levels m, the existence of maps from M k2 (m, χL ξ) to Jk,L such that T(`2 ) on the left corresponds to ξ(`) T(`) on the Jacobi form side. We shall construct in this thesis examples for such maps. 1.b. Basis of simultaneous Hecke eigenforms (chapter 3): It is easy rk(L) to see (for weights greater than 2 + 2) that the space of Jacobi forms is a direct sum of the subspaces spanned by Eisenstein series and cusp forms,

xix which will be introduced in section 3.3. We shall prove that our Hecke operators T(`) (` coprime to lev(L)) leave these subspaces invariant, and are Hermitian with respect to a suitably defined Petersson scalar product on the subspace of cusp forms. Also, we shall prove that the orthogonal group O(D L ) of the discriminant module D L of L acts naturally on the spaces of Jacobi forms with index L, and that the corresponding operators W(α) (α ∈ O(D L )) are Hermitian too and commute with Hecke operators. By the spectral theory of such operators, one has the following theorem. Theorem (see Theorem 3.2.13). The space of Jacobi cusp forms S k,L has a basis of simultaneous eigenforms for all operators T(`) (gcd(`, lev(L)) = 1) and for all operators W(α)(α ∈ O(D L )). Let D L = (L# /L, β) be the associated discriminant module with the lattice fl ' “ L (see Definition 1.2.10). We set Iso(D L ) := x ∈ L# /L fl β(x) ∈ Z . Let k be a rk(L) positive integer with k > 2 + 2. For each r ∈ Iso(D L ), we define a Jacobi Eisenstein series of weight k and index L, in terms of Jacobi theta series ϑL,r which are naturally associated to L (see section 2.3), as follows: E k,L,r := 12

X A ∈SL2 (Z)∞ \ SL2 (Z)

fl ϑL,r+L flk,L A.

Eis We shall use Jk,L for the subspace in Jk,L that spanned by all Eisenstein series E k,L,r .

Theorem (see Theorem 3.3.18). The series E k,L,x,ξ :=

X d ∈Z× N

ξ(d)E k,L,dx , x

where x runs through a set of representatives for the orbits in the orbit space – Iso(D L ) Z× , and ξ runs through all primitive Dirichlet characters mod F lev(L) Eis with F | N x such that ξ(−1) = (−1)k , form a basis of Hecke eigenforms of Jk,L . More precisely: ξ,ξ (`)E k,L,x,ξ 1 −1 ξ,χL T(`)E k,L,x,ξ = ξ(`) σk −1 (`2 )E k,L,x,ξ 2

T(`)E k,L,x,ξ = σk

if rk(L) is odd, if rk(L) is even,

xx


for all ` ∈ N such that gcd(`, lev(L)) = 1. Here, k1 = 2k − 1 − rk(L), k2 = k − and for any two Dirichlet characters ξ and χ we use ξ,χ

σk (`) :=

rk(L) 2 ,

X ¡`¢ ξ d χ(d)d k . d |`

Note that this theorem supports the interpretations of the last theorem in 1.a, which we proposed at the end of that section. See Remark 3.3.19 for a detailed discussion. 2. Lifting to Elliptic Modular Forms (chapter 4): A natural question in view of the preceding theorems is whether we can construct in general a relation between Jacobi forms of lattice index of odd rank and elliptic modular forms which extends the work of Skoruppa and Zagier [SZ88] for the case of scalar index (see Equation (1)). For obtaining at least partially such lifts we apply two methods: 2.a. Lifting via Shimura correspondence for half integral weight (section 4.1): We know that every Jacobi form of lattice index has a theta expansion which implies, for odd rank index, a connection to half integral weight modular forms. We can try to use the Shimura correspondence for half integral weight forms to map Jacobi forms of lattice index to modular forms of integral weight. For this let L be of odd rank, and k be a positive integer such that 2k − rk(L) − 1 ≥ 2. Let x ∈ L# and D ∈ Q such that D ≡ β(x) mod Z. Assuming that N x 2 D is a square free negative integer, where N x is the smallest positive integer such that N x x ∈ L. For a Jacobi cusp form φ ∈ S k,L , set S D,x (φ) =

∞ ‡X X

a k−d

rk(L) 2 e−1

`=1 a|` ξ S D,x (φ) =

‡

(−1)k N x 2

·

X

χL (D, a)C φ

¢· ` D, x e (`τ) , a a2

¡ `2

¡ ¢ ξ(s) S D,sx (φ) ⊗ ξ ,

s mod N x D ≡β(sx) mod Z

where ξ(·) = , and S D,sx (φ) ⊗ ξ denotes the function obtained from · S D,sx (φ) by multiplying its n-th Fourier coefficient by ξ(n).

xxi ξ Theorem (See Theorem 4.1.4). S D,x (φ) is an elliptic modular form of weight 2 2k − 1 − rk(L) on Γ0 (lev(L)N x /2), and, in fact, a cusp form if 2k − 1 − rk(L) > 2. Moreover, one has ξ

ξ

(2)

T(p)S D,x (φ) = ξ(p)S D,x (T(p)φ)

for all primes p with gcd(p, lev(L)) = 1 Indeed, this theorem supports the interpretations of the last theorem in 1.a. Namely, if S D,x takes Jk,L to elliptic modular forms on Γ0 (lev(L)/2), then ξ the twisted version S D,x takes Jk,L to elliptic modular forms on Γ0 (N x 2 lev(L)/2), which we proved indeed. If in addition, S D,x commutes with Hecke operators, ξ we deduce for S D,x the relation (2), which is again what we proved. 2.b. Lifting via stable isomorphisms between lattices (section 4.2): Two even lattices L 1 = (L 1 , β1 ) and L 2 = (L 2 , β2 ), are said to be stably isomorphic if and only if there exists even unimodular lattices U1 ,U2 such that L 1 ⊕ U1 ∼ = L 2 ⊕ U2 . If L 1 , L 2 are stably isomorphic, then one can show that there is an isomorphism I L 2 ,L 1 : Jk+drk(L 2 )/2e,L 2 −→ Jk+drk(L 1 )/2e,L 1

(see Theorem 4.2.2 and Theorem 4.2.4). Note that stably isomorphic lattices have the same level and determinant. We shall show that this isomorphism commutes with the Hecke operators T(`) (see Theorem 4.2.4). Also, we obtain the following important result: Theorem (see Theorem 4.2.5). If the lattice L = (L, β) is stably isomorphic to ¡ ¢ the lattice Z, (x, y) 7→ det(L)x y , then there is a Hecke-equivariant isomorphism ¡ ¢− ∼ = Jk,L −→ M2k−1−rk(L) lev(L)/4 ,

where M2k−1−rk(L) lev(L)/4 is the Certain Space inside M2k−1−rk(L) lev(L)/4 which was introduced in [SZ88, 3], and where the "−" sign denotes the sub¡ ¢ space of all f ∈ M2k−1−rk(L) lev(L)/4 such that f | Wlev(L)/4 = −(−1)k/2 f . ¡

¢

¡

¢

xxii


3. Lifting from elliptic modular forms (section 5.1): We saw above that, for even rank index, we expect liftings from spaces M k2 (m, χL ξ) to Jk,L in the sense that T(`2 ) corresponds to ξ(`) T(`). We shall study this in the case that the determinant of the lattice L = (L, β) is an odd prime. In this case we shall explicitly construct a lifting ¡ ¢ S : M k− rk(L) lev(L), χL → Jk,L 2

(see Theorem 5.1.2). The map S will turn out to be surjective. However, in general, it will not be injective, but we shall see that we can restrict S to a ¢ ¡ natural subspace of M k− rk(L) lev(L), χL , invariant under all T(`2 ), so that we 2 obtain an isomorphism. To complete the picture, we will discuss briefly the relation between the Hecke operators on Jacobi forms and the Hecke operators on the vectorvalued components (see section 5.2). We also construct some numerical examples using the method of theta blocks (see section 6.1).

Chapter 1 Preliminaries We denote by N, Z, Q, R, and C the set of all strictly positive natural numbers, the ring of rational integers, the rational number field, the real number field, and the complex number field. Also we put S1 = { z ∈ C, | z| = 1}. For a prime number p, Q p denotes the field of p-adic numbers, while Z p denotes the subring of p-adic integers. This should not to be confused with the ring of integers modulo p which we denote by Z p := Z/pZ. For complex valued function f , f denotes its complex conjugate. We set p i = −1. We shall always take the branch of the square root having argument p 1 in (−π/2, π/2]. Thus for z ∈ C\{0}, we define z = z 2 so that −π/2 < arg(z1/2 ) ≤ π/2, p p and we set 0 = 0. The function z on the complex plane takes positive reals to positive reals, complex numbers in the upper half-plane to the first quadrant, and complex numbers in the lower half-plane to the fourth p quadrant. Its restriction to C\R≤0 is holomorphic. Further, we put z k/2 = ( z)k for every k ∈ Z. Let n be an integer, with prime factorization u · p 1 e 1 · · · p k e k , where u = ±1 ¡a¢ and the p i are primes. Let a be an integer. The Kronecker symbol is n ‡ · ¡ a ¢ ¡ a ¢ Qk ‡ a · e a i defined to be n = u i=1 p i . For odd p i , the number p i is simply the ¡ ¢ usual Legendre symbol. This leaves the case when p i = 2. We define a2 to be 0 if a is even, 1 if a ≡ ±1 (mod 8), and −1 if a ≡ ±3 (mod 8). The quantity 1

1. Preliminaries ¡a¢ u

is 1 when u = 1. When u = −1, we define it by ‡a· −1

8 0

.

These definitions extend the Legendre symbol for all pair of integers a, n. Let p be a prime number. The p-adic order or p-adic valuation for Z is defined as ord p : Z → N by: 8 2 and b ≡ 0 mod 2 where a¯ ∈ Z such that aa t = 2 and b 6≡ 0 mod 2, and is 0 otherwise.

Proof. Follows from [BEW98, Theorem 1.2.2]. Lemma 2.1.9. Let t = 2v for some positive integer v, and d be an odd integer. One has ‡− t · X ¡ ¢ e t d(x2 + x y + y2 ) = − t . (2.9) 3

x,y mod t

Proof. One has X

e t d(x2 + x y + y2 ) = ¡

¢

x,y mod t

X

e t dx2 ¡

¢ X

x mod t

=

X

e t d(x y + y2 ) ¡

¢

y mod t

¡

¢ 2

e t dx · G(d, dx, t).

x mod t

17

2. Hecke Theory of Jacobi Forms of Lattice Index First, we assmue that t 6= 2. By Lemma 2.1.8, the right-hand side (RHS) of the above equation equals 8 1 ‡ 2 2· ¡− t ¢ ¯ < 2 X t (1 + i) d ²(d)e t −d 4dx ¡ 2¢ RHS = e t dx × :0 x mod t

if x ≡ 0 mod 2, otherwise,

¯ ≡ 1 mod 4t. Thus where d¯ is an integer such that dd

e t d(x2 + x y + y2 ) =

X

¡

x,y mod t

¡

¢

x mod t x≡0 mod 2

‡− t · 1 ²(d) = t 2 (1 + i) d

By replacing

‡ 2 2· ‡− t · ¯ ²(d)e t −d 4dx

1

e t dx2 t 2 (1 + i)

X

¢

x 2

X

d

et

x mod t x≡0 mod 2

¡3

4 dx

2

¢

.

with y we see, using Lemma 2.1.8, that

X

et

‡ −t · ¢ 1 X ¢ 1 1 ¡ 2 2 2 (1 + i) dx = e 3d y = t ²(3d). t 4 2 2

¡3

x mod t x≡0 mod 2

3d

y mod t

Inserting this into the last formula for x,y mod t e t d(x2 + x y + y2 ) , we obtain the claimed result. Now we assume that t = 2. One has P

X

¡

¢

‡−2· X X ¢ ¡ 2¢ ¡ 2¢ . e2 d(x + x y + y ) = e2 dx · G(d, dx, 2) = 2 e2 dx = −2 ¡

2

2

x,y mod 2

x mod 2

x mod 2 x odd

3

The proof is complete. Lemma 2.1.10. Let t = 2v for some positive integer v, and d be an odd integer. One has X e t (dx y) = t. (2.10) x,y mod t

Proof. One has X

e t (dx y) =

x,y mod t

X x,y mod t x≡0 mod t

=

X x(t) x≡0 mod t

as stated in the lemma. 18

e t (dx y) +

X x,y mod t x6≡0 mod t

t+0 = t

e t (dx y)

2.1. Some Lemmas on Lattices and Gauss Sums Definition 2.1.11. For integers `, c ∈ Z, the Ramanujan sum G ` (c) is defined as follows: X G ` (c) = e` (cd). (2.11) d mod ` (d,`)=1

It is known that the Ramanujan sum G ` (c) is multiplicative when considered as a function of ` for a fixed value of c. Definition 2.1.12. For integers `, c ∈ Z where ` is positive and odd, we set F(c, `) =²(`)

X ‡d· d mod ` `

(2.12)

e` (cd) .

Lemma 2.1.13. The function F(c, `) for fixed c is multiplicative in `, i.e., (2.13)

F(c, `1 `2 ) = F(c, `1 )F(c, `2 )

for all c ∈ Z and odd positive integers `1 , `2 such that (`1 , `2 ) = 1. Proof. Let `1 , `2 be two relatively prime integers. Any integer d mod `1 `2 can be written with the Chinese Remainder Theorem as d = b2 `1 + b1 `2 , where b1 runs through integers mod `1 and b2 runs mod `2 . Then

= ²(`1 `2 )

`1 ` 2

b 1 mod `1 b 2 mod `2

= ²(`1 `2 )

b 2 `1 + b 1 `2

b 1 `2

X

= ²(`1 `2 )

b 1 mod `1

= ²(`1 `2 )

`2 `1

¶

`1 `2

¶

¶

`1

b 1 mod `1

e`1 (cb1 ) b1

X b 1 mod `1

= ²(`1 `2 )²(`1 )−1 ²(`2 )−1

`2 `1

`1

¶

¶

e`1 `2 (c(b2 `1 + b1 `2 ))

e`1 `2 (cb1 `2 )

¶

`1

e`1 `2 (cd)

b 2 `1 + b 1 `2

X

X

·

` 1 `2

d mod `1 `2

X

d

‡

X

F(c, `1 `2 ) = ²(`1 `2 )

b 2 mod `2

¶

e`1 (cb1 ) `1 `2

¶

`2

b 2 mod `2 b 2 `1

X

b 2 `1 + b 1 `2

X

`2

¶

e`1 `2 (cb2 `1 )

¶

e`2 (cb2 )

X b 2 mod `2

b2 `2

¶

e`2 (cb2 )

F(c, `1 )F(c, `2 ).

`1 Using the quadratic reciprocity, we see that ²(`1 `2 )²(`1 ) `2 = −1 −1 1. Namely, ²(`1 `2 )²(`1 ) ²(`2 ) equals +1 if ‡`1 ·‡ ≡ `·2 mod 4 and −1 other`2 ` 1 wise, where by quadratic reciprocity equals `1 `2 . It follows F(c, `1 `2 )

−1

` ²(`2 )−1 `21

‡ ·‡ ·

= F(c, `1 )F(c, `2 ).

19

2. Hecke Theory of Jacobi Forms of Lattice Index Lemma 2.1.14 ([BEW98],[And87]). Let p be an odd prime, v ∈ N be an odd positive integer, and c ∈ Z. One has v

F(c, p ) =

1 − c/p v−1 p v− 2 p

¶ δ(p v−1 k c).

(2.14)

Recall that L = (L, β) denotes a positive definite even lattice over Z. Definition 2.1.15. For integers `, s ∈ Z where ` is positive, we set X

SL (s, `) :=

¡

x∈L/`L

(2.15)

e` sβ(x) . ¢

Lemma 2.1.16. Let s ∈ Z, `1 , `2 ∈ NL such that gcd(`1 , `2 ) = 1. One has (2.16)

SL (s, `1 `2 ) = SL (s`2 , `1 )SL (s`1 , `2 ).

Proof. For `1 , `2 ∈ NL with gcd(`1 , `2 ) = 1 we have a Z-module homomorphism ϕ : L/`2 L ⊕ L/`1 L → L/`1 `2 L

given by ϕ(x + `2 L, y + `1 L) = `1 x + `2 y + `1 `2 L. ϕ is surjective: Let z ∈ L. Since gcd(`1 , `2 ) = 1, then there are h, k ∈ Z such that

1 = `1 h + `2 k, so z = `1 hz + `2 kz. Thus ϕ(hz + `1 `2 L, kz + `1 `2 L) = z + `1 `2 L. ϕ is injective: ϕ(x + `2 L, y + `1 L) = 0 + `1 `2 L if and only if `1 x + `2 y ∈ `1 `2 L,

that is `1 x + `2 y = `1 `2 z for some z ∈ L, which implies `2 y = `1 (x − `2 z). Write P P P x = i a i v i , y = i b i v i and z = i c i v i , where {v i | 1 ≤ i ≤ rk(L)} is a basis for L. Then `2 b i = `1 (a i − `2 c i )

(1 ≤ i ≤ rk(L)).

Since gcd(`1 , `2 ) = 1, we have `1 | b i . Thus y = `1 g for some g ∈ L, so `2 y = `1 `2 g ∈ `1 `2 L. In a similar way we find that `1 x ∈ `1 `2 L. Thus SL (s, `1 `2 ) =

X

¡

x∈L/`1 `2 L

=

X

e`1 `2 sβ(x) X

x1 ∈L/`2 L x2 ∈L/`1 L

=

X x1 ∈L/`2 L

¢

e`1 `2 sβ(`1 x1 + `2 x2 )

e`2 s`1 β(x1 ) ¡

¡

X

¢

x2 ∈L/`1 L

=SL (s`1 , `2 )SL (s`2 , `1 )

as stated in the lemma. 20

¢

e`1 s`2 β(x2 ) ¡

¢

2.1. Some Lemmas on Lattices and Gauss Sums Lemma 2.1.17. Let p ∈ NL be a prime number, v ∈ N, and s ∈ Z. Assume that p - s. One has 8 ‡ rk(L) · rk(L) 0

Thus by Equation (2.68), one has (`/s)k−2−b

X

C T(`)φ (D, r) =

s|`

rk(L) 2 c

C T0 (s)φ (D, r).

`/s=perfect square

By using Theorem 2.6.8, which describes the action of the operators T0 (`) on Fourier coefficients, we can write C T(`)φ (D, r) =

(`/s)k−2−b

X s|`

` s =perfect

square

rk(L) 2 c

X

¡ ¢ ¡ 2 ¢ g k−2 W lev(L)2 D, g C φ gs 2 D, sg0 r .

g | s2 g | s lev(L)D 2

2

(2.69) We want to simplify Equation (2.69). For that, we set a := g `s and b := ag = `s . When s runs over all positive divisors of ` such that `s is a perfect-square and g runs over the positive divisors of s2 such that g2 | s2 lev(L)D , then a runs 2 over all positive divisors of `2 and b runs over all positive divisors of gcd(a, à ). Moreover, if b0 is an integer such that bb0 ≡ 1 mod lev(L), we set a0 := b0 g0 . It 50

2.6. The Action of Hecke Operators on Fourier Coefficients 2

2

is obvious that aa0 ≡ 1 mod lev(L) and Cφ a`2 D, à0 r = Cφ gs 2 D, sg0 r . Thus we can reorder the summation in Equation (2.69) as follows: ¡

X

C T(`)φ (D, r) =

Cφ

¡ `2

a|`2 a |` lev(L)D 2

a2

¢ D, à0 r a k−2

¢

¡

b−b

X

rk(L) 2 c

2

¢

¢ ¡ W lev(L)2 D, ab .

b|(a, à )

2

b=perfect square

To complete the proof, we need to simplify the inner sum in the last equa2 tion. Setting x = (a,a`) , then (a, à ) = a/x2 . The condition a|`2 and a2 |`2 lev(L)D implies that

lev(L)2 D x2

∈ Z. By using Lemma 2.1.22, we obtain

¡ ¡ ¢ ¢ W lev(L)2 D, ab = x2−rk(L) W lev(L)2 D/x2 , bxa 2 .

Thus b−b

X

rk(L) 2 c

2

rk(L) ¡ ¢ ¢ ¡ W lev(L)2 D, ab =a1−d 2 e x · WI lev(L)2 D/x2 , a/x2

b|(a, à ),b=perfect square

=a1−d

rk(L) 2 e

¡ ¢ WI lev(L)2 D, a ,

where in the last step we used Proposition 2.1.3 (see Equation (2.5)). Inserting this into the last formula of C T(`)φ (D, r) gives a k−1−d

X

C T(`)φ (D, r) =

a|`2 a2 |`2 lev(L)D

rk(L) 2 e

¢ ¡ 2 ¢ ¡ WI lev(L)2 D, a C φ a`2 D, à0 r ,

Again, by using Proposition 2.1.3 and gcd(lev(L), a) = 1, we obtain the claimed formula. Proof of Theorem 2.6.3. Recall that for even rk(L) the operators T(`) and T0 (`) are related as follows (see Definition 2.5.7): χL (s)(sd 2 )k−

X

T(`) :=

rk(L) 2 −2

d,s>0 sd 2 |`, s square-free

T0 ( sd` 2 ).

(2.70)

Using Equation (2.70) and Theorem 2.6.8, which gives a closed formula for the action of T0 (`) in terms of Fourier coefficients, we can write C T(`)φ (D, r) = `k−

rk(L) 2 −2

X

X

`1 |` s|`1 `/`1 = perfect square s is square free

51

2. Hecke Theory of Jacobi Forms of Lattice Index X

×

χL (s)(`1 /s)

rk(L) 2 +2− k

d k−2 W (lev(L)2 D, d)C φ

d |(`1 /s)2 s2 d 2 |`1 2 lev(L)D

¡

`1 0 ¢ `1 2 D, 2 2 s d r . s d

We want to simplify the above equation. Similarly as in the proof of the preceding theorem (see proof of Theorem 2.6.1), we can reorder the summation as follows (the substitutions here are a := `s`1 d , b := a/d = s`/`1 ): a k−2 C φ

X

C T(`)φ (D, r) =

¡ `2

a|`2 a |` lev(L)D 2

a2

D, à0 r

b−

¢ X

rk(L) 2

2

¡ ¢ X W lev(L)2 D, ab χL (s). s| b s square-free sb= perfect square

b|(a, à )

2

The inner sum s χL (s) contains exactly one term. In fact, it equals χL (b) (since χL (s) = χL (b)). This gives P

a k−2 C φ

X

C T(`)φ (D, r) =

a|`2 a |` lev(L)D 2

¡ `2

a2

D, à0 r

2

2

a2 | `2 lev(L)D imply that C T(`)φ (D, r) =

lev(L)2 D x2

X

a . x2

¡ ¢ W lev(L)2 D, ab .

The conditions (`, lev(L)) = 1, a | `2 , and

∈ Z. Thus

a k−2 C φ

a|`2 a2 |`2 lev(L)D

=

rk(L) 2

b|(a, à )

2

Setting x := (a,a`) , one has (a, à ) =

χL (b)b−

¢ X

X a|gcd(`2 ,lev(L)D)

a k−

¡ `2

a2

¢ ¡ lev(L)2 D a ¢ D, à0 r x2−rk(L) WII , x2 x2

rk(L) 2 −1

χL (a)C φ

¡ `2

a2

¢ D, à0 r ,

where the first identity follows from Equation (2.21) and Equation (2.5), and the second using Proposition 2.1.4. Now, the proof is complete.

2.7

Hecke Operators and Euler Products

The eigenfunctions for the Hecke operators T(`) (` ∈ NL ) correspond naturally to Dirichlet series having Euler product expansions. These Dirichlet series, the L-functions of eigenfunctions, will express the connection between Jacobi forms and elliptic modular forms. 52

2.7. Hecke Operators and Euler Products Definition 2.7.1. A Jacobi form φ ∈ Jk,L is called a Hecke eigenfunction for the operator T(`) if there exists λ(`, φ) ∈ C with T(`)φ = λ(`, φ)φ.

The number λ(`) = λ(`, φ) known as the eigenvalue of T(`) with corresponding eigenfunction φ. Definition 2.7.2. Let φ be a Jacobi form of weight k and index L = (L, β) which is an eigenfunction of the Hecke operators T(`) for all ` ∈ NL . The formal L-function of φ in s is defined as follows: L(s, φ) =

(2.71)

λ(`)`−s .

X `∈NL

Definition 2.7.3. Let φ be a Jacobi form of weight k and index L = (L, β) which is an eigenfunction of the Hecke operator T(`) for ` ∈ NL . For a pair (D, r) ∈ Supp(L) such that lev(r)D is a square free integer, we set F(s, D, r, φ) :=

X `∈NL

2.7.1

(2.72)

C φ (`2 D, ` r)`−s .

Even Rank Lattices Case

In this subsection we assume that the rank of the lattice L = (L, β) is even. First, we will study the multiplicative properties of the operator T(`), then we shall use these multiplicative properties to determine the Euler product of the L-functions. According to Theorem 2.6.3, the operator T(`) acts on each φ ∈ Jk,L in terms of Fourier of coefficients as follows: C T(`)φ (D, r) =

X a|`2 ,lev(L)D

a k−

rk(L) 2 −1

χL (a)C φ

¡ `2

a2

¢ D, à0 r ,

(2.73)

Recall that for any positive integer a we use a0 to denote an integer such that aa0 ≡ 1 mod lev(L).

53

2. Hecke Theory of Jacobi Forms of Lattice Index Theorem 2.7.4. The Hecke operators T(`) on Jk,L satisfy the following multiplicative relation: T(m) · T(n) =

d k−

X

rk(L) 2 −1

χL (d)T

d | m2 ,n2

(2.74)

¡ mn ¢ d

for every m, n ∈ NL . To prove Theorem 2.7.4 we need the following lemmas. Lemma 2.7.5. Let m, n ∈ NL be coprime positive integers. Then one has T(m)T(n) = T(mn).

Proof. Let φ ∈ Jk,L . By Equation (2.73) one has C T(m)(T(n)φ) (D, r) =

where k2 = k −

rk(L) 2 ,

X a|(m2 ,∆)

X

(ab)k2 −1 χL (ab)C φ

b|(n2 ,m2 ∆/a2 )

¡ n2 m2 a2 b 2

¢ D, nma0 b0 r ,

and ∆ = lev(L)D . First we will prove that (2.75)

(n2 , ∆) = (n2 , m2 ∆/a2 )

for all a|(m2 , ∆). Clearly if d |(n2 , m2 ∆/a2 ), then d | m2 ∆, and as d | n2 , we have (d, m2 ) = 1. It follows that d | (n2 , ∆), i.e., (n2 , m2 ∆/a2 ) | (n2 , ∆). Conversely, 2 suppose that d |(n2 , ∆). Since a | m2 we have (d, a) = 1. Now d | m2 ∆ = a2 ma2∆ , 2 and thus d | ma2∆ . This shows that (n2 , ∆)|(n2 , m2 ∆/a2 ). Thus Equation (2.75) holds true and (m2 n2 , ∆) = (m2 , ∆)(n2 , ∆) = (m2 , ∆)(n2 , m2 ∆/a2 ), where the two factors (m2 , ∆) and (n2 , m2 ∆/a2 ) are coprime. Therefore when a runs over positive divisors of (m2 , ∆) and b runs over positive divisors of (n2 , m2 ∆/a2 ), then ab runs over all positive divisors of (m2 n2 , ∆). This gives C T(m)(T(n)φ) (D, r) =

X g|(m2 n2 ,∆)

g k2 −1 χL (g)C φ

¡ n2 m2 g2

¢ D, nmg0 r = C T(mn)φ (D, r).

Now, the proof is complete. Lemma 2.7.6. Let r be a positive integer. For a prime p ∈ NL one has T(p r ) · T(p) = T(p r+1 ) + p k−

54

rk(L) 2 −1

χL (p)T(p r ) + p2(k−

rk(L) 2 −1)

T(p r−1 ).

(2.76)

2.7. Hecke Operators and Euler Products Proof. Let φ ∈ Jk,L . Then, by Equation (2.73), the action of the left-hand side of Equation (2.76) on φ in terms of Fourier coefficients is given by C T(p r )(T(p)φ) (D, x) = X

=

a|(p2r ,∆)

=

X e

X b|(p2 ,

X a|(p2r ,∆)

a k2 −1 χL (a)C T(p)φ (

p2r ∆, p r a0 x) a2

p2r+2

(ab)k2 −1 χL (ab)C φ ( a2 b2 ∆, p r+1 a0 b0 x)

p2r ∆) a2

N(e)e k2 −1 χL (e)C φ (

p2r+2 ∆, p r+1 e0 x), e2

rk(L)

where k2 = k − 2 , ∆ = lev(L)D , and N(e) is the number of ways of writing e as ab in the preceding sum. If such a decomposition exists then e | p2 a and hence a = (e,pe 2 ) δ for some integer δ; writing down the conditions on a and b = e/a we find formula ¡ p2 p2r+2 ¢ N(e) = number of divisors δ of ∆, p2 , e, ∆ e , e ,

where N(e) = 0 unless e|(∆ p2 , p2r+2 ), e2 |∆ p2r+2 . The action of the right-hand side of Equation (2.76) on φ is given by X d | p2r ,p2

p2r+2

X

X

d | p2r ,p2

a|(p2r+2 /d 2 ,∆)

= =

d k2 −1 χL (d)C T(p r+1 /d)φ (D, x)

X e

(ad)k2 −1 χL (ad)C φ ( a2 d 2 ∆, p r+1 a0 d 0 x)

N"(e)e k2 −1 χL (e)C φ (

p2r+2 ∆, p r+1 e0 x), e2

where now N"(e) counts the decomposition of e as ad satisfying the conditions in the sum; from e|∆d we find that (∆e,e) divides d , and writing d as (∆e,e) δ we obtain for N"(e) the same formula as for N(e). Thus the action of both sides of Equation (2.76) are equal and the proof is complete. Lemma 2.7.7. Let r, s be positive integers. For each prime p ∈ NL one has T(p r )T(p s ) =

X

d k−

rk(L) 2 −1

χL (d)T(p r+s /d)

d | p2r ,p2s

=

2 min(r,s) X

p v(k−

rk(L) 2 −1)

χL (p v )T(p r+s−v ).

(2.77)

v=0

55

2. Hecke Theory of Jacobi Forms of Lattice Index Proof. We fix r and prove Equation (2.77) inductively for prime powers p s . It is obvious that Equation (2.77) holds true for s = 0 since T(p r )T(p0 ) = T(p r ). For s = 1 we refer to Lemma 2.7.6. Assume that s ≥ 2 and Equation (2.77) has been proven for 1, p, · · · , p s . We want to prove it for p s+1 . By definition of T(p s+1 ) and Lemma 2.7.6 T(p r )T(p s+1 ) =T(p r )T(p s )T(p) − p k2 −1 χL (p)T(p r )T(p s ) − p2(k2 −1) T(p r )T(p s−1 ),

where k2 = k −

rk(L) 2 .

By inductive hypothesis we obtain

T(p r )T(p s+1 ) =

min(2r,2s) X

p v(k2 −1) χL (p v )T(p r+s−v )T(p)

v=0

−p

k 2 −1

χL (p)

− p2(k2 −1)

min(2r,2s) X

v=0 min(2r,2s X −2)

p v(k2 −1) χL (p v )T(p r+s−v )

p v(k2 −1) χL (p v )T(p r+s−1−v ).

v=0

Again, using Lemma 2.7.6, we obtain T(p r )T(p s+1 ) =

min(2r,2s) X

p v(k2 −1) χL (p v )T(p r+s+1−v )

v=0

+ p2(k2 −1)

min(2r,2s) X

p v(k2 −1) χL (p v )T(p r+s−1−v ).

v>min(2r,2s−2)

If s > r , then the second sum in the right-hand of the above equality is empty and the desired formula follows since min(2r, 2s) = min(2r, 2(s + 1)). If s ≤ r , then this second sum contains exactly two terms (v ∈ {2s − 1, 2s}). Thus T(p r )T(p s+1 ) =

min(2r,2s) X

p v(k2 −1) χL (p v )T(p r+s+1−v )

v=0

+ p(2s+1)(k2 −1) χL (p)T(p r−s ) + p(2s+2)(k2 −1) T(p r−s+1 ) =

min(2r,2(s X +1)) v=0

which is what we wanted. 56

p v(k2 −1) χL (p v )T(p r+s+1−v )

2.7. Hecke Operators and Euler Products Proof of Theorem 2.7.4. Let m = p p r and n = powers of prime numbers. Then by Lemma 2.7.5 Q

d k−

X

rk(L) 2 −1

χL (d)T

¡ mn ¢

d | m2 ,n2

d

=

Q

X Y ‡ 2 min(r,s) p

p v(k−

p

p s be the expressions as

rk(L) 2 −1)

· χL (p v )T(p r+s−v ) .

v=0

Therefore we have only to prove Theorem 2.7.4 when m and n are powers of a prime p. For this we refer to Lemma 2.7.7. Now, the proof is complete. As an easy consequence of this theorem, we obtain an important insight into the arithmetic proprieties of the eigenfunctions. Proposition 2.7.8. Let φ be a Jacobi form of weight k and index L = (L, β) which is an eigenfunction of the Hecke operators T(`) for all ` ∈ NL . For each prime number p ∈ NL we set: g p (φ) := λ(p) − p k−

rk(L) 2 −1

χL (p).

The L-function L(s, φ) has the product expansion of the form L(s, φ) =

L lev(L) (s − k +

rk(L) 2

+ 1, χL )

‡ ·−1 rk(L) 1 − g p (φ)p−s + p2(k− 2 −1−s) .

Y

L lev(L) (2s − 2k + rk(L) + 2, χ2L )

p∈NL p prime

Proof. Since λ(`1 ) · λ(`2 ) = λ(`1 `2 ) if (`1 , `2 ) = 1, we can write L(s, φ) =

Y

L p (s, φ),

p∈NL , p prime

where for a prime p, the p-local zeta function of φ is given by L p (s, φ) =

∞ λ(p v ) X . vs v=0 p

One has, using Theorem 2.7.4, the following multiplicative relation: λ(p) · λ(p v ) = λ(p v+1 ) + p k−

rk(L) 2 −1

χL (p)λ(p v ) + p2(k−

rk(L) 2 −1)

λ(p v−1 ).

Thus λ(p)L p (s, φ)

57

2. Hecke Theory of Jacobi Forms of Lattice Index =λ(p) +

∞ X

v=1 ∞ X

λ(p)λ(p v ) (p v )s

∞ ∞ λ(p v ) X X rk(L) λ(p v−1 ) 2(k− 2 −1) + p s (p v )s v v=1 v=1 v=1 (p ) rk(L) ¡ ¡ ¢ λ(p) ¢ =λ(p) + L p (s, φ) − 1 − p s p s + p k− 2 −1 χL (p) L p (s, φ) − 1

=λ(p) +

+ p2(k−

λ(p v+1 ) (p v )s

rk(L) 2 −1)

+ p k−

rk(L) 2 −1

χL (p)

p−s L p (s, φ).

Hence rk(L) rk(L) rk(L) ¡ ¢¡ ¢−1 L p (s, φ) = 1 + p k− 2 −1−s χL (p) 1 + p k− 2 −1−s χL (p) − λ(p)p−s + p2(k− 2 −1−s)

=

1− p

2(k−

rk(L) 2 −1− s) χL (p)2

rk(L) k− 2 −1− s χL (p) 1− p

rk(L) rk(L) ¡ ¡ ¢ ¢−1 1 − λ(p) − p k− 2 −1 χL (p) p−s + p2(k− 2 −1−s)

as stated in the proposition. Theorem 2.7.9. For each r ∈ L# and D ≤ 0 such that D ≡ β(r) mod Z and rk(L) lev(r)D is a square-free integer. Setting k 2 := k − 2 , one has ‡

χL (`)`k2 −1−s

X

·

X `≥1 (`,lev(L))=1

`|lev(L)D (`,lev(L))=1

= C φ (D, r)

¡ ¢ C φ `2 D, ` r `−s

1 + χL (p)p k2 −1−s ¢ . k 2 −1 χ (p) p− s + p2(k 2 −1− s) L p-lev(L) 1 − λ(p) − p Y

¡

The products are over all primes p not dividing lev(L). Proof. According to Theorem 2.6.3 if lev(r)D is square-free, we have a k−

X

rk(L) 2 −1

a|`,lev(L)D

χL (a)C φ

¡ `2

a2

¢ D, à0 r = λ(`)C φ (D, r).

(2.78)

Thus by Definition 2.7.2 of the Lfunction of φ , one has C φ (D, r)

X `∈NL

λ(`)`−s =

X X `∈NL a|`

a k−

rk(L) 2 −1

χL (a)δ(a | lev(L)D)C φ

¡ `2

a2

¢ D, a` r `−s

‡ X ·‡ X · rk(L) ¡ ¢ = `k− 2 −1 χL (`)δ(` | lev(L)D)`−s C φ `2 D, ` r `−s . `∈NL

`∈NL

Now, Proposition 2.7.8,which describes L(s, φ) as an Euler product, completes the proof. 58

2.7. Hecke Operators and Euler Products Remark 2.7.10. If we think of an elliptic modular form of weight k2 = rk(L) k − 2 , with nebentypus, say, χL ξ, and with Hecke eigenvalues γ(`), then P 2 −s (taken over all ` coprime to lev(L)) equals L(φ, s) if we replace ` ξ(`)γ(` )` 2 λ(p) by ξ(p)γ(p ). This suggests, for each ξ and suitable levels m, the existence of maps from M k− rk(L) (m, χL ξ) to Jk,L such that T(`2 ) on the left corresponds to 2 ξ(`) T(`) on the Jacobi form side. We shall construct in this thesis examples for such maps.

2.7.2

Odd Rank Lattices Case

In this subsection we assume that the rank of the lattice L = (L, β) is odd. Theorem 2.7.11. The Hecke operators T(`) on Jk,L satisfy the following multiplicative relation: T(m) · T(n) =

X d | m,n

d

2k−rk(L)−2

T

‡

mn d2

·

(2.79)

for every m, n ∈ NL . To prove this theorem we need the following lemmas Lemma 2.7.12. Let r be a positive integer. For a prime p ∈ NL one has T(p r )T(p) = T(p r+1 ) + p2k−rk(L)−2 T(p r−1 ).

(2.80)

Proof. The action of the left-hand side of Equation (2.80) in terms of Fourier coefficients is given by Theorem 2.6.1 as follows: C T(p r )(T(p)φ) (D, s) =

XX rk(L) ¡ p2r p2 ¢ p2r (ab)k−d 2 e−1 %(D, a)%( a2 D, b)C φ a2 b2 D, p r p(ab)0 r . a

b

In the first sum a runs over all positive divisors of p2r such that a2 | p2r lev(L)D . In the second sum b runs over all positive divisors of p2 such that b2 | rk(L) p2r+2 lev(L)D . We set h := k −d 2 e−1, and ∆ := lev(L)D . The previous equation a2 can be written as C T(p r )(T(p)φ) (D, s) = Ω1 + Ω2 + Ω3 ,

59

2. Hecke Theory of Jacobi Forms of Lattice Index where Ω1 =

X

a h %(D, a)C φ

a| p2r

Ω2 =

X a| p2r

Ω3 =

X a| p2r

¡ p2r+2 a2

¢ D, p r+1 a0 s δ(a2 | p2r+2 D),

p2r

(ap)h %(D, a)%( a2 D, p)C φ p2r

¡ p2r a2

(ap2 )h %(D, a)%( a2 D, p2 )C φ

¢ D, p r a0 s δ(a2 | p2r ∆),

¡ p2r−2 a2

¢ D, p r−1 a0 s δ(a2 | p2r−2 ∆).

The first sum can be written as Ω1 = C T(p r+1 )φ (D, s) − A 1 − B1 , where A 1 = p(2r+1)h %(D, p2r+1 )C φ

D , p r+1 (p2r+1 )0 s p2r

δ(p2r |∆), ¡ ¢ B1 = p(2r+2)h %(D, p2r+2 )C φ p2rD+2 , p r+1 (p2r+2 )0 s δ(p2r+2 |∆).

¡

¢

Also, the third sum Ω3 can be written as Ω3 = A 3 + B3 + C3 , where A 3 = p(2r+1)h %(D, p2r−1 )%( p2rD−2 , p2 )C φ ( pD2r , p r+1 (p2r+1 )0 s)δ(p2r |∆), B3 = p(2r+2)h %(D, p2r )%( pD2r , p2 )C φ ( p2rD+2 , p r+1 (p2r+2 )0 s)δ(p2r+2 |∆) X ¢ ¡ p2r−2 p2r C3 = (ap2 )h %(D, a)%( a2 D, p2 )C φ a2 D, p r−1 a0 s δ(a2 | p2r−2 ∆). a| p2r−2

The condition p2r |∆ in A 3 implies that %(D, p2r−1 ) = 0, and then A 3 = 0. The 2r 2r condition a2 | p2r−2 ∆ in C3 implies that ( pa2 ∆, p2 ) = p2 , %( pa2 D, p2 ) = p, and then C 3 = p2k−rk(L)−2 C T(p r−1 φ) (D, s). Inserting this into the last formula of Ω3 gives Ω3 = p2k−rk(L)−2 C T(p r−1 φ) (D, s) + B3 .

In fact, the condition p2r+2 |∆ in B3 and B1 implies that %(D, p2r )%( pD2r , p2 ) = p r p = %(D, p2r+2 ), and then B1 = B3 . Thus C T(p r )(T(p)φ) (D, s) = Ω1 + Ω2 + Ω3 = C T(p r−1 φ) (D, s) + p2k−rk(L)−2 C T(p r−1 φ) (D, s) + Ω2 − A 1 .

To complete the proof we still need to show that Ω2 = A 1 . It is obvious that both A 1 and Ω2 are 0 unless ord p (∆) = 2r . If ord p (∆) = 2r , then the sum over a in Ω2 contains exactly one term (a = p2r ), i.e., Ω2 = p(2r+2)h %(D, p2r )%( pD2r , p)C φ ( pD2r , p r+1 (p2r+1 )0 s)δ(ord p (∆) = 2r).

Since ord p (∆) = 2r , one has %(D, p2r )%( pD2r , p) = p r = %(D, p2r+1 ), and then A 1 = Ω2 . The proof is complete. 60

2.7. Hecke Operators and Euler Products Lemma 2.7.13. Let m, n ∈ NL be coprime positive integers. Then one has T(m)T(n) = T(mn).

Proof. We omit the proof, which is similar to the proof of Lemma 2.7.5. Proof of Theorem 2.7.11. Let m = p p r and n = p p s be the expressions as powers of prime numbers. Then by Lemma 2.7.13 one has Q

X

d 2k−rk(L)−2 T

‡

d | m,n

mn d2

·

=

Q

Y ‡ min(r,s) X p

· p v(2k−rk(L)−2) T(p r+s−2v ) .

v=0

Therefore we have only to prove Theorem 2.7.11 when m = p r and n = p s are powers of a prime p. We fix r and prove inductively for prime powers p s . For s = 1 we refer to Lemma 2.7.12. Assume that s ≥ 2 and the equation T(p r )T(p s ) =

min(r,s) X

p v(2k−rk(L)−2) T(p r+s−2v )

v=0

has been proven for 1, p, · · · , p s . We want to prove it for p s+1 . One has ¡ ¢ T(p r )T(p s+1 ) = T(p r ) T(p s )T(p) − p2k−rk(L)−2 T(p s−1 ) =

min(r,s) X

p

v(2k−rk(L)−2)

T(p

r + s−2v

)T(p) + p

2k−rk(L)−2

v=0

min(r,s X −1)

p v(2k−rk(L)−2) T(p r+s−1−2v ),

v=0

again, by using Lemma 2.7.12, we obtain T(p r )T(p s+1 ) =

min(r,s) X

p v(2k−rk(L)−2) T(p r+s+1−2v )

v=0

+

min(r,s) X

p(v+1)(2k−rk(L)−2) T(p r+s−1−2v ),

v>min(r,s−1) 2v+1≤ r + s

where the second sum contains exactly one term (v = s) if r > s, and is empty otherwise. This gives T(p r )T(p s+1 ) = =

min(r,s) X

p v(2k−rk(L)−2) T(p r+s+1−2v ) + p(s+1)(2k−rk(L)−2) T(p r−s−1 )

v=0 min(r,s X +1)

p v(2k−rk(L)−2) T(p r+s+1−2v ).

v=0

Now, the proof is complete. 61

2. Hecke Theory of Jacobi Forms of Lattice Index Lemma 2.7.14. Let (D, r) ∈ supp(L). If lev(r)D is a square-free integer, then for all ` ∈ NL we have the following multiplicative relation: X rk(L) ¡ ¢ C φ `2 D, ` r =C φ (D, r) a k−d 2 e−1 χL (D, a)µ(a)λ(`/a). a|`

(2.81)

Proof. By Remark 2.6.2 we have X

a k−d

rk(L) 2 e−1

a|`

χL (D, a)C φ

¡ `2

a2

¢ D, a` r = λ(`)C φ (D, r).

(2.82)

Now, the claimed formula follows from this via Möbius inversion. Proposition 2.7.15. Let φ be a Jacobi form of weight k and index L = (L, β) which is an eigenfunction of the Hecke operators T(`) for all ` ∈ NL . One has the product expansion of the form L(s, φ) =

Y

¡

1 − p−s λ(p) + p2k−rk(L)−2−2s

¢−1

.

(2.83)

p∈NL p prime

Proof. Since λ(m) · λ(n) = λ(mn) if (m, n) = 1, we can write Y

L(s, φ) =

L p (s, φ),

p∈NL

p prime

where for a prime p, the p-local L-function of φ is given by L p (s, φ) =

∞ X

λ(p v )p−vs .

v=0

By Theorem 2.7.11, we have λ(p)L p (s, φ)

=λ(p) +

∞ λ(p)λ(p v ) X p vs v=1

=λ(p) + p s

∞ λ(p v+1 ) X

+ p(2k−1−rk(L))−1−s

∞ λ(p v−1 ) X

(v−1)s p(v+1)s v=1 p ¡ λ(p) ¢ =λ(p) + − 1 − p s p s + p s L p (s, φ) + p(2k−1−rk(L))−1−s L p (s, φ). v=1

This gives L p (s, φ) = 1 − p−s λ(p) + p2(k− ¡

62

rk(L) 2 −1− s)

¢−1

, which completes the proof.

2.7. Hecke Operators and Euler Products Remark 2.7.16. If the φ in Proposition 2.7.15 lifts to an elliptic modular form f of weight k 1 = 2k − 1 − rk(L), then L(s, φ) should be (up to a finite number of Euler factors) the L-series of f . We observe that the right-hand side of Equation (2.83) has indeed the right shape (compare with Equation (1.8)). Theorem 2.7.17. Let (D, r) ∈ Supp(L) such that lev(r)D is a square-free integer. One has the following factorization: ¡ ¢ L lev(L) s − k + drk(L)/2e + 1, χL (D, ·) F(s, D, r, φ) = C φ (D, r) L(s, φ).

(2.84)

Proof. By Lemma 2.7.14 one has F(s, D, r, φ) =

X `∈NL

C φ (`2 D, ` r)`−s

=C φ (D, r)

X ‡X

δ

k−d

rk(L) 2 e−1

·

χL (D, δ)µ(δ)λ(`/δ) `−s ,

`∈NL δ|`

which can be written using the Dirichlet convolution as ‡ X ·‡ X · rk(L) F(s, D, r, φ) =C φ (D, r) `k−d 2 e−1 χL (D, `)µ(`)`−s λ(`)`−s `∈NL

`∈NL

=C φ (D, r) L(s, φ) L lev(L) s − k + d ¡

¢−1 rk(L) e + 1, χ (D, · ) . L 2

In the last step we applied the Möbius inversion to pull out the reciprocal of the Dirichlet L-function.

63


Chapter 3 Basis of Simultaneous Eigenforms The aim of this chapter is to show that there exist bases of simultaneous Hecke eigenforms (i.e., bases consisting of functions, which are eigenforms to all Hecke operators T(`) (` coprime to lev(L))) both of the subspace of Jacobi cusp forms S k,L and of the subspace of Jacobi-Eisenstein series E k,L (to be defined in section 3.3). To that end we will define a scalar product on S k,L , called the Petersson Inner Product, and show that all Hecke operators are Hermitian with respect to that product. The rest will follow readily via some linear algebra.

3.1

The Action of the Orthogonal Group

Let L = (L, β) be a positive definite even lattice. Let D L = (L# /L, β) be the associated discriminant form. The orthogonal group O(D L ) consists of all automorphisms α of L# /L such that β ◦ α = β. Proposition 3.1.1. O(D L ) acts on Jk,L from left as follows: (α, φ) 7→ W(α)φ,

where, for φ(τ, z) =

X

h x (τ)ϑL,x (τ, z),

x∈L# /L

65

3. Basis of Simultaneous Eigenforms we set X ¡ ¢ W(α)φ (τ, z) := h α(x) (τ)ϑL,x (τ, z). x∈L# /L

Proof. We prove, first of all, that W(α)φ is a Jacobi form of weight k and index L = (L, β). The action of W(α) maps the collection of tuples (h x ) x∈L# /L ¡ ¢ ¡ ¢ into itself by the permutation h x x∈L# /L 7→ hα(x) x∈L# /L . Since β ◦ α = β, it is clear that the matrix representations in Proposition 2.4.9 are preserved under this permutation, i.e., for each α ∈ O(D L ) X ¡ ¢ W(α)φ (τ, z) = h α(x) (τ)ϑL,x (τ, z) x∈L# /L

is a Jacobi form of weight k and index L = (L, β). Next, we prove the group axioms. Let 1 ∈ O(D L ) denote the trivial automorphism. It is clear that ¡ ¢ W(1)φ (τ, z) = φ(τ, z).

For α, γ ∈ O(D L ) we have ¡ ¡ ¢¢ W(α) W(γ)φ (τ, z) X h α(γ(x)) (τ)ϑL,x (τ, z) = x∈L# /L

X

=

x∈L# /L

h (αγ)(x) (τ)ϑL,x (τ, z)

¡ ¢ = W(αγ)φ (τ, z).

Now, the proof is complete. Theorem 3.1.2. The action of the orthogonal group on the vector space of Jacobi forms commutes with the action of the Hecke operators T(`) (` ∈ NL ),i.e., ¡ ¢ ¡ ¢ T(`) W(α)φ = W(α) T(`)φ

(φ ∈ Jk,L )

for all ` ∈ NL and α ∈ O(D L ). Proof. For each pair (D, x) ∈ supp(L) we have the identity CW(α)φ (D, x) = C φ (D, α(x)). If rk(L) is even, one has CW(α)(T(`)φ) (D, x) = C T(`)φ (D, α(x))

66

3.2. Jacobi Cusp Forms and Hecke Operators X

=

a k−1−

rk(L) 2

a k−1−

rk(L) 2

χL (a)C φ

¡ `2

a|gcd(`2 ,lev(L)D)

X

=

a2

D, à0 α(x)

χL (a)CW(α)φ

¡ `2

a|gcd(`2 ,lev(L)D)

a2

¢

D, à0 x

¢

= C T(`)(W(α)φ) (D, x).

The case of odd rank lattices can be proved exactly in the same way, so we will omit its proof.

3.2

Jacobi Cusp Forms and Hecke Operators

For τ ∈ H, z ∈ L ⊗Z C, let τ = u + iv and z = x + i y be the decompositions into real and imaginary parts. We define a volume element on H × (L ⊗Z C) by dVL,(τ,z) := v−rk(L)−2 dudvdxd y.

The volume element dVL,(τ,z) is invariant under the action of JL (Q) on H × (L ⊗Z C) (see e.g. [Zie89, p. 202]). Definition 3.2.1. Let φ and ψ be flk,L -invariant under a subgroup Λ of JL (Z) of finite index. We set fl

−1

ωφ,ψ (τ, z) := φ(τ, z)ψ(τ, z)v k e−4πβ(y).v .

Lemma 3.2.2. One has ωφflfl

fl

g,ψflk,L g k,L

(3.1)

(τ, z) = ωφ,ψ (g(τ, z)) for all g ∈ JL (Q). In

particular the function ωφ,ψ (τ, z) is Λ-invariant (i.e., ωφ,ψ (M(τ, z)) = ωφ,ψ (τ, z) for all M ∈ Λ). Proof. See e.g. [Bri04, Lemma 2.23]. Definition 3.2.3. We set ' “ F JL (Z) = (τ, z) ∈ H × (L ⊗C) | τ ∈ FΓ , z in a fundamental mesh for L ⊗C/τL + L ' “ / (τ, z) 7→ (τ, − z) ,

where FΓ := {τ ∈ H | − 1/2 ≤ Re τ ≤ 1/2, and |τ| ≥ 1}

denotes the classical fundamental domain for the operation of Γ = SL2 (Z). 67

3. Basis of Simultaneous Eigenforms Lemma 3.2.4. F JL (Z) is a fundamental domain of the action of JL (Z) on H × (L ⊗Z C) Proof. See e.g. [Bri04, Remark 2.25]. Definition 3.2.5. Let φ and ψ be flk,L -invariant under a subgroup Λ of finite index in JL (Z). Suppose that φ or ψ is a cusp form. We define the Petersson scalar product of φ and ψ with respect to Λ by fl

< φ, ψ >Λ := [J (1Z):Λ] L

Z FΛ

ωφ,ψ (τ, z)dVL,(τ,z) ,

(3.2)

where FΛ denotes a fundamental domain of the action of Λ on H × (L ⊗Z C). Remark 3.2.6. Writing JL (Z) = ∪M Λ M as decompositions into classes, then 1 L (Z):Λ]

< φ, ψ >Λ = [J

X M ∈Λ\ JL (Z)

Z M F JL (Z)

ωφ,ψ (τ, z)dVL,(τ,z) .

(3.3)

Proposition 3.2.7. The integral in (3.2) is absolutely convergent. The scalar product (3.2) does not depend on the choice of the fundamental domain FΛ . Moreover, the scalar product (3.2) is positive definite. Proof. See e.g. [Zie89, p 202-203]. Proposition 3.2.8. The scalar product (3.2) does not depend on the choice of Λ, i.e., if Λ0 is another subgroup of finite index in JL (Z) such that φ and ψ fl are flk,L -invariant under Λ0 , then < φ, ψ >Λ =< φ, ψ >Λ0 .

Proof. Let FΛ∪Λ0 be a fundamental domain of the action of Λ ∪ Λ0 . Write Λ ∪ Λ0 = ∪v Λ M v = ∪v0 Λ0 M v0 as decompositions into classes. Then FΛ :=

[ v

M v FΛ∪Λ0 ,

FΛ0 :=

[ v0

M v0 FΛ∪Λ0

are fundamental domains of the action of Λ and Λ0 respectively. This gives 1 < φ, ψ >Λ = [JL (Z) : Λ]

68

Z FΛ

ωφ,ψ (τ, z)dVL,(τ,z)

3.2. Jacobi Cusp Forms and Hecke Operators X 1 = [JL (Z) : Λ] v

Z M v FΛ∪Λ0

ωφ,ψ (τ, z)dVL,(τ,z)

0 Z [Λ ∪ Λ0 : Λ] [JL (Z) : Λ ∪ Λ ] ωφ,ψ (τ, z)dVL,(τ,z) [JL (Z) : Λ] [JL (Z) : Λ ∪ Λ0 ] FΛ∪Λ0 XZ 1 ωφ,ψ (τ, z)dVL,(τ,z) = [JL (Z) : Λ ∪ Λ0 ][Λ ∪ Λ0 : Λ0 ] v0 Mv0 FΛ∪Λ0 Z 1 = ωφ,ψ (τ, z)dVL,(τ,z) [JL (Z) : Λ0 ] FΛ0

=

= < φ, ψ >Λ0

which completes the proof. Definition 3.2.9 (Notation). Since < φ, ψ >Λ does not depend on the choice of Λ (see Proposition 3.2.8), we often write it simply < φ, ψ >. Proposition 3.2.10. Let φ=

X

h x ϑL,x

,

ψ=

X

g x ϑL,x

x∈L# /L

x∈L# /L

be two Jacobi forms in Jk,L and at least one of φ and ψ is a cusp form, then < φ(τ, z), ψ(τ, z) >= 2

−

rk(L) 2

¡

det(L)

¢− 1

Z

X

2

SL2 (Z)\H x∈L# /L

h x (τ)g x (τ)v k−

rk(L) 2 −2

dudv.

(3.4) In other words, the Petersson scalar product of φ and ψ is equal (up to constant) to the Petersson product in the usual sense of the vector-valued modular forms rk(L) ~ h = (h x ) x∈L# /L and ~ g = (g x ) x∈L# /L of weight k − 2 . rk(L)

Remark 3.2.11. Note that x∈L# /L h x (τ)g x (τ)v k− 2 −2 is invariant under SL2 (Z), which follows from the fact the Weil representation ρ L is unitary. P

Proof of Proposition 3.2.10. In the scalar index case, a proof can be found in [EZ85, P.61]. For the higher-rank case, one can prove (using the computations in [BK93, P.504]) the following statement: in a fixed fiber (τ ∈ H fixed), the scalar product of ϑL,µ , ϑL,λ (µ, λ ∈ L# /L) is equal to Z

−1

–

(L⊗Z C) (L+τL)

ϑL,µ (τ, z)ϑL,λ (τ, z)e−4πβ(y).v dxd y

69

3. Basis of Simultaneous Eigenforms = 2−

rk(L) 2

¡ ¢− 1 rk(L) det(L) 2 v 2 δ(µ − λ ∈ L).

(3.5)

Hence, the proposition immediately follows from the above statement. Now, we state the main results of this section. Theorem 3.2.12. Let φ and ψ be Jacobi forms of weight k and index L = (L, β) such that one of them at least is a Jacobi cusp form. Then for each ` ∈ NL we have (3.6)

< T(`)φ, ψ > JL (Z) =< φ, T(`)ψ > JL (Z) .

Theorem 3.2.13. The space of Jacobi cusp forms S k,L has a basis of simultaneous eigenforms for all operators T(`) and all operators W(α) (` ∈ NL , α ∈ O(D L )). The rest of this section deals with the proof of the theorems. For that, we need the following lemmas: Lemma 3.2.14. Let g ∈ JL (Q). Assuming that φ, ψ, φflk,L g, ψflk,L g−1 are flk,L invariant under a subgroup Λ g of finite index in JL (Z). Then fl

fl

fl

fl fl < φflk,L g, ψ >Λ g =< φ, ψflk,L g−1 >Λ g .

Proof. Recall that the Petersson scalar product of φ, ψ is independent of the choice of the fundamental domain (see Proposition 3.2.7). Thus fl < φfl

1 k,L g, ψ >Λ g = [JL (Z) : Λ g ]

Z

1 = [JL (Z) : Λ g ]

Z

FΛ g

ω flfl φ

g−1 FΛ g

k,L

g,ψ

ω flfl φ

(τ, z) dVL,(τ,z)

g,ψ k,L

(g−1 (τ, z)) dVL,g−1 (τ,z) .

The volume element dVL,(τ,z) is invariant under the action of JL (Q) (see e.g. [Zie89, p. 202]). Using this and Lemma 3.2.2 we obtain fl < φfl

1 k,L g, ψ >Λ g = [JL (Z) : Λ g ]

The proof is complete. 70

Z FΛ g

ω

fl

φ,ψflk,L g−1

fl (τ, z) dVL,(τ,z) =< φ, ψflk,L g−1 >Λ g .

3.2. Jacobi Cusp Forms and Hecke Operators Lemma 3.2.15. Let g ∈ JL (Q). Then JL (Z) and JL (Z) ∩ g−1 JL (Z)g are commensurable. Proof. One has JL (Z)gJL (Z) =

JL (Z)gs

[ s∈(JL (Z)∩ g−1 JL (Z)g)\ JL (Z)

as a disjoint union, and ¡ ¢ [JL (Z) : JL (Z) ∩ g−1 JL (Z)g] = # JL (Z)\ JL (Z)gJL (Z) .

Write g = (A, x). Let γ : JL (Z)gJL (Z) → Γ A Γ be the canonical map (M, h) 7→ M , † † and let γ∗ : JL (Z) JL (Z)gJL (Z) → Γ Γ A Γ be the induced map. Each class in the fibre γ−∗ 1 (SL2 (Z)M) has a representative of the form (M, xB1 h), where h ∈ H L (Z) and B1 ∈ A −1 Γ M ∩ Γ. Write x = (λ, µ, ξ). Let N be a positive integer such that (N λ, N µ, 1) ∈ H L (Z), ` ∈ N such that ` M ∈ GL2 (Z), let B2 ∈ A −1 Γ M ∩ Γ with B1 ≡ B2 mod ` N , then ¡

¢¡ ¢−1 ¡ ¡ −1 ¢ −1 ¢ M, xB1 h M, xB2 h = 1, (xB1 (xB2 )−1 ) M = 1, (λ, µ, 1)(B1 −B2 )M ∈ JL (Z).

Thus (M, xB2 h) is also a representative of the same coset. The number of such possibilities mod ` N is finite since A −1 Γ M ∩Γ mod ` N ⊆ SL2 (Z/` N Z). Let † M ∈ Γ Γ A Γ, B ∈ A −1 Γ M ∩ Γ, and h 1 , h 2 ∈ H L (Z). The two classes JL (Z)(M, xB h 1 ) and JL (Z)(M, xB h2 ) are equal if and only if ‡ · h 1 ≡ h 2 mod (M, xB )−1 H L (Z)(M, xB ) ∩ H L (Z).

Hence in the view of the well-known fact that Γ Γ A Γ is finite, and A −1 Γ M ∩ Γ mod ` N is finite, it is enough to show that †

(H L (Z) ∩ (M, xB )−1 H L (Z)(M, xB ))\ H L (Z)

is finite. But this is immediate, since n

o fl ¡ ¢ fl (` N λ, ` N µ, 1) λ, µ ∈ L ⊆ H L (Z) ∩ (M, xB )−1 H L (Z)(M, xB ) .


3. Basis of Simultaneous Eigenforms Proof of Theorem 3.2.12. By Definition 2.5.7, the operator T(`) can be written as a linear combination of the double coset Hecke operator T0 (`), so it suffices to prove that (3.7)

< T0 (`) φ, ψ > JL (Z) =< φ, T0 (`) ψ > JL (Z) .

For each g ∈ JL (Z)\ JL (Z) `0 0` JL (Z) we set Λ g := (g−1 JL (Z)g) ∩ (gJL (Z)g−1 ) ∩ JL (Z). By Lemma 3.2.15 we conclude that Λ g is a subgroup of finite index fl fl in JL (Z). Moreover, by [Zie89, Lemma 1.4], we see that φ, ψ, φflk,L g, ψflk,L g−1 fl are flk,L -invariant under Λ g . The intersection ¡

−1

¢

Λ=

Λg

\ · −1 g∈ JL (Z)\ JL (Z) ` 0 JL (Z) 0 ` ‡

is a subgroup of finite index in JL (Z). This gives < T0 (`) φ, ψ > JL (Z) =< T0 (`) φ, ψ >Λ = `k−2−rk(L)

X ‡ −1 · g∈ JL (Z)\ JL (Z) ` 0 JL (Z) 0 `

= `k−2−rk(L)

X ‡ −1 · g∈ JL (Z)\ JL (Z) ` 0 JL (Z) 0 `

=< φ, `k−2−rk(L)

fl < φflk,L g, ψ >Λ fl < φ, ψflk,L g−1 >Λ

X ‡ −1 · g∈ JL (Z)\ JL (Z) ` 0 JL (Z) 0 `

fl ψflk,L g−1 >Λ

=< φ, T0 (`) ψ > JL (Z) .

The first identity follows from the fact that is independent of the choice of Λ (see Proposition 3.2.8), the second from the bilinearity of , the third from Lemma 3.2.14, the fourth from the anti-linearity of in the second ¡ −1 ¢ ¡ ¢ argument, the last from the fact that JL (Z) `0 0` JL (Z)= JL (Z) `0 `0−1 JL (Z). Thus for each ` ∈ NL , the operators T0 (`) T(`) is a self-adjoint operator with respect to the Petersson scalar product. We recall a lemma from basic linear algebra. Lemma 3.2.16. Let V be a finite-dimensional complex vector space equipped with a positive definite Hermitian form . 72

3.3. Jacobi-Eisenstein Series and Hecke Operators 1. Let f : V → V be a linear map which is Hermitian; in other words, if v, w ∈ V then = . Then V has a basis consisting of eigenvectors for f . 2. Let f 1 , f 2 , · · · be a sequence of Hermitian operators sending V to V which commute with each other. Then V has a basis consisting of vectors that are eigenvectors for all of the f i . Proof of Theorem 3.2.13. It is known from [BS14] that the space S k,L is a finite-dimensional Hilbert space under the Petersson scalar product. If α ∈ O(D L ), then the action of W(α) permutes the functions h x (x ∈ L# /L). It follows from Proposition 3.2.10 that W(α) is Hermitian. Also by Theorem 3.2.12, we see if ` ∈ NL then T(`) is Hermitian with respect to the Petersson scalar product. Therefore, we can apply Lemma 3.2.16 to the set of T(`) (` ∈ NL ) and to the set of W(α) (α ∈ O(D L )). The first part of Lemma 3.2.16 says that there exist eigenforms which form an orthonormal basis for S k,L . These need not be simultaneous eigenforms for all T(`),W(α). However, since the operators T(`) (resp. W(α)) commute with each other, the second part of Lemma 3.2.16 shows that S k,L has an orthonormal basis consisting of simultaneous eigenforms. Each of these can be multiplied by a constant factor to get a new basis of simultaneous normalized eigenforms.

3.3

Jacobi-Eisenstein Series and Hecke Operators

As in the usual theory of modular forms, we will obtain an example of the Jacobi form of lattice index by constructing Jacobi-Eisenstein series. Let L = (L, β) be an even positive definite lattice over Z. Recall that each φ ∈ Jk,L has Fourier expansion φ(τ, z) =

X

¡ ¢ C φ (D, r)e nτ + β(r, z)

(τ, z) ∈ h × (L ⊗Z C).

(D,r)∈supp(L)

73

3. Basis of Simultaneous Eigenforms The Jacobi form φ is called a cusp form if Cφ (0, r) = 0 for all r such that β(r) ∈ Z. By S k,L as before, we mean the subspace of Jacobi forms in Jk,L consisting of cusp forms. Definition 3.3.1. Let D L be the associated discriminant form with L. We set fl ¡ ¢ ' “ Iso D L := x ∈ L# /L fl β(x) ∈ Z .

Definition 3.3.2. For r ∈ L# , we set (3.8)

¡ ¢ g L,r (τ, z) := e τβ(r) + β(r, z) .

Definition 3.3.3. We set JL (Z)∞ :=

n ¡¡

1 n 0 1

¢

o

(3.9)

, (0, µ, 1) ; n ∈ Z, µ ∈ L . ¢

Proposition 3.3.4. Let γ∞ = a0 db , (0, µ, 1) ∈ JL (Z)∞ . For each r ∈ Iso D L and k ∈ N, the function g L,r satisfies the following: ¡¡

¢

¢

¡

¢

(3.10)

fl g L,r flk,L γ∞ = g L,r fl g L,r flk,L − I 2 = (−1)k g L,−r

(3.11)

Proof. The following identity proves Equation (3.10): ¡

fl fl ¢ ¡ z+µ ¢ g L,r flk,L γ∞ (τ, z) = (g L,r flk,L A)(τ, z + µ) = g L,r A τ, w(τ)2 = g L,r (τ, z + µ) = g L,r (τ, z).

One also has ¡

fl ¡ ¢¢ ¡ ¢ g L,r flk,L −01 −01 (τ, z) = (−1)k e τβ(r) + β(r, − z) = (−1)k g L,−r (τ, z)

which proves Equation (3.11). rk(L)

Definition 3.3.5 (Jacobi-Eisenstein series). Let k ∈ N with k > 2 + 2. For ¡ ¢ r ∈ Iso D L we define the Jacobi-Eisenstein series of weight k and index L as follows: X fl E k,L,r := 21 g L,r flk,L γ. (3.12) γ∈ JL (Z)∞ \ JL (Z)

74

3.3. Jacobi-Eisenstein Series and Hecke Operators Proposition 3.3.6. For each r ∈ Iso D L , k ∈ N with k > E k,L,r is absolutely uniformly convergent on compact sets. ¡

¢

rk(L) 2

+ 2, the series

Proof. See e.g. [BK93, P.503]. Lemma 3.3.7. The sum in Equation (3.12) is independent of the choice of coset representatives for JL (Z)∞ \ JL (Z). Proof. Let γ∞ ∈ JL (Z)∞ , γ ∈ JL (Z). One has fl fl fl fl g L,r flk,L γ∞ γ = g L,r flk,L γ∞ flk,L γ = g L,r flk,L γ.

Hence, each term in Equation (3.12) does not depend on the choice of the representative, but only on the JL (Z)∞ -orbit of γ. Proposition 3.3.8. The series E k,L,r given by Definition 3.3.5 transforms like a Jacobi form of weight k and index L. Moreover, one has E k,L,r = (−1)k E k,L,−r . Proof. Any A ∈ JL (Z) permutes JL (Z)∞ \ JL (Z) by right multiplication, i.e. the map ψ : JL (Z)∞ \ JL (Z) → JL (Z)∞ \ JL (Z),

JL (Z)∞ γ 7→ JL (Z)∞ γ A

is bijective. Thus one has fl E k,L,r flk,L A = 12 = 12

X γ∈ JL (Z)∞ \ JL (Z)

fl g L,r flk,L γ A

X γ A ∈ JL (Z)∞ \ JL (Z)

fl g L,r flk,L γ A

=E k,L,r .

Hence, E k,L,r transforms like a Jacobi form of weight k and index L. The equality E k,L,r = (−1)k E k,L,−r follows from Equation (3.11). Proposition 3.3.9. One has < φ, E k,L,r >= 0

(3.13)

for all φ ∈ S k,L . 75

3. Basis of Simultaneous Eigenforms Proof. according to [BK93, Lemma 1], there is λk,r ∈ C such that < φ, E k,L,r >= λk,r C φ (0, r). Since φ is a cusp form, we have C φ (0, r) = 0. Thus < φ, E k,L,r >= 0. To write the Jacobi-Eisenstein series in more explicit form, we need to find a set of coset representatives for JL (Z)∞ \ JL (Z). One can take the set n‡

· o A, (λ, 0, 1) A | A ∈ SL2 (Z)∞ \ SL2 (Z), λ ∈ L

(3.14)

as a complete set of coset representatives for JL (Z)∞ \ JL (Z) (see [BK93, P.504]). Proposition 3.3.10. The series E k,L,r given by Definition 3.3.5 can be written in terms of theta series as follows: E k,L,r = 21

X A ∈SL2 (Z)∞ \ SL2 (Z)

(3.15)

fl ϑL,r flk,L A.

Proof. As a set of coset representatives for JL (Z)∞ \ JL (Z) we take the set which is given by Equation (3.14). One has E k,L,r = 12

X

X

A ∈SL2 (Z)∞ \ SL2 (Z)

λ∈L

fl fl g L,r flk,L (λ, 0, 1)flk,L A.

This can be written, using the definition of the slash operator and Definition 3.3.1 of the function g L,r , as E k,L,r (τ, z) = 12 = 21

X

X ‡

A ∈SL2 (Z)∞ \ SL2 (Z) λ∈L

X A ∈SL2 (Z)∞ \ SL2 (Z)

e A τβ(r + λ) + β(r + λ, w

¡X λ∈L

· ‡

z 2) A (τ)

e

− cβ(z) w A (τ)2

·

w A (τ)−2k

¢fl g k,L,r+λ flk,L A(τ, z).

Comparing the inner sum with Definition 2.3.2 of the theta series, we see P that λ∈L g k,L,r+λ = ϑL,r . Now, the proof is complete. Definition 3.3.11. Let φ be a Jacobi form of weight k and index L = (L, β). The singular term of φ is defined as ¡

¢ singular-term(φ) (τ, z) :=

X s∈ L # β(s)≡0 mod Z

76

¡ ¢ C φ (0, s)e τβ(s) + β(s, z) .

(3.16)

3.3. Jacobi-Eisenstein Series and Hecke Operators Proposition 3.3.12. The singular term of the series E k,L,r is given by ‡ · ¡ ¢ singular-term(E k,L,r ) (τ, z) = 12 ϑL,r (τ, z) + (−1)k ϑL,−r (τ, z) X ¡ ¢ = C E k,L,r (0, s)e τβ(s) + β(s, z) ,

(3.17)

s∈ L # β(s)≡0 mod Z

where ‡ · C E k,L,r (0, s) = 12 δ(s ≡ r mod L) + (−1)k δ(s ≡ − r mod L) .

(3.18)

Remark 3.3.13 (and Definition). If φ ∈ Jk,L , then X

φ−

C φ (0, r)E k,L,r ∈ S k,L .

¡ ¢ r ∈Iso D L

Thus, in the view of Proposition 3.3.9, we may write Eis Jk,L = S k,L ⊕ Jk,L , Eis where Jk,L is the Jacobi-Eisenstein subspace of the space of the Jacobi forms of weight k and index L = (L, β) which consists of all functions E k,L,r ¡ ¢ ( r ∈ Iso D L ).

Proof of Proposition 3.3.12. As a complete system of coset representatives for SL2 (Z)∞ \ SL2 (Z) one takes the set n

A=

¡a b¢ c d

o ∈ SL2 (Z) | (c, d) = 1 .

Thus we can split the sum (3.15) into two parts, according to whether c = 0 or not. If c = 0, then these terms give a contribution of the singular term, which is ‡ · singular-term(E k,L,r ) = 21 ϑL,r + (−1)k ϑL,−r .

Now, Definition 2.3.2 of the theta function completes the proof. Lemma 3.3.14. The dimension of the vector space of the Jacobi-Eisenstein series is given by ‡ n o· ¡ ¢ ¡ ¢ flfl Eis dim Jk,L = 21 | Iso D L | + (−1)k # r ∈ Iso D L fl 2r ∈ L .

(3.19) 77

3. Basis of Simultaneous Eigenforms Proof. If k is even, then E k,L,r = E k,L,−r . In this case, the space of Jacobi Eisenstein series is spanned by E k,L,s where s runs through a set ∆ of representatives for Iso(D L ) modulo the action of {±1}. One has |∆| = 12 #{r ∈ Iso(D L ) | 2r 6∈ L} + #{ r ∈ Iso(D L ) | 2r ∈ L}. If k is odd, then E k,L,r = −E k,L,−r , and Eis E k,L,r = 0 if 2r ∈ L. In this case, Jk,L is spanned by E k,L,s where s runs through †' “ ∆ r ∈ ∆ | 2r ∈ L . The linearity independence of the Jacobi theta series im¡ ¢ plies that singular-term(E k,L,s ) = 12 ϑL,s + (−1)k ϑL,−s , where s is as above, are linearly independent. Thus E k,L,s are linearly independent. Remark 3.3.15. For even k, the dimension formula (3.19) can also be found in [Bru02, p.26]. Definition 3.3.16 (Notation). For each positive integer F , we use Pr(F) to denote the set of all primitive Dirichlet characters mod F . We now state our main results of this section Definition 3.3.17. Let k ∈ N such that k > E k,L,x,ξ :=

X d ∈Z× N

rk(L) 2

+ 2. We set

ξ(d)E k,L,dx . x

Here x ∈ R Iso where R Iso is a set of representatives for the orbits in the orbit – space Iso(D L ) Z×lev(L) , N x is the smallest positive integer such that N x x ∈ L, and ξ is a primitive Dirichlet character modulo F with F | N x and ξ(−1) = (−1)k . Theorem 3.3.18. In the same notations as in Definition 3.3.17, the series E k,L,x,ξ , where x runs through the set R Iso and ξ runs through all primitive Dirichlet characters mod F with F | N x such that ξ(−1) = (−1)k , form a basis Eis of Hecke eigenforms of Jk,L . More precisely, for all ` ∈ N with gcd(`, lev(L)) = 1, one has (3.20) T(`)E k,L,x,ξ = λ(`, E k,L,x,ξ ) E k,L,x,ξ , where λ(`, E k,L,x,ξ ) :=

8 ξ,ξ > :ξ(`) σ rk(L)

k−

2

−1

(`2 )

if rk(L) is odd, if rk(L) is even. ξ,χ

Here, for any two Dirichlet characters ξ and χ we set σk (`) := 78

P

` d |` ξ d

¡ ¢

χ(d)d k .

3.3. Jacobi-Eisenstein Series and Hecke Operators Remark 3.3.19. Note that Theorem 3.3.18 is in complete accordance with Remark 2.7.16 and Remark 2.7.10. More precisely, let N, h be two positive integers where h ≥ 3. For any two Dirichlet characters ξ modulo u and χ modulo v such that uv = N and ξχ(−1) = (−1)h and ξ is primitive, we consider the Eisenstein series ∞ ξ,χ

E h (τ) =

ξ,χ

X

n=0

σh−1 (n)q n ,

ξ,χ σh−1 (0) = 12 L(1 − h, χ)

where or = 0 according to whether u = 1 (ξ is the trivial ξ,χ character modulo 1) or not. It is known that E h ∈ M h (Γ0 (N), ξχ) (see e.g. [DS06, Theorem 4.5.1]). Let k, x, ξ as in Theorem 3.3.18. We set S (E k,L,x,ξ )(τ) :=

X `∈NL

λ(`, E k,L,x,ξ )e (`τ) .

Then, for odd rk(L), one has ξ,ξ

S (E k,L,x,ξ ) =E 2k−rk(L)−1 ⊗ ϕ,

where ϕ(·) :=

‡

lev(L)2

·

·

. Also, for even rk(L), it is obvious that

ξ(`)λ(`, E k,L,x,ξ ) = the `2 -th coefficient of E

ξ,χL

k−

rk(L) 2

.

To prove Theorem 3.3.18 we need the following lemma: Lemma 3.3.20. In the same notations as in Definition 3.3.17, one has X

X

x∈R Iso F | N x

fl n o fl Eis # ξ ∈ Pr(F) fl ξ(−1) = (−1)k = dim Jk,L .

(3.21)

Proof. First, we assume that k is even. If −1 ∈ StabZ×lev(L) (x), i.e. if Orb(x) consists of elements s such that 2s ∈ L, then X F |Nx

fl n o fl # ξ ∈ Pr(F) fl ξ(−1) = (−1)k = ϕ(N x ) = | Orb(x)|.

If −1 6∈ StabZ×lev(L) (x) ( N x 6= 1, 2), then half of the characters are even and half of them are odd. Thus X F |Nx

fl n o fl # ξ ∈ Pr(F) fl ξ(−1) = (−1)k = 12 ϕ(N x ) = 21 | Orb(x)|.

79

3. Basis of Simultaneous Eigenforms We therefore conclude by the Orbit-Stabilizer theorem that X

X

x∈R Iso F | N x

fl n o X X fl | Orb(x)| | Orb(x)| + # ξ ∈ Pr(F) fl ξ(−1) = (−1)k = 12 x∈R Iso 2x∈L

x∈R Iso 2x6∈L

n o n o ¡ ¢ flfl ¡ ¢ flfl = 21 # x ∈ Iso D L fl 2x 6∈ L + # x ∈ Iso D L fl 2x ∈ L ‡ n o· ¡ ¢ ¡ ¢ flfl = 21 | Iso D L | + (−1)k # x ∈ Iso D L fl 2x ∈ L Eis . = dim Jk,L

Now, we assume that k is odd. One has ξ(−1) = −1. Since there is no odd characters modulo 1 or 2, and N x 6∈ {1, 2} (2x 6∈ L). This gives X F |Nx

fl n o fl # ξ ∈ Pr(F) fl ξ(−1) = (−1)k = 21 ϕ(N x ) = 12 | Orb(x)|.

We therefore conclude X

X

x∈R Iso F | N x

fl n o X fl # ξ ∈ Pr(F) fl ξ(−1) = (−1)k = 12 | Orb(x)| x∈R Iso 2x6∈L

n o ¡ ¢ flfl = 21 # x ∈ Iso D L fl 2x 6∈ L ‡ n o· ¡ ¢ ¡ ¢ flfl = 21 | Iso D L | + (−1)k # x ∈ Iso D L fl 2x ∈ L Eis . = dim Jk,L

Now, the proof is complete. Proof of Theorem 3.3.18. From Lemma 3.3.20 we have X

X

x∈R Iso F | N x

fl n o fl Eis # ξ ∈ Pr(F) fl ξ(−1) = (−1)k = dim Jk,L .

Eis To prove that the set of series E k,L,x,ξ x,ξ is a basis for Jk,L we still need to show that they are linearly independent. Recall that each Jacobi-Eisenstein series E k,L,x,ξ is determined by its singular-term, which is given by

'

“

X ¢ ¡ singular-term E k,L,x,ξ = ξ(d)ϑL,dx , d ∈Z× N

x

or equivalently by its Fourier coefficients C E k,L,x,ξ (0, s) (β(s) ≡ 0 mod Z), which are given by X C E k,L,x,ξ (0, s) = ξ(d)C E k,L,dx (0, s) = Ξ x,ξ (s), (3.22) d ∈Z× N

80

x

3.3. Jacobi-Eisenstein Series and Hecke Operators where 8 1.

Moreover, St commutes with Hecke operators, i.e., St (T(p2 ) f ) = T(p)St ( f ). Remark 4.1.2. In [Shi73] Shimura proved the cited theorem apart from the fact that St ( f ) has a level 2N , which only conjectured. The latter was proven by Niwa in [Niw75, 3] and later by Cipra in [Cip83]. We know that every Jacobi form of lattice index has a theta expansion which implies, for odd rank index, a connection to half integral weight modular forms. We can try to use the Shimura correspondence for half integral weight forms to map Jacobi forms of lattice index to modular forms of integral weight. Definition 4.1.3. Let L be of odd rank, and k be a positive integer. For φ ∈ S k,L , x ∈ L# , and D 0 ∈ Q such that D 0 ≡ β(x) mod Z, set S D 0 ,x (φ) := ξ S D ,x (φ) := 0

with ξ(·) =

‡

(−1)k N x 2

∞ ‡X X

a k−d

rk(L) 2 e−1

`=1 a|`

X

χL (D 0 , a)C φ

¢· ` D , x e (`τ) . 0 a a2

¡ `2

¡ ¢ ξ(s) S D 0 ,sx (φ) ⊗ ξ ,

s mod N x D 0 ≡β(sx) mod Z

·

where N x is the order of x + L ∈ L# /L ( N x x ∈ L), and S D 0 ,sx (φ) ⊗ ξ denotes the function obtained from S D 0 ,sx (φ) by multiplying its n-th Fourier coefficient by ξ(n). ·

Theorem 4.1.4. Let the notations be as in Definition 4.1.3. Assume that ξ 2k − rk(L) − 1 ≥ 2, and N x 2 D 0 is a square free negative integer. Then S D ,x (φ) 0 84

4.1. Lifting via Shimura Correspondence for Half Integral Weight is an elliptic modular form of weight 2k − 1 − rk(L) on Γ0 (lev(L)N x 2 /2), and, in fact, a cusp form if 2k − 1 − rk(L) > 2. Moreover, one has ξ

T(p)S D

ξ

0 ,x

(φ) = ξ(p)S D

0 ,x

(4.3)

(T(p)φ)

for all primes p with gcd(p, lev(L)) = 1. Proof. By Proposition 2.4.7, φ can be written as a sum X

φ(τ, z) =

h r (τ)ϑL,r (τ, z),

r ∈L# /L

where X

h r (τ) =

C φ (D, r) e (−D τ) .

D ∈Q (D,r)∈supp(L)

For x ∈ L# , we set (4.4)

ξ(s)h sx (N x 2 τ).

X

Fφ (τ) :=

s mod N x

According to Proposition 2.4.7, which gives a closed formula for the Fourier expansion of h sx , we see that X‡

Fφ (τ) =

D ∈Q

Let A =

¡a b¢ c d

· ¡ ¢ ξ(s)C φ (D, sx) e −DN x 2 τ .

X

(4.5)

s mod N x (D,sx)∈supp(L)

∈ Γ0 (lev(L)N x 2 ). One has

j(A, τ)−2(k−

rk(L) 2 )

Fφ (A τ) = j(A, τ)−2(k−

rk(L) 2 )

X s mod N x

= j(A, τ)

rk(L) −2(k− 2 )

X

¡ ¢ ξ(s)h sx N x 2 acττ++db ¡ ¢ ξ(s)h sx BN x 2 τ ,

s mod N x

where B := j(A, τ)

‡

a Nx 2 b c/N x 2 d

−2(k−

rk(L) 2 )

·

∈ Γ0 (lev(L)). It is obvious that j(A, τ) = j(B, N x 2 τ). Thus

Fφ (A τ) =

X

ξ(s)

(−1)k 2 det(L) d

s mod N x 2

=

N x 2 det(L) d

¶

X s mod N x

¶

h asx (N x 2 τ) 2

ξ(as)h asx (N x τ) =

2 det(L) d

¶

Fφ (τ),

where the first identity follows from Theorem 2.3.4 and Proposition 2.3.1, and the second identity using that ad ≡ 1 mod 4. Thus Fφ defines an element 85

4. Lifting to Elliptic Modular Forms in S k− rk(L) lev(L)N x 2 , ψ , where ψ(·) = ¡

¢

‡2 det(L)·

2

λ = k−d

·

(a quadratic character). Set

t = − N x 2 D 0 , and ψ(tλ) (m) = ψ(m)

rk(L) 2 e,

¡−1¢λ ¡ t ¢ m

m

= ξ(m)χL (D 0 , m). One

has ξ

SD

0 ,x

X

=

X

(φ)(τ) =

¡ ¢ ξ(s) S D 0 ,sx (φ) ⊗ ξ (τ)

s mod N x (D 0 ,sx)∈supp(L) ‡X ∞ X

ξ(s)

ξ(`)

a k−d

rk(L) 2 e−1

χL (D 0 , a)C φ

¢· ` D , sx e (`τ) a2 0 a

¡ `2

`=1 a|` s mod N x (D 0 ,sx)∈supp(L) ∞ ‡X X X rk(L) ¡ 2 ¢· ξ( a` s)C φ a`2 D 0 , a` sx e (`τ) . = a k−d 2 e−1 ψ(tλ) (a) `=1 a|` s mod N x (D 0 ,sx)∈supp(L)

Now, inserting the Fourier expansion of the Fφ as given in Equation (4.5) gives ξ

SD

0

(φ)(τ) = ,x

∞ ‡X X `=1 a|`

· 2 aλ−1 ψ(tλ) (a)c Fφ (t a`2 ) e (`τ) = S t (Fφ )(τ).

Thus by Shimura correspondence (see Theorem 4.1.1), we obtain 8 < M2k−rk(L)−1 (Γ0 (lev(L)N x 2 /2))

if k =

S 2k−rk(L)−1 (Γ0 (lev(L)N x 2 /2))

if k >

ξ S D ,x (φ) ∈ 0 :

rk(L)+3 , 2 rk(L)+3 . 2

Moreover, for each prime integer p ∈ NL ξ

T(p)S D

ξ

0 ,x

(φ) = T(p)S t (Fφ ) = S t (T(p2 )Fφ ) = S t (Fξ(p)(T(p)φ) ) = ξ(p)S D

0 ,x

(T(p)φ)

as stated in the theorem. Remark 4.1.5. In fact, Theorem 4.1.4 supports our expectations. Namely, if S D 0 ,x takes Jk,L to elliptic modular forms on Γ0 (lev(L)/2), then the twisted version S Dξ0 ,x takes Jk,L to elliptic modular forms on Γ0 (N x 2 lev(L)/2), which we proved indeed. If in addition, S D 0 ,x commutes with Hecke operators, we deduce for S Dξ0 ,x the relation (4.3), which is again what we proved. 86

4.2. Lifting via Stable Isomorphisms between Lattices

4.2

Lifting via Stable Isomorphisms between Lattices

For lattices L 1 = (L 1 , β1 ), L 2 = (L 2 , β2 ) over Z we define their orthogonal sum by L 1 ⊕ L 2 := (L 1 ⊕ L 2 , f ), where f (x1 ⊕ x2 , y1 ⊕ y2 )= β1 (x1 , y1 ) + β2 (x2 , y2 ). If G 1 and G 2 are Gram matrices associated to L 1 and L 2 respectively, then the block sum G 1 ⊕ G 2 is a Gram matrix associated to L 1 ⊕ L 2 . Two even lattices L 1 and L 2 are said to be stably isomorphic if and only if there exists even unimodular lattices U1 and U2 such that L 1 ⊕ U1 ∼ = L 2 ⊕ U2 . We shall show in this section that Jacobi form whose index is stably isomorphic to a rank one even lattice lift to elliptic modular forms. For this we need the following theorems: Theorem 4.2.1 ([Nik80, Theorem 1.3.1]). Two even integral lattices L 1 and L 2 are stably isomorphic if and only if their discriminant modules D L 1 and D L 2 are isomorphic. Theorem 4.2.2 ([BS14, Theorem 2.3]). Let L 1 and L 2 be two even positive ∼ =

definite lattices over Z. Assume that j : D L 2 −→ D L 1 is an isomorphism of finite quadratic modules. Then the map Ij : J

k+d

rk(L 2 ) 2

e,L 2

−→ J

k+d

rk(L 1 ) 2

e,L 1

given by φ(τ, z) =

X r ∈L 2 # /L 2

h r (τ)ϑL 2 ,r (τ, z) 7→ I j (φ) =

X r ∈L 2 # /L 2

h r (τ)ϑL 1 , j−1 (r) (τ, z)

is an isomorphism. Remark 4.2.3. In terms of Fourier coefficients we have C I j (φ) (D, s) = C φ (D, j(s)).

(4.6)

Here by abuse of language, we use Cφ (D, j(s)) for Cφ (D, s0 ), where s0 + L = j(s + L). Recall that C φ (D, s) depends only D and on s mod L. 87

4. Lifting to Elliptic Modular Forms It is an interesting question whether the isomorphism of Theorem 4.2.2 commutes with Hecke operators. We shall show that is the case. Theorem 4.2.4. The isomorphism Ij : J

k+d

rk(L 2 ) 2

∼ =

e,L 2

−→ J

k+d

rk(L 1 ) 2

e,L 1

,

which is defined in Theorem 4.2.2, commutes with the Hecke operators T(`), i.e., T(`)I j = I j T(`) (4.7) for all ` ∈ NL . Proof. Let φ be a Jacobi form of index (L 2 , β2 ) and of weight k + d According to Remark 4.2.3 we have the identity

rk(L 2 ) 2 e.

(4.8)

C I j (φ) (D, s) = C φ (D, j(s))

for all (D, s) ∈ supp(L). We will prove Equation (4.7) for Jacobi forms of even rank lattices; the case of odd rank lattices can be verified similarly. One has C I j (T(`)φ) (D, s) =C T(`)φ (D, j(s)) X ¢ ¡ 2 = a k−1 χL (a)C φ a`2 D, à0 j(s) a

=

X a

a k−1 χL (a)C I j (φ)

¡ `2

a2

D, à0 s

¢

=C T(`)(I j (φ)) (D, s).

Here the third identity follows from Equation (4.8). The main result of this section is the following: Theorem 4.2.5. If the lattice L = (L, β) is stably isomorphic to the lattice ¡ ¢ Z, (x, y) 7→ det(L)x y , then there is a Hecke-equivariant isomorphism ¡ ¢− ∼ = Jk,L −→ M2k−1−rk(L) lev(L)/4 ,

(4.9)

where M2k−1−rk(L) lev(L)/4 is the Certain Space inside M2k−1−rk(L) lev(L)/4 which was introduced in [SZ88, 3], and where the "−" sign denotes the sub¡ ¢ space of all f ∈ M2k−1−rk(L) lev(L)/4 such that f | Wlev(L)/4 = −(−1)k/2 f . ¡

88

¢

¡

¢

4.2. Lifting via Stable Isomorphisms between Lattices Proof. Since L = (L, β) is stably isomorphic to the lattice B = Z, (x, y) 7→ ¢ det(L)x y , then according to Theorem 4.2.1 and Theorem 4.2.4, there is an ∼ = isomorphism Jk,L −→ Jk−d rk(L) e+1,B which commutes with the action of the 2 Hecke operators. Now, the result follows by applying the main theorem of [SZ88] on the vector space Jk−d rk(L) e+1,B and the fact that in this case 2 lev(L) = 2 det(L). ¡

89


Chapter 5 Lifting from Elliptic Modular Forms Recall that we use the notation L = (L, β) to denote a positive definite even lattice over Z. We shall use lev(L), rk(L), and det(L) to denote the level, the rank, and the determinant of the lattice L, as defined at section 1.2 respectively. We observed in the previous chapters that the multiplicative properties of Jacobi forms of weight k and index L are similar to the multiplicative rk(L) properties of elliptic modular forms of weight k − 2 if the rank rk(L) is even. In this chapter we will provide some examples of this relation between the two vector spaces. The first obvious example is the case of Jacobi forms whose indexes are unimodular lattices. Let L be unimodular, and φ ∈ Jk,L . The function φ has an expansion φ(τ, z) = h0 (τ)ϑL,0 (τ, z), where h0 (τ) ∈ M k− rk(L) (SL2 (Z)). The Fourier 2 P expansion of h0 is given by h0 (τ) = n≥0 a h0 (n)e (nτ) , with a h0 (n) := Cφ (−n, 0). For every positive integer ` one has C T(`)φ (− n, 0) =

X

a k−

rk(L) 2 −1

Cφ

¡ −`2

a|`2 ,n

a2

¢ n, 0 = a T(`2 )h0 (n).

Namely (T(`)φ)(τ, z) = (T(`2 )h 0 )(τ)ϑL,0 (τ, z)

which support the mentioned expectation. 91

5. Lifting from Elliptic Modular Forms

5.1

Jacobi Forms of Prime Discriminant and Hecke Operators

In this section we assume that the determinant of the lattice L = (L, β) is an odd prime p. It is known from Lemma 1.2.16 that rk(L) is even and rk(L) ¡ ¢ (−1) 2 p ≡ 1 mod 4. The application a 7→ χ(a) := χL (a) = ap defines a Dirichlet character modulo p. According to Theorem 1.2.3 there is an isomorphism ϕ : ∼ 2 = D L → (Z/pZ, x 7→ αpx ), where α ∈ Z with gcd(α, p) = 1. We know from Milgram’s formula that X ¡ ¡ ¢ ¢ p (5.1) e β(x) = p e rk(L)/8 . x∈L# /L

By Lemma 2.1.7, which gives the well-known formula for Gauss sum, the left-hand side of Equation (5.1) is equal to X

p

e β(x) = χ(α)²(p) p, ¡

¢

x∈L# /L

where, as usual, ²(p) =

r‡ · −4 p

. Thus we find that e rk(L)/8 = χ(α)²(p). In fact, ¡

¢

we obtain the following table for rk(L) modulo 8: p ≡ 1 mod 4

p ≡ 3 mod 4

χ(α) = 1

0

2

χ(α) = −1

4

6

Table 5.1: rk(L) modulo 8 For a positive even integer k we set k2 := k − a map

rk(L) 2 .

We shall explicitly construct

¡ ¢ S : M k2 p, χ → Jk,L .

Note that M k2 p, χ = 0 unless k is even. The map S will turn out to be surjective. However, in general, it will not be injective, but we shall see that ¡ ¢ we can restrict S to a natural subspace of M k2 p, χ , invariant under all T(`2 ), so that we obtain an isomorphism. ¡

¢

Definition 5.1.1. For t ∈ {±1} and k ∈ N we shall use M kt p, χ for the subspace of elliptic modular forms f of integral weight k on Γ0 (p) with nebentypus χ ¡

92

¢

5.1. Jacobi Forms of Prime Discriminant and Hecke Operators whose Fourier expansion is of the form (5.2)

a f (n)q n .

X

f (τ) =

n≥0 χ(− n)6=− t

It is known that M k p, χ = M k+1 p, χ M k−1 p, χ (see e.g. [BB03, p.3]). ¡ ¢ Moreover, it is easy to see that the spaces M k±1 p, χ are invariant under the action of the Hecke operators T(`2 ) for all positive integers ` such that (`, p) = 1. ¡

¢

¡

¢L

¡

¢

Theorem 5.1.2. Let k be an even positive integer, and k2 := k − applications f 7→ 21

X A ∈Γ0 (p)\ SL2 (Z)

rk(L) 2 .

The (5.3)

fl Θflk,L A,

where fl Θ(τ, z) := ( f flk2 Wp )(τ)ϑL,0 (τ, z),

and φ 7→ (−1) χ(α) ¡

rk(L) 2

(5.4)

h 0 |k2 Wp , χ(α) ¡

define maps S : M k2 p, χ → Jk,L and Ω : Jk,L → M k2 maps S and Ω are mutually inverse isomorphisms. ¢

¢ p, χ respectively. The

To prove this theorem, we will need the next lemmas. Lemma 5.1.3. The application Ω defined by Equation (5.4) maps Jk,L to ¢ χ(α) ¡ Mk p, χ . 2 Proof. Given φ = x∈L# /L h x ϑL,x ∈ Jk,L then we deduce by Theorem 2.3.4 that ¡ ¢ ¡ ¢ h 0 ∈ M k2 p, χ . Since the Fricke involution is an automorphism of M k2 p, χ , then h0 |k2 Wp also belongs to the same vector space. For showing that Ω(φ) is ¡ ¢ indeed in the right subspace, we write Wp = S 0p 01 . So that P

Ω(φ)(τ) = (−1)

rk(L) 2

‡

h 0 |k2 S |k2

¡ p 0¢· (τ). 0 1

Applying the transformation laws from Proposition 2.4.9 and Proposition 2.2.8, we obtain Ω(φ)(τ) = i

−rk(L) 2

p(k2 −1)/2

X

h x (pτ).

x∈L# /L

93

5. Lifting from Elliptic Modular Forms Now, inserting the Fourier expansion of the h x as given in Proposition 2.4.7 gives X X −rk(L) Ω(φ)(τ) = i 2 p(k2 −1)/2 C φ (− n/p, x)e (nτ) . n≥0

x∈L# /L − n/p≡β(x) mod Z

Then the n-th Fourier expansion of Ω(φ) is 0 unless there exists an x ∈ L# such that − n/p ≡ β(x) mod Z. But this condition equivalent, in view of 2 the isomorphism D L ∼ = (Z/pZ, t 7→ αpt ), to the exists of t ∈ Z such that − n ≡ ‡

α t mod p, i.e., it is equivalent to ‡ · ¢ χ(α) ¡ p, χ . − αp . Thus Ω(φ) ∈ M k 2

−n p

·

=

α p

‡ ·

or p | n, i.e. equivalent to

‡

−n p

·

6=

2

Lemma 5.1.4. The application S defined by Equation (5.3) is well-defined. (i.e. the sum on the right does not depend on the choice of the representative A .) Proof. For G =

¡a b¢ c d

∈ Γ0 (p), one has fl fl fl Θflk,L G = ( f flk2 Wp G) ϑL,0 fl rk(L) G. 2

Now By Theorem 2.3.4 and Proposition 2.3.1 we have fl ϑL,0 fl rk(L) G = χ(d)ϑL,0 . 2

fl fl fl ¡ ¢ Moreover, f flk2 Wp G = χ(d) f flk2 Wp since f flk2 Wp is in M k2 p, χ (see preceding

proof.). Now, the identity χ(d)2 = 1 completes the proof. rk(L)

Proposition 5.1.5. Let k be an even positive integer, k2 := k − 2 , and f = ¡ ¢ ¢ P χ(α) ¡ p, χ then n∈N a f (n)e (nτ) ∈ M k 2 p, χ . Then S ( f ) ∈ Jk,L . Moreover, if f ∈ M k 2 S ( f )(τ, z) =

X

h x, f (τ)ϑL,x (τ, z),

x∈L# /L

with h x, f (τ) =

X

C S ( f ) (D, x) e (−D τ) ,

D ∈Q,D ≤0 D ≡β(x) mod Z

where

rk(L)

1− k 2

i 2 p 2 a f (− pD). C S ( f ) (D, x) = 1 + δ(x 6∈ L)

94

(5.5)

5.1. Jacobi Forms of Prime Discriminant and Hecke Operators Proof. First, we want to show that S ( f ) is a Jacobi form of weight k and index L. Any B ∈ SL2 (Z) permutes Γ0 (p)\ SL2 (Z) by right multiplication. That is, the map ϕ : Γ0 (p)\ SL2 (Z) → Γ0 (p)\ SL2 (Z) given by Γ0 (p)A 7→ Γ0 (p)AB

is well defined and bijective. So if { A i } is a set of representatives for Γ0 (p)\ SL2 (Z) then { A i B} is a set of representatives for Γ0 (p)\ SL2 (Z) as well. Thus fl S ( f )flk,L B = 12

X AB∈Γ0 (p)\ SL2 (Z)

fl Θflk,L AB = S ( f ).

Let (λ, µ, 1) ∈ H L (Z), the identity ¡ ¢ ϑL,0 (τ, z + λτ + µ)e τβ(λ) + β(λ, z) = ϑL,0 (τ, z)

implies that S ( f )flk,L (λ, µ, 1) = S ( f ). Therefore S ( f ) transforms like a Jacobi form of weight k and index L = (L, β). Now, we want to prove the claimed expansion, which also implies the holomorphicity of S ( f ) at infinity. As a set of representatives for Γ0 (p)\ SL2 (Z) we take the elements fl

ˆ

1 0

!

0 1

j

ST =

,

ˆ ! 0 −1

j

1

for 0 ≤ j < p.

Therefore 2S (f ) = Θ+

X ¡ fl ¢¡ ¢ f flk2 Wp ST j ϑL,0 | rk(L) ST j . 2

j mod p

By applying the transformation ‡ · laws of the Jacobi theta series (see Theo−1 − j j rem 2.3.3) and WP ST = 0 − p , we obtain 2 S ( f )(τ, z) = Θ(τ, z) + i

rk(L) 2

p−

k2 1 2 −2

X j mod p

= Θ(τ, z) + i

rk(L) 2

p

1− k 2 2

X x∈L# /L

f

¡ τ+ j ¢ X p

ϑL,x (τ, z)

¡ ¢ ϑL,x (τ, z)e j β(x)

x∈L# /L

X n≥0 − n/p≡β(x) mod Z

a f (n)e

‡

nτ p

·

,

where, in the second identity, we used X j mod p

‡

·

e ( np + β(x)) j = p δ(− np ≡ β(x) mod Z). 95

5. Lifting from Elliptic Modular Forms χ(α) ¡

If f ∈ M k2

¢ p, χ then by [Kri91, P.672] one has ¡

1− k 2 rk(L) ¢ f |k2 Wp (τ) = i 2 p 2

X

a f (n)e

‡

n∈N n≡0 mod p

nτ p

· .

(5.6)

Inserting this into the last formula for S ( f ) gives rk(L)

S ( f )(τ, z) =

X

X

ϑL,x (τ, z)

n≥0 − n/p≡β(x) mod Z

x∈L# /L

1− k 2

‡ · i 2 p 2 a f (n)e npτ . 1 + δ(x 6∈ L)

Now, the proof is complete. χ(α) ¡

Proof of Theorem 5.1.2. Let f ∈ M k2 sion X

¢ p, χ , and let φ ∈ Jk,L with an expan-

h x (τ)ϑL,x (τ, z).

x∈L# /L

By virtue of Lemma 5.1.3 and Proposition 5.1.5 we have only to prove that S (Ω(φ)) = φ

and

Ω(S ( f )) = f .

By Equation (5.3) and Equation (5.4) we have S (Ω(φ)) = (−1)

rk(L) 2

S (h 0 |k2 Wp ) =

X x∈L# /L

h x,h0 |k2 Wp (τ)ϑL,x (τ, z),

where rk(L)

h x,h0 |k2 Wp (τ) = (−1) =

1 1 + δ(x 6∈ L)

rk(L) 2

p

1− k 2 2

X

i 2 1 + δ(x 6∈ L) X

X n≥0 − n/p≡β(x) mod Z

a h0 |k2 Wp (n)e (nτ/p)

C φ (− n/p, y)e (nτ/p).

n≥0 y∈L# /L − n/p≡β(x) mod Z − n/p≡β(y) mod Z

Since det(L) = lev(L) = p is an odd prime and k is even, the condition β(x) ≡ β(y) mod Z is equivalent to C φ (− n/p, y) = C φ (− n/p, x). Thus h x,h0 |k2 Wp (τ) =

#{ y ∈ L# /L | β(y) ≡ β(x) mod Z} hx = hx. 1 + δ(x 6∈ L)

Thus the claimed identity S (Ω(φ)) = φ holds true. Also the second claimed identity Ω(S ( f )) = f holds true since Ω(S ( f )) = (−1)


rk(L) 2

h 0, f |k2 Wp = (−1)

rk(L) 2

f |k2 Wp 2 = f .

5.1. Jacobi Forms of Prime Discriminant and Hecke Operators χ(α) ¡

Theorem 5.1.6. The isomorphism S : M k2 p, χ → Jk,L , which is defined in Theorem 5.1.2, commutes with the action of the Hecke operators. More ¢ χ(α) ¡ precisely, for each f ∈ M k2 p, χ and ` ∈ NL , one has ¢

¡ ¢ T(`) (S ( f )) = S T(`2 ) f .

Proof. Using the explicit action of the operator T(`) on Fourier coefficients (see Theorem 2.6.3, one has X

C T(`)S ( f ) (D, x) =

a k2 −1 χL (a)C S ( f )

¡ `2

a|`2 ,pD

a2

¢ D, à0 x ,

where a0 ∈ Z with aa0 ≡ 1 mod p, and rk(L)

CS ( f )

1− k 2

i 2 p 2 2 D, ` a x = a f (− a`2 pD) 2 0 a 1 + δ(à x 6∈ L)

¡ `2

0

¢

(see Proposition 5.1.5).

Since (`, p) = 1, δ(à0 x 6∈ L) = δ(x 6∈ L). This gives rk(L)

1− k 2

X i 2 p 2 2 C T(`)S ( f ) (D, x) = a k2 −1 χL (a)a f (− a`2 pD). 1 + δ(x 6∈ L) a|`2 ,pD

Now by using Equation (1.7)), which gives a closed formula for the action of Hecke operators on modular forms of integral weights in terms of Fourier coefficients, we obtain rk(L)

1− k 2

i 2 p 2 a 2 (− pD) = C S (T(`2 ) f ) (D, x) C T(`)S ( f ) (D, x) = 1 + δ(x 6∈ L) T(` ) f

as claimed in the theorem. Example 5.1.7. Consider the Gram matrix F = 21 12 . One has det F = 3. ¡ ¢ The application a 7→ χ(a) := a3 defines a Dirichlet Character mod 3. Let L = (Z2 , (x, y) 7→ x t F y). One has D L ∼ = (Z3 , x 7→ α x2 /3), where α ∈ Z with (α, 3) = 1. By Table 5.1 we have χ(α) = +1 (since rk(L) = 2). The vector space M3 (3, χ) is two dimensional as can be easily seen by running the following Sage session ¡

¢

Listing 5.1: Sage input e = DirichletGroup(3, RationalField()).gen() M=ModularForms(e,3,prec=20);print M

97

5. Lifting from Elliptic Modular Forms Listing 5.2: Sage output Modular Forms space of dimension 2, character [-1] and weight 3 over Rational Field 2

2

The theta function θ (τ) := ϑL,0 (τ, 0) = x,y∈Z q x +x y+ y is an element of M1 (3, χ) ¡ ¢ (see Proposition 2.3.1 and Theorem 2.3.4). Clearly A := θ 3 ∈ M3 3, χ . Also, by using modular derivative ( see e.g. [Lan76, Sec. 10.5]) we see that ¡ ¢ P d θ ∈ M3 3, χ (E 2 (τ) = 1 − 24 n≥1 σ1 (n)q n ). The functions A and B := θ E 2 − 12q dq ¡ ¢ B are linearly independent. Since M3 3, χ is two-dimensional (see Listing 5.1 and Listing 5.2), A and B form a basis. The first Fourier coefficients of A and B are P

A =1 + 18q + 108q2 + 234q3 + 234q4 + 864q5 + 756q6 + 900q7 + + 1836q8 + 2178q9 + O(q10 ),

B =1 − 90q − 216q2 − 738q3 − 1170q4 − 1728q5 − 2160q6 − 4500q7 − 3672q8 − 6570q9 + O(q10 ),

which can be computed using the following Sage (a computer algebra system [S+ 11]) code Listing 5.3: Sage input ec = lambda n : sum([d for d in divisors(n) if 1 == gcd(3,d)]) R. = PowerSeriesRing(ZZ) E = 1 + 12 * sum( [ec(n)*q^n for n in range( 1, 11)]) + O(q^10) th = sum( q^(x^2+x*y+y^2) for x in range(-19,20) for y in range (-19,20)) + O(q^10) E2 = 1 - 24*sum( sigma(n)*q^n for n in range( 1, 11)) + O(q^10) # Note E*th == th^3 A = th^3 B = -12*(q*th.derivative() - 1/12 * E2*th) print "A= ",A print "B= ",B

Listing 5.4: Sage output A=

98

1 + 18*q + 108*q^2 + 234*q^3 + 234*q^4 + 864*q^5 + 756*q^6 + 900*q^7 + 1836*q^8 + 2178*q^9 + O(q^10)

5.1. Jacobi Forms of Prime Discriminant and Hecke Operators B=

1 - 90*q - 216*q^2 - 738*q^3 - 1170*q^4 - 1728*q^5 - 2160*q^6 -4500*q^7 - 3672*q^8 - 6570*q^9 + O(q^10)

Next, we would like to find the Hecke eigenfunctions with respect to all Hecke P operators T(`) (` ∈ N with gcd(`, 3) = 1). Let ` = 2. For all f = n≥1 a f (n)q n ∈ ¡ ¢ M3 3, χ one has a T(2) f (n) = a f (2n) − 4a f (n/2)δ(n is even).

(5.7)

Thus we find T(2)A = −3 + 108q + O(q2 ), and T(2)B = −3 − 216q + O(q2 ). The matrix M(2) ¶of the action of T(2) satisfies (T(2)A, T(2)B) = (A, B)M(2), i.e., M(2) =

− 32 − 29 − 32

3 2

. The eigenvalues of M(2) are {3, −3}. The normalized eigen-

functions are −B e3 := A108 = q + 3q2 + 9q3 + 13q4 + 24q5 + 27q6 + 50q7 + 51q8 + O(q9 ),

e−3 := 3A2+B = 1 − 9q + 27q2 − 9q3 − 117q4 + 216q5 + 27q6 − 450q7 + 459q8 O(q9 ). Of course, by Equation (5.7) for the Hecke action we deduce that e−3 = 1 − 9

X ¡X

¢ d 2 χ(d) q n ,

e3 =

X ¡X

¢ d 2 χ(n/d) q n .

n≥1 d | n

n≥1 d | n

It is obvious that f + :=9e3 + e−3 = 1 + 9

X ‡X

· d 2 (χ(n/d) − χ(d)) q n

n≥1 d | n

=1 + 54q2 + 72q3 + 432q5 + 270q6 + 918q8 + 720q9 + O(q10 )

is in the subspace M3+1 3, χ , and ¡

¢

−

f :=9e3 − e−3 = −1 + 18

X ‡X

·

d (χ(n/d) + χ(d)) q n 2

n≥1 d | n

= − 1 + 18q + 90q3 + 234q4 + 216q6 + 900q7 + 738q9 + O(q10 )

is in the subspace M3−1 3, χ . Since M3 3, χ = M3+1 3, χ ⊕ M3−1 3, χ is two dimensional, we observe that f ± are Hecke eigenforms for all Hecke operators T(`) (gcd(`, 3) = 1). The corresponding eigenvalue λ(`, f + ) of f + equals P 2 a|` χ(a)a (using Proposition 1.1.11). Now we construct the Jacobi form ¡

¢

¡

¢

¡

¢

¡

¢

99

5. Lifting from Elliptic Modular Forms S ( f + ) ∈ J4,L using Proposition 5.1.5. Note that dim J4,L = 1. Thus S ( f + ) is a

Hecke eigenfunction for all Hecke operators T(`) (gcd(`, 3) = 1) with Hecke eigenvalues λ(`, S ( f + )) given by the equation λ(`, S ( f + ))C S ( f + ) (D, 0) =

X

a2 χL (a)C S ( f + )

¡ `2

a|`2

a2

¢ D, 0 .

If we choose D = 0, then CS ( f + ) (0, 0) = constant · a f + (0) 6= 0 and λ(`, S ( f + )) =

X

a2 χL (a) = λ(`2 , f + )

a|`2

as claimed.

5.2

Operators on the Vector-Valued Components

In this section we recall the definition of the vector-valued modular forms associated with Weil representation. Then we define Hecke operators T(`) on these forms. Next, we relate these operators to the ones which developed in [BS07]. Again, we shall use L = (L, β) to denote an even positive definite lattice over Z (see section 1.2), and ρ L for the Weil representation ρ L : SL2 (Z) → GL(C[L# /L]) associated with L (see section 2.3). Definition 5.2.1. A vector-valued function h : H → C[L# /L] is a vector-valued modular form on SL2 (Z) of weight k ∈ 21 Z and type ρ L if e for all A e ∈ SL2 (Z). 1. h |k Ae = ρ L ( A)h

2. f is holomorphic on H and at the cusp ∞. We shall use M k (ρ L ) to denote the vector space of all such vector-valued modular forms. Let h ∈ M k (ρ L ). We denote the components of h by h x , so that h=

X x∈L# /L

100

h x δx .

5.2. Operators on the Vector-Valued Components The invariance of h under |k ( 10 11 , 1) implies that e β(x) h x are periodic with period 1. Thus each h x has a Fourier expansion ¡

hx =

¢

¡

X

¢

(5.8)

C h (D, x) e (−D τ) .

D ∈Q (D,x)∈supp(L)

Recall that the vector-valued Jacobi theta series Θ=

X

ϑL,x δ x

x∈L# /L

is a vector-valued modular form of weight rem 2.3.3).

rk(L) 2

and type ρ L (see Theo-

For a positive integer k it is well-known from [BS14] that M k− rk(L) (ρ L ) is 2 isomorphic to Jk,L as C-linear spaces via the correspondence J · | ΘK : M k− rk(L) (ρ L ) → Jk,L 2

given by

h 7→ J h | ΘK.

(5.9)

Definition 5.2.2. Let ` ∈ N such that gcd(`, lev(L)) = 1. We define a Hecke operator T(`) acting on the vector space M k− rk(L) (ρ L ) via 2

JT(`)h | ΘK := T(`)J h | ΘK

(h ∈ M k− rk(L) (ρ L )). 2

(5.10)

It is obvious from the definition that for all pairs (D, x) ∈ supp(L) one has C T(`)h (D, x) = C T(`)Jh

| ΘK (D,

x) ,

(5.11)

where the Fourier coefficients C T(`)Jh | ΘK (D, x) of the Jacobi form T(`)J h | ΘK are given by Theorem 2.6.3 for even rank lattice, and by Theorem 2.6.1 for odd rank lattice. Moreover, the operator T(`) on M k− rk(L) (ρ L ) has a multiplicative 2 property as in Theorem 2.7.11 for odd rk(L) and as in Theorem 2.7.4 for even rk(L). The main result of [BS07] is a well-defined double coset operator acting on the space of vector-valued modular forms of weight k and type ρ L . In the next lines we will recall their construction briefly and then we shall show the relation between our Hecke operators T(`) and those operators. 101

5. Lifting from Elliptic Modular Forms In the rest of this section we assume that rk(L) is even. It is well known that ρ L factor through SL2 (Z). Moreover, it is trivial on the subgroup Γ(lev(L)). Recall that the map ϕ : SL2 (Z)/Γ(lev(L)) → SL2 (Zlev(L) ) given by A Γ(lev(L)) 7→ A mod lev(L)

is an isomorphism of groups. Therefore, ρ L factor through the finite group SL2 (Zlev(L) ). Let N be a positive integer. We set n o Q(N) = (M, r) ∈ GL2 (Z N ) × Z N × ; det(M) ≡ r 2 mod N .

with a product defined component-wise. For (M, r) ∈ Q(N) the assignment ¡ ¢−1 (M, r) 7→ (M 0r 0r , r) defines an isomorphism Q(N) ∼ = SL2 (Z N ) × Z N × . Proposition 5.2.3. Let (M, r) ∈ Q(lev(L)). Then (M, r) acts on C[D L ] by Weil representation ρ L as follows: ‡ ¡ ¢−1 · ρ L ((M, r)) · δγ = χL (r)ρ L ϕ−1 (M 0r 0r ) · δγ

for all γ ∈ D L . Proof. See [BS07, (3.5)] and [BS07, Proposition 3.3]. Definition 5.2.4 ([BS07, Definition 4.1]). Consider the Hecke pair (Q(lev(L)), Γ) in sense [Shi71]. For each (M, r) ∈ Q(lev(L)), the corresponding double cosets decompose into a finite union of left cosets Γ(M, r)Γ =

[ γ∈Γ\Γ M Γ

Γ(γ, r).

We define the corresponding double coset Hecke operator H (M, r) that is acting on the vector space of vector-valued modular forms Mk (ρ L ) by fl hflH (M, r) =

X γ∈Γ\Γ M Γ

fl 1 ρ L (γ, r)−1 hflk,L det(γ)− 2 γ.

(5.12)

Remark 5.2.5. The operator H (M, r) is exactly the operator T(M, r) in [BS07, Definition 4.1] but without the normalization term. Now we are ready to introduce our result: 102

5.2. Operators on the Vector-Valued Components Theorem 5.2.6. Let L = (L, β) be an even positive definite lattice of even rank. Let ` ∈ N such that (`, lev(L)) = 1. For h ∈ M k (ρ L ) one has T(`)h = `

k−2

X

X

`0 |` s|`0 `/`0 = s is square-free

fl ‡‡ 0 2 · 0 · χL (s) h fl H (` 0/s) 01 , ` /s .

(5.13)

Proof. We omit the proof, which is similar to the proof of Theorem 2.6.3. Remark 5.2.7. Here, in the last part of this section, we restricted ourselves to lattices of even rank. A similar treatment for lattices of odd rank can be done by replacing SL2 (Z) with its metaphoric cover SL2 (Z), the double coset Hecke operator H with the one which is defined in [BS07, 4.21], and χL with the character which is defined in [McG03, Lemma 4.5].

103


Chapter 6 Appendix 6.1

Examples Using the Method of Theta Blocks

The Fourier expansion of theta blocks can be computed from the Dedekind eta function X ‡24· n2 1 Y n 24 η(τ) := q (1 − q ) = q , q = e2π iτ (6.1) n∈N

n∈Z

n

which is a modular form of weight 12 with a multiplier system on SL2 (Z), and the Jacobi theta function (a Jacobi form of weight and index 21 ) ϑ(τ, z) =

X ‡−4· r ∈Z

r

r2

r

(6.2)

q 8 ζ2 .

Theorem 6.1.1 (The construction method [GSZ]). Let α : L → Zm be an isometric embedding into the m-fold orthogonal sum of Z. Then the function (6.3)

ϑ(τ, α1 (z))ϑ(τ, α2 (z)) . . . ϑ(τ, αm (z))η(τ) t

defines an element of Jk,L , where α i is the i -th coordinate function of α, t ∈ N such that t + 3m ≡ 0 (mod 24), and k = t+2m . Gram Matrix ¡2 1¢ 12

Jacobi form

weight 15

ϑ(τ, z2 )ϑ(τ, z1 )ϑ(τ, z1 + z2 )η(τ)

105

9

6. Appendix

¡2 1¢

14 1 2 0 0 B C @ 0 2 0 A 0 0 2

ϑ(τ, z2 )ϑ(τ, z2 )ϑ(τ, z2 )ϑ(τ, z1 )ϑ(τ, z1 + z2 )η(τ)9

7

ϑ(τ, z3 )ϑ(τ, z3 )ϑ(τ, z1 + z2 )ϑ(τ, z1 − z2 )η(τ)12

8

0

Table 6.1: Examples of Jacobi forms using the method of Theta Blocks Note that in the first row of Table 6.1 we used the embedding (z1 , z2 ) 7→ (z1 , z2 , z1 + z2 ), in the second row we used the embedding (z1 , z2 ) 7→ (z1 , z2 , z2 , z2 , z1 + z2 ), in the third we used the embedding (z1 , z2 , z3 ) 7→ (z3 , z3 , z1 + z2 , z1 − z2 ). m Lemma 6.1.2. The maximal even sub-lattice Zev in Zm is m Zev

n

= (x1 , x2 , . . . , xm ) |

X

x i ∈ 2Z

o

(m ≥ 1).

(6.4)

i m m is even. Assuming that Zev is not the maximal Proof. We observe that Zev m even sub-lattice in Zm ,i.e., there exists an even lattice such that Zev ⊂ L ⊆ Zm . P m Let z = (z1 , · · · , z m ) be an element in L which is not in Zev , i.e., i z i is odd. Then one has z t z = z12 + · · · + z2m is odd. This contradicts the assumption that L is even. Hence, the assumption is false and the lemma is true. m# m m m# One has|Zev /Zev | = 4 (since Zev ⊆index 2 Zm = Zm # ⊆index 2 Zev ) . As a set of m# m coset representatives of Zev /Zev we take the set

n ∆ = [0] = (0, 0, · · · , 0),

[1] = ( 12 , 12 , · · · , − 21 ), [2] = (0, 0, · · · , 1), o [3] = ( 12 , 12 , · · · , 21 ) . t

# x x m = (Z m /Z m ,Q : x + Z m 7→ We have D Zev ev ev ev 2 mod Z). Let t ∈ N such that t + 3m ≡ 0 (mod 24). Set k := t+2m . According to Theorem 6.1.1, the product t m (τ, (z1 , · · · , z m )) := ϑ(τ, z1 )ϑ(τ, z2 ) . . . ϑ(τ, z m )η(τ) Θk,Zev

(6.5)

m . It is easy to see that the function Θ m given by Equation belongs to Jk,Zev k,Zev (6.5) has an expansion of the form m (τ, z) = Θk,Zev

106

X x=(x1 ,··· ,xm )∈ 12 Zm

¡ ¢ h x (τ)e β(x, z) + β(x)τ ,

6.1. Examples Using the Method of Theta Blocks where h x (τ) =

‡

−4

(2x1 )···(2xm )

· η(τ) t .

m is a cusp form (since h x equal 0 or ±η t , which are all cusp Note that Θk,Zev forms).

For odd m we observe that m [0] ≡0 · [1] mod Zev , m [1] ≡1 · [1] mod Zev , m [2] ≡2 · [1] mod Zev , m [3] ≡3 · [1] mod Zev . m# m m /Zev =< [1] >∼ Thus Zev 7→ x + 4Z). Moreover, if we endow Z4 = Z4 (By ϕ : [x] + Zev mx2 with the quadratic form Q : x + 4Z 7→ 8 mod Z, then Q ◦ ϕ = Q , i.e., m ∼ D Zev = (Z4 , x + 4Z 7→

mx2 mod Z) 8

(6.6)

as finite quadratic modules. We consider for each of these odd m (m = 1, 3, 5, 7) the smallest t ∈ N such that t ≡ −3m mod 24 m

1

3

5

7

t

21

15

9

3

11

9

7

5

2k − m − 1

20

14

8

2

dim M2k−m−1 (Γ0 (2))

6

4

3

1

dim S 2k−m−1 (Γ0 (2))

4

2

1

0

new dim S 2k (Γ (2)) − m−1 0

2

2

1

0

1

1

1

1

k=

t+ m 2

m dim Jk,Zev

Table 6.2 m ∼ Note that the last line in Table 6.2 is obtained by using Jk,Zev = M k− m2 (Z4 , x + ¢ 2 4Z 7→ − mx 8 mod Z) (using also Equation (6.6)) and the dimension formula for the vector-valued modular form as in [ES95, p.12]. (For m = 7 we need an additional argument, e.g. dim J9,Z7ev = 1 and dim J5,Z7ev · E 4 ⊆ J9,Z7ev ). Note also

¡

107

6. Appendix that Θ5,Z7ev is a cusp form, but its h x equal 0 or 3

h x (8τ) = ±η(8τ) = ±

∞ ‡−4· X

n≥1

n

nq n

2

which are "trivial" cusp forms of weight 3/2 in the sense of [Shi73, Proposition 2.2] and therefore, in the view of [Shi73, (c) p.478] and Remark 2.7.16, Θ5,Z7ev should lift to an Eisenstein series. This is in complete accordance with the fact that M2 (Γ0 (2)) = C(E 2 (τ) − 2E 2 (2τ)). m ( m = 1, 3, 5, 7) are Hecke eigenfuncFrom Table 6.2 we see that all Θk,Zev m ) of T(`): tions. We calculate the first eigenvalues λ(`, Θk,Zev m) Table 6.3: The first eigenvalues λ(`, Θk,Zev

`

λ(`, Θ11,Z1ev )

λ(`, Θ9,Z3ev )

λ(`, Θ7,Z5ev )

λ(`, Θ5,Z7ev )

1

1

1

1

1

3

−53028

−1836

12

4

5

−5556930

3990

−210

6

7

−44496424

−433432

1016

8

9

1649707317

1776573

−2043

13

11

6320674932

1619772

1092

12

13

−33124973098

−10878466

1382

14

15

294672884040

−7325640

−2520

24

17

−722355252174

60569298

14706

18

19

−1312620671860

−243131740

−39940

20

21

2359556371872

795781152

12192

32

23

3379752742152

−606096456

68712

24

25

11805984696775

−1204783025

−34025

31

27

−25848278533800

−334611000

−50760

40

29

−29378097714810

5258639310

−102570

30

31

131976476089952

−1824312928

227552

32

33

−335172750294096

−2973901392

13104

48

35

247263513418320

−1729393680

−213360

48

Continued on next page 108

6.1. Examples Using the Method of Theta Blocks Table 6.3 – continued from previous page `

λ(`, Θ11,Z1ev )

λ(`, Θ9,Z3ev )

λ(`, Θ7,Z5ev )

λ(`, Θ5,Z7ev )

37

−466464103652194

−3005875402

160526

38

39

1756551073440744

19972863576

16584

56

41

1889447681239482

−49704880758

10842

42

43

−4323507451065388

58766693084

−630748

44

45

−9167308081056810

7088526270

429030

78

47

12103384387771536

−42095878032

472656

48

49

−9418963436585367

90974288217

208713

57

51

38305054312282872

−111205231128

176472

72

53

−30593935900444338

−181140755706

−1494018

54

55

−35123548149878760

6462890280

−229320

72

57

69605648987392080

446389874640

−479280

80

59

9908742512283780

206730587820

2640660

60

61

−91638145794467098

−124479015058

827702

62

63

−73406076253134408

−770023588536

−2075688

104

65

184073156757469140

−43405079340

−290220

84

67

−103349440678278244

95665133588

−126004

68

69

−179221528410836256

1112793093216

824544

96

71

285448322456957592

−371436487128

−1414728

72

73

875008267167254042

−1800576064726

980282

74

75

−626047756500584700

2211981633900

−408300

124

77

−281247431740443168

−702061017504

1109472

96

79

−1081394522969090320

1557932091920

−3566800

80

81

−546708732286707639

−2218085399079

3858921

121

83

−665085275193888948

2492790917604

5672892

84

85

4014077571463265820

241671499020

−3088260

108

87

1557861765620944680

−9654861773160

−1230840

120

89

−2020985164277790390

2994235754490

−11951190

90

91

1473942847957201552

4715075275312

1404112

112

93

−6998448574097974656

3349438535808

2730624

128

Continued on next page 109

6. Appendix Table 6.3 – continued from previous page `

λ(`, Θ11,Z1ev )

λ(`, Θ9,Z3ev )

λ(`, Θ7,Z5ev )

λ(`, Θ5,Z7ev )

95

7294141190078989800

−970095642600

8387400

120

97

−12825578365118067934

4382492665058

8682146

98

99

10427263683698877444

2877643201356

−2230956

156

The first column in Table 6.3 ( m = 1) is covered by Theorem 4.2.5, i.e., Θ11,Z1ev lift to elliptic modular form f of weight 20 on Γ0 (2) with f | W2 = − f . To new (Γ0 (2)). Recall determine this f , we use Sage to find the generators of S20 new that the dim S20 (Γ0 (2)) = 2 (see Table 6.2) Listing 6.1: Sage input f,g=Newforms(2,20) print "f=",f.qexp(12) print "g=",g.qexp(12)

Listing 6.2: Sage input f= q - 512*q^2 - 13092*q^3 + 262144*q^4 + 6546750*q^5 ^6 +96674264*q^7 - 134217728*q^8 - 990861003*q^9 q^10 +11799694452*q^11 + O(q^12) g= q + 512*q^2 - 53028*q^3 + 262144*q^4 - 5556930*q^5 q^6 -44496424*q^7 + 134217728*q^8 + 1649707317*q^9 2845148160*q^10 +6320674932*q^11 + O(q^12)

+ 6703104*q 3351936000* - 27150336* -

new According to Theorem 1.1.15, the newform f ∈ S20 (Γ0 (2)) given by (see Listing 6.2)

f (τ) = q + 512q2 − 53028q3 + 262144q4 − 5556930q5 − 27150336q6 − 44496424q7 + 134217728q8 + 1649707317q9 − 2845148160q10 + O(q11 )

is a normalized Hecke eigenfunction. We compute its Atkin-Lehner eigenvalue. Again, using Theorem 1.1.15 one has 20

f | W2 = −21− 2 a f (2) f = − f .

110

6.1. Examples Using the Method of Theta Blocks In fact, one has λ(`, f ) = λ(`, Θ11,Z1ev ) for all odd ` as follows from Theorem 4.2.5 and as can be verified numerically by comparing the eigenvalues λ(`, f ) = a f (`) of f with the eigenvalues λ(`, Θ11,Z5ev ) from Table 6.3. m lifts to an elliptic modular form f of weight Recall that if the Θk,Zev P m) = m )`− s should be (up to a k 1 = 2k − 1 − m then L(s, Θk,Zev ` is odd λ(`, Θ k,Zev finite number of Euler factors) the L-series of f (see Remark 2.7.16). • m = 3: The vector space S 14 (Γ0 (2)) is of dimension 2. The generators of this space are g 1 = q − 300q3 + 4096q4 − 26730q5 + 98304q6 − 184600q7 + 854973q9 − 1966080q10 + 2042172q11 + O(q12 ),

g 2 = q2 + 24q3 − 480q5 − 300q6 + 3888q7 + 4096q8 − 14400q9 − 26730q10 + 6600q11 + O(q12 ),

which can be easily seen by running the following Sage session Listing 6.3: Sage input S=CuspForms(2,14,prec=12) print S g1,g2=S.gens() print "g1=",g1 print "g2=",g2

Next, we would like to find the Hecke eigenfunctions with respect to all Hecke operators T(`) (` ∈ N with gcd(`, 2) = 1). Let ` = 3. Using Sage, we compute the matrix M(3) of the action of T(3), i.e. the matrix M(3) which satisfies (T(3)g 1 , T(3)g 2 ) = (g 1 , g 3 )M(3). One has M(3) =

ˆ ! −300 98304

24

−300

.

The eigenvalues of M(3) are {1236, −1836}, and the corresponding eigenfunctions are f ± := g 1 ± 64g 2 .

The first Fourier coefficients of f + and f − are: f − (τ) := q − 64q2 − 1836q3 + 4096q4 + 3990q5 + 117504q6 − 433432q7

111

6. Appendix − 262144q8 + 1776573q9 − 255360q10 + 1619772q11 + O(q12 )

f + (τ) := q + 64q2 + 1236q3 + 4096q4 − 57450q5 + 79104q6 + 64232q7 + 262144q8 − 66627q9 − 3676800q10 + 2464572q11 + O(q12 ). new old In fact, f + , f − ∈ S14 (Γ0 (2))(since S 14 (Γ0 (2)) = 0). Also we can verify this by running the following Sage session

Listing 6.4: Sage input f_minus,f_plus=Newforms(2,14) print "f-=",f_minus.qexp(12) print "f+=",f_plus.qexp(12)

Listing 6.5: Sage input f-= q - 64*q^2 - 1836*q^3 + -433432*q^7 - 262144*q^8 1619772*q^11 +O(q^12) f+= q + 64*q^2 + 1236*q^3 + 64232*q^7 + 262144*q^8 ^11 + O(q^12)

4096*q^4 + 3990*q^5 + 117504*q^6 + 1776573*q^9 - 255360*q^10 + 4096*q^4 - 57450*q^5 + 79104*q^6 + 66627*q^9 - 3676800*q^10 + 2464572*q

By the implementation of Sage, it is guarantied that f ± = n a f ± (n)q n are simultaneous normalized eigenfunctions for all Hecke operators T(`) with eigenvalues λ(`, f ± ) = a f ± (`). The function f − has Atkin-Lehner eigenvalue equals 1, and for each odd ` we observe that the eigenvalue λ(`, f − ) of f − equals λ(`, Θ9,Z3ev ) from Table 6.3. • m = 5: The cuspidal subspace S 8 (Γ0 (2)) is of dimension 1. The generaQ n 8 2n 8 tor of this subspace is f (τ) = η(τ)8 η(2τ)8 = q ∞ n=1 (1 − q ) (1 − q ) . The first coefficients of f are P

f (τ) = q − 8q2 + 12q3 + 64q4 − 210q5 − 96q6 + 1016q7 − 512q8 − 2043q9 + 1680q10 + 1092q11 + O(q12 ).

Since dim S8 (Γ0 (2)) = 1, the function f is a normalized Hecke eigenfunction with respect to the operator T(`). In fact, f has Atkin-Lehner eigenvalue equals 1, and for odd ` the Hecke eigenvalue λ(`, f ) equals λ(`, Θ7,Z5ev ). This 112

6.1. Examples Using the Method of Theta Blocks can be verified by comparing the eigenvalues λ(`, f ) with λ(`, Θ7,Z5ev ) from Table 6.3. • m = 7: The vector space M2 (Γ0 (2)) is of dimension 1. The generator of this space is f (τ) =24(E 2 (τ) − 2E 2 (2τ)) = 1 + 24

X¡ `≥1

X

¢ d q`

d |` d is odd

=1 + 24q + 24q2 + 96q3 + 24q4 + 144q5 + 96q6 + 192q7 + 24q8 + 312q9 + 144q10 + 288q11 + O(q12 ).

The function f has Atkin-Lehner eigenvalue equals − sign(a f (2)) = −1. Since dim M2 (Γ0 (2)) = 1, the function f (τ) is a Hecke eigenfunction with eigenvalues P ¡ ¢ λ(`, f ) = d |` d4 d . In fact, one has λ(`, f ) = λ(`, Θ5,Z7ev ) for all odd positive integers ` which can be verified by comparing the eigenvalues λ(`, f ) with λ(`, Θ5,Z7ev ) from Table 6.3.

6.1.1

Final remarks

It is obvious that the previous examples support our expectations in section 4.1. Namely, that the function S D 0 ,x given by Definition 4.1.3 take Jacobi cusp forms of weight k and index L (rk(L) is odd) to elliptic modular forms of weight 2k − 1 − rk(L) on Γ0 (lev(L)/2). In fact, the level is determined by our examples to be lev(L)/4 instead of lev(L)/2. Moreover, we observe that the elliptic modular forms ‡rk(L)·that are lifts of Jacobi forms have Atkin-Lehner eigenvalue equal to − . Thus we conjecture the following: 2

Conjecture 6.1.3. Let L be a positive definite even odd rank lattice over Z. Setting εL = −1 if rk(L) ≡ 1 or 3 mod 8, and 1 otherwise. There is a Heckeequivariant isomorphism ¢ε ¡ ∼ = Jk,L −→ M2k−1−rk(L) lev(L)/4 L ,

where M2k−1−rk(L) lev(L)/4 is the Certain Space inside M2k−1−rk(L) lev(L)/4 which was introduced in [SZ88, 3], and where the "εL " denotes the subspace ¡ ¢ of all f ∈ M2k−1−rk(L) lev(L)/4 such that f | Wlev(L)/4 = εL (−1)k/2 f . ¡

¢

¡

¢

113


List of Notations ¡ ¢ M kt p, χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

(a, b) := gcd(a, b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 < φ, ψ > the Petersson scalar product of φ and ψ . . . . . . . . . . . . . . . . . . . . . . . . . 68

B L (c, t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 β(·, ·) non-degenerate symmetric Z−bilinear form . . . . . . . . . . . . . . . . . . . . . . . 10 β(x)

= 12 β(x, x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

χL

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

∆`

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

e (x)

= e2π ix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

WI (c, a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ²(a)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

er (x) = e2π ix/r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 WII (c, a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Γ

= SL2 (Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

GL+ 2 (Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 H L (Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 H L (Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 JL (Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

115

6. Appendix JL (Z) SL2 (Z) n H L (Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 L = (L, β) a Lattice over Z, where L is free Z-module of finite rank, and β is a

non-degenerate symmetric bilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . 12 d xe

= min { n ∈ Z | n ≥ x} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

¡a¢ n

the Kronecker symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

b xc

= max { n ∈ Z | n ≤ x} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

NL

the set of all positive integer ` ∈ N with (`, lev(L)) = 1 . . . . . . . . . . . . . . . 14

N, Z, Q, R, Z p , Z p , Q p , and C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

H

the upper half plane (the Poincaré half plane) . . . . . . . . . . . . . . . . . . . . . . 3

SL (s, `) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 |k

Petersson slash operator of weight k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

lev(L), lev(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 ord p (n) the p-adic order or p-adic valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Pr(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 W (c, a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 fl fl

k,L

Jacobi slash operator of weight k and index L . . . . . . . . . . . . . . . . . . . . . 31

supp(L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 T(`)

a Hecke operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

T0 (`) a double coset Hecke operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ϑL,x

the Jacobi theta series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

a0

Let a be a positive integer such that a coprime to the level of the lattice L. We shall use a0 to denote an integer such that aa0 ≡ 1 mod lev(L) 45

DL

the discriminant form associated to the lattice L = (L, β) . . . . . . . . . . . 13

E k,L,r a Jacobi-Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

116

6.1. Examples Using the Method of Theta Blocks G r (c) the Ramanujan sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Jk,L

the space of Jacobi forms of weight k and index L . . . . . . . . . . . . . . . . . 34

L(χ, s) =

P∞

n=1 χ(n)n

−s

.......................................................6

L(s, φ) L-function of φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 L#

the dual Lattice of the lattice L = (L, β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

L N (χ, s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 S1

......................................................................1

S k,L

the subspace of Jacobi forms in Jk,L consisting of cusp forms. . . . . . 36

vθ

theta multiplier system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

117


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Hecke Operators on Jacobi Forms of Lattice Index and the Relation to

Hecke Operators on Jacobi Forms of Lattice Index and the Relation to

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