Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields

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Lecture Notes in Mathematics 2130

Hatice Boylan

Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields

Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis (ZRurich) Mario di Bernardo (Bristol) Alessio Figalli (Austin) Davar Khoshnevisan (Salt Lake City) Ioannis Kontoyiannis (Athens) Gabor Lugosi (Barcelona) Mark Podolskij (Heidelberg) Sylvia Serfaty (Paris and NY) Catharina Stroppel (Bonn) Anna Wienhard (Heidelberg)

2130

More information about this series at http://www.springer.com/series/304

Hatice Boylan

Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields

123

Hatice Boylan Matematik BRolRumRu R ˙Istanbul Universitesi ˙Istanbul, Turkey

ISBN 978-3-319-12915-0 ISBN 978-3-319-12916-7 (eBook) DOI 10.1007/978-3-319-12916-7 Springer Cham Heidelberg New York Dordrecht London Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2014957642 Mathematics Subject Classification (2010): 11F50, 11F27 © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To my beloved father, Mustafa Boylan

Foreword

The current research monograph presents a breakthrough in at least three ways. Firstly, it introduces the simple but striking idea that the index of a Jacobi form over a totally real number field should be a lattice of rank one over the ring of integers rather than a number. The classical theory of Jacobi forms over the rational numbers uses positive integers for the index. Accordingly the first attempts to extend this theory to number fields tried to define the index by totally positive numbers in the different of the field. It soon turned out that this is too restrictive to obtain a satisfactory theory. However, if one views the index of classical Jacobi forms as (one half of) the Gram matrix of a rank one lattice over the rational integers, it becomes clear why, for general number fields, scalar indices will not suffice to capture all Jacobi forms. Indeed, as soon as the ring of integers of a given number field is no longer a principal ideal domain, we lose the one-to-one correspondence between lattices of rank one and numbers. The missing right notion of index blocked for a long time the research on Jacobi forms over number fields. As shown in this monograph, the consequent use of lattices as indices leads finally to a smooth and consistent theory. Secondly, the development of a theory of Jacobi forms over number fields was also blocked by the lack of concrete or interesting examples. Again this monograph breaks this spell. It shows that there are indeed interesting examples. More precisely, it gives us in the last chapter a complete description of all Jacobi forms of singular weight on the full Hilbert modular group over any given totally real number field. These new functions generalize to number fields the classical Jacobi theta function which occurs in the Jacobi triple product, and which is essentially equal to the Weierstrass sigma function. One might expect that Boylan’s theta functions will play a similar important role in the theory of Hilbert modular forms, the theory of abelian varieties or algebraic number theory as the classical Jacobi theta function, or equivalently, the Weierstrass sigma function. Finally, indispensable for the study of Jacobi forms over the rationals is their realization as vector-valued modular forms. These modular forms take values in Weil representations. Again the theory of Jacobi forms over number fields was blocked since the corresponding objects for number fields were only vaguely known vii

viii

Foreword

as part of Weil’s general and abstract theory of what is known nowadays as Weil representations. For concrete considerations the Weil representations of Hilbert modular groups were apparently considered as too complicated. Again this research monograph surprises by showing that this is not at all true. It develops from scratch an appealing complete theory of finite quadratic modules and their associated Weil representations over arbitrary (not necessarily totally real) number fields. It describes the decomposition into irreducible parts of those Weil representations which are important for the Jacobi forms considered in this monograph. This theory alone provides already a valuable and indispensable tool for future research not only on Jacobi forms but also for the representation theory of Hilbert modular groups. The theory of Jacobi forms over number fields is far from being at a stage as the corresponding theory over the rationals. However, the beauty of this theory already glimpses through if one looks at its basics and the concrete examples as they are presented in this book. This monograph will serve well as the cornerstone for building up a complete arithmetic theory of the newly introduced Jacobi forms. Indeed, there is currently already various work in progress. In [SS14] the authors calculate the dimension of the spaces of vector-valued Hilbert modular forms with special emphasis on deriving explicit formulas for the dimensions of spaces of Jacobi forms over number fields of weight greater than 2. The approach is based on a general Eichler–Selberg trace formula for vectorvalued Hilbert modular forms, and on the theory of Weil representations and their connection to Jacobi forms as developed in this monograph. p The article [BHS14] determines the structure of the ring of Jacobi forms over Q. 5/ as module over the ring of Hilbert modular forms. The research project [SW14] aims to develop the theory of Hecke operators for Jacobi forms over number fields, and to study the connection of this new type of Jacobi forms to Siegel–Hilbert modular forms. The article [Boy14] calculates the Fourier coefficients of Jacobi Eisenstein series over number fields and gives thereby the first concrete examples for a deep arithmetic connection between the new Jacobi forms and Hilbert modular forms. In [BS14b] the authors plan to summarize the results of these (and possibly other) research activities, to give further explicit examples of liftings from Jacobi forms over number fields to Hilbert modular forms, and, in particular, to present a complete corresponding Hecke theory. The main interest for constructing a theory for Jacobi forms over number fields arose from the fact that we expect several deep results from the theory of elliptic modular forms and Jacobi forms over Q to hold true for the number field case too. In particular, we expect liftings from Jacobi forms over number fields to Hilbert modular forms. Moreover, the Fourier coefficients of the Jacobi forms should encode the vanishing at the critical point of twisted L-functions associated with Hilbert modular forms. This is, in particular, interesting in the context of a generalized Birch and Swinnerton–Dyer conjecture for elliptic curves over number fields.

Foreword

ix

This monograph is an important step towards such a theory. It opens the door to a new and fascinating world of so far unseen functions. I hope that it will stimulate more researchers to follow this invitation to a new exciting subject. Bonn, Germany September 2014

Nils-Peter Skoruppa

Preface

In analogy to the theory of classical Jacobi forms which has proven to have various important applications ranging from number theory to physics, we develop in this research monograph a theory of Jacobi forms over arbitrary totally real number fields. However, we concentrate here mainly on the connection of such Jacobi forms and the theory of Weil representations, leaving out important topics like Hecke theory and liftings to Hilbert modular forms, which still have to be developed. We hope to come back to those topics in later publications, but that the present work stimulates already further interest in this rich new theory. Here, we develop, first of all, a theory of finite quadratic modules over number fields and their associated Weil representations. Next we develop in detail the basics of the theory of Jacobi forms over number fields and the connection to Weil representations. As a main application of our theory, we are able to describe explicitly all singular Jacobi forms over arbitrary totally real number fields whose indices have rank one. We expect that these singular Jacobi forms play a similar important role in this newly founded theory of Jacobi forms over number fields as the Weierstrass sigma function does in the classical theory of Jacobi forms. I thank the Max-Planck Institute for Mathematics in Bonn for its hospitality and the beautiful research environment which they provided during the year 2013 when I was working on finalizing this book and on related topics. I also thank ˙Istanbul Üniversitesi for allowing me to spend the year 2013 in Bonn. I thank the Mathematical Sciences Research Institute in Berkeley for hosting me in spring 2011 at a still early stage of this research project, and Tongji University in Shanghai and Chennai Mathematical Institute for giving me the possibility to lecture in fall 2009 on various very early results on Jacobi forms over number fields. Finally, I thank Nils-Peter Skoruppa for having me introduced to this fascinating subject, for showing constantly interest in my work, and for being always ready to share his profound vision of mathematics. Last but not least, I thank all those who cannot all be named here explicitly without omissions but have been influential for my career. Bonn, Germany and ˙Istanbul, Turkey November 2013

Hatice Boylan

xi

Contents

1 Finite Quadratic Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Finite Quadratic O-Modules . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Cyclic Finite Quadratic O-Modules.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Some Lemmas Concerning Quotients O=a . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 6 14

2 Weil Representations of Finite Quadratic Modules . .. . . . . . . . . . . . . . . . . . . . 2.1 Review of Representations of Groups.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 The Weil Representation W .M / . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Decomposition of Weil Representations .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Complete Decomposition of Cyclic Representations . . . . . . . . . . . . . . . . . 2.5 The One Dimensional Subrepresentations .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 The Number of Irreducible Components.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.1 The First Approach . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6.2 The Second Approach . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

19 20 27 30 39 43 47 48 56

3 Jacobi Forms over Totally Real Number Fields . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 O-Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Algebraic Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 The Metaplectic Cover QR of R . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 The Jacobi Group of an O-Lattice . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 The Jacobi Theta Functions . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Definition and Basic Properties of Jacobi Forms ... . . . . . . . . . . . . . . . . . . . 3.7 Jacobi Forms as Vector-Valued Hilbert Modular Forms . . . . . . . . . . . . . . Appendix: Jacobi Forms of Odd Index .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

65 66 69 71 72 81 91 94 99

4 Singular Jacobi Forms.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Characterization of Singular Jacobi Forms . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Theta Functions and Weil Representations . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Decomposition of the Q -Modules L . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

103 103 104 107

xiii

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Contents

4.4 The Singular Jacobi Forms of Rank One Index . . .. . . . . . . . . . . . . . . . . . . . 109 4.5 Constructing Jacobi Forms of Non-Singular Weight .. . . . . . . . . . . . . . . . . 115 Appendix . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 123 Glossary . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 127 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 129

Introduction

A Jacobi form of weight k and index m (both half integral) on the full modular group SL.2; Z/ is a holomorphic function .; z/ on the product H C of the complex upper half plane H with the set of complex numbers C such that .; x; y/ WD 2 .; x C y/ e 2 i mx satisfies the following properties: (i) The function .; x; y/ is quasi-periodic in the real variables x and y with period 1. (ii) For fixed rational x; y, the map 7! .; x; y/ defines an elliptic modular form of weight k (possibly with character) on the principal congruence subgroup .a/ of SL.2; Z/, where a denotes the square of the least common multiple of the denominators of x and y. The first property implies that, for fixed , the map z 7! .; z/ defines ı a theta function (a holomorphic section of a line bundle) on the elliptic curve C Z C Z . If we study n-dimensional abelian varieties whose endomorphism ring contains the ring of integers O of a totally real number field K of degree n over Q, then we find naturally analogs of Jacobi forms. We will call them Jacobi forms over the number field K. A careful analysis shows, however, that we have to replace the index m by a totally positive definite integral O-lattice of rank one. Such a lattice can always be represented by a pair .c; !/, where c is a fractional O-ideal and ! a totally positive element in K such that c2 ! is contained in the inverse different of K (see Proposition 3.10). If K is the field of rational numbers and O is the ring of integers Z, then such a lattice can always be represented by a pair .Z; 2m/, i.e. by the Z-module Z equipped with the Z-bilinear form .x; y/ 7! 2mxy, where m is a positive integer. The main difference here is that, for a number field K of class number greater than 1, a finitely generated, torsion free O-module is not in general isomorphic to O, but only to a fractional O-ideal, whose ideal class might be not trivial. A Jacobi form over K of half integral weight k and index L D .c; !/ is a holomorphic function .; z/ on Hn Cn such that the function 1 2 .; x; y/ WD .; x C y/ e 2 i tr. 2 M.!/x / satisfies: xv

xvi

Introduction

(i) The function .; x; y/ is quasi-periodic in the variables x and y in Rn with respect to the O-sublattice M.c/. (ii) For fixed x and y in M.K/, the map 7! .; x; y/ defines a Hilbert modular form of weight k (possibly with character) on the principal congruence subgroup .a/ of SL.2; O/, where a is the square of the least common multiple of the denominators of ac1 and of bc1 with x D M.a/ and y D M.b/. Here M denotes the Minkowski embedding of K into Rn , which maps a to the vector whose j -th component equals j .a/, where we use a fixed enumeration 1 , . . . , n of the embeddings of K into R. Moreover, when writing x Cy or M.!/x 2 , we view Cn as a ring with respect to component-wise multiplication. Finally, tr.z/, for z in Cn , denotes the sum of the components of z. Note that the first property expresses the fact that, for fixed , the map z 7! .; z/ defines a theta section in a line bundle) on the ı function (a holomorphic abelian variety Cn M.O/ CM.O/ . For a precise definition of Jacobi forms over number fields we refer the reader to Definition 3.45. A justification of the informal description given here can be found in the Appendix of Chap. 3. Later, it will also be more convenient to use C ˝Q K instead of Cn since the first object carries naturally several algebraic structures which we shall make use of, and it allows for coordinate independent calculations. One of the first steps into an interesting theory of Jacobi forms is, of course, to exhibit explicit examples. As it turns out, for number fields different from Q, it is in fact already not trivial and challenging to construct examples. In this monograph, after developing a sufficiently general theory of Jacobi forms over number fields, we determine explicitly all singular Jacobi forms over number fields, i.e. all Jacobi forms over number fields whose weight equals 1=2 (see Definition 3.47 and Proposition 4.1). The singular Jacobi forms over Q have been determined by Skoruppa in [Sko85, p. 27]. Namely, for 2 H and z 2 C, set #.; z/ WD

X

4 s

qs

2 =8

s=2

n q ./ WD e 2 i n ; n .z/ WD e 2 i sz :

s2Z

4

(Here s denotes the nontrivial Dirichlet character modulo 4). The function # is a Jacobi form over Q on the full modular group of weight 1=2 and index 1=2. In particular, # is a singular Jacobi form. Skoruppa [Sko85, p. 27] showed that #.; d z/ and # .; d z/, where # .; z/ WD

X 2 #.; 2z/ 12 .z/ D q s =24 s=2 ; s #.; z/ s2Z

and where d is a positive integer, are the only singular Jacobi forms over Q on the full modular group. What makes the singular Jacobi forms interesting is that they occur in various important areas of mathematics. First of all, #.; z/ is, up to normalization, the

Introduction

xvii

Weierstrass’ sigma-function .; z/ associated with the elliptic curve C Namely, we have #.; z/ D ./3 e z

2q d dq

log . /

ı Z C Z .

.; z/;

Q where ./ D q 1=24 n1 .1 q n / is the Dedekind’s eta function. As such #.; z/ is the basic functions out of which can be constructed all theta functions on elliptic curves. In the arithmetic theory of elliptic curves, it shows up as the Green’s function ı for the elliptic curve C Z C Z . Moreover, #.; z/ and # .; z/ show up in the theory of Kac–Moody algebras via the famous triple and quintuple identity, respectively. For example, the Jacobi triple product identity #.; z/ D q 1=8 . 1=2 1=2 /

Y

.1 q n /.1 q n /.1 q n 1 /

n1

can be interpreted as the Weyl–Kac denominator identity for a certain affine Kac– Moody algebra. In view of the indicated importance of the function #.; z/, ıit is natural to ask whether such functions exist also for the abelian varieties Cn M.O/ C M.O/ mentioned above. It is then also natural to expect that they are also singular Jacobi forms, which explains our interest in determining all singular Jacobi forms over number fields. We explain our main results concerning singular Jacobi forms (see Theorems 4.2 and 4.3 for more precise statements). Theorem There exist nonzero singular Jacobi forms over K if and only if 2 splits completely in K and the principal genus of K contains an ideal of the form gd1 , where g is a (possibly empty) product of pairwise different prime ideals of degree 1 over 3, and where d denotes the different of K. Recall that the principal genus of K is the set of fractional O-ideals a which represent a square in the narrow ideal class group ClC .K/ of K, i.e. for which there exist a fractional O-ideal c and a totally positive ! in K such that a D c2 !. A theorem of Hecke [Hec81, Theorem 177] states that the different d is a square in the ideal class group of K. However, it need not necessarily to be a square inpthe narrow ideal class group. A counterexample is provided by the number field Q. 47/. Note that 2 splits completely in this number field. Theorem Suppose 2 splits completely in K. If c is a fractional O-ideal and ! is a totally positive element in K such that g WD c2 !d is a (possibly empty) product of pairwise different prime ideals of degree 1 over 3, then #.c;!/ .; z/ WD

X s2cg1

4g .s 0 /e 2 i tr

1 2 8 M.!s /

e 2 i tr

1 2 M.!s/z

xviii

Introduction

defines a Jacobi form over K of singular weight 1=2 and index .c; !/. Here s 0 2 O is such that s s 0 mod 4c, where C 4c is a generator for cg1 =4c. By 4g , we denote the totally odd Dirichlet character modulo 4g (see Definition 2.44). Vice versa, every nonzero singular Jacobi form over K is (up to multiplication by a constant) of this form. If .c; !/ is an index as in the theorem, then a1 c; a2 ! , for any nonzero a in K, is also such an index. Two indices are isomorphic if and only if one can be obtained from the other in this way, i.e. by multiplying with a suitable a (see Proposition 3.9). Note that the singular Jacobi forms associated with isomorphic lattices differ only in a trivial way. Namely, we have #.a1 c;a2 ! / .; z/ D #.c;!/ .; M.a/z/. We shall see (Proposition 4.7) that the number of indices modulo isomorphism which admits a nonzero singular Jacobi form equals j F.K/j j ClC .K/Œ2 j, where F.K/ is the subset of the principal genus consisting of ideals of the form gd1 with g as in the last theorem, and where ClC .K/Œ2 is the kernel of the squaring map of the narrow ideal class group. For the field of rational numbers this number equals 2. The two classes of indices admitting a nonzero Jacobi form are represented by .Z; 1/ and .Z; 3/ and, indeed, we rediscover the forms from Skoruppa’s theorem: #.Z;1/ D # and #.Z;3/ D # . We explain the other main themes of the book. In Chap. 3 we shall develop a general theory of Jacobi forms over number fields whose indices are arbitrary O-lattices. In Chap. 4, we shall see that singular Jacobi forms correspond to one-dimensional submodules of certain (projective) SL.2; O/-modules of theta functions (see Proposition 4.1) which turn out to be isomorphic to Weil representations associated with certain finite quadratic modules over number fields. A theory of finite quadratic modules over number fields and a theory of Weil representations associated with finite quadratic modules over number fields have not yet been worked out in the literature. Therefore we shall develop these theories in Chaps. 1 and 2, respectively. In Chap. 2, we decompose, in particular, the spaces of cyclic Weil representations into irreducible subrepresentations (see Theorem 2.5). This will give us the clue for determining explicitly all singular Jacobi forms whose indices are O-lattices of rank one, since these correspond to the one-dimensional subrepresentations of cyclic Weil representations (see Theorem 2.6). Translating these results back to the language of Jacobi forms, we can then determine in Chap. 4 explicitly all singular Jacobi forms whose indices are O-lattices of rank one. In the last section of this chapter we show how to construct explicitly Jacobi forms over number fields of non-singular weight, and we give examples. Finally, in the Appendix, we present tables concerning the first number fields which admit nonzero singular Jacobi forms.

Notations

In general, if the number field K is clear from the context, we often drop the subscript K, i.e. we write O, d, tr.a/, N.a/, etc. for OK , dK , trK=Q .a/, and NK=Q .a/. Let K be number field with ring of integers O. A Dirichlet character modulo an integral O-ideal a is a map from O to C defined by (

.r/ D

0 .r C a/ if .r; a/ D 1 0

otherwise;

where 0 is a group homomorphism from .O=a/ to C . An exact divisor b of an integral O-ideal a is the ideal so that b C ab1 D O. Let a be a fractional O-ideal and p be a prime ideal of the number field K. We use vp .a/ for the valuation of a at p, i.e. for the exponent of the exact power of p occurring in the prime ideal factorization of a. Note that vp .a/ can be negative for some a. If a is integral, we P have vp .a/ 0, for all p. In expressions like bja , where a is an integral O-ideal, it is always understood, if not otherwise stated, that the integral O-ideals Q b runs through Q dividing a. Similarly, in expressions like pja or pa ka it is understood that p runs through the prime ideals or exact prime ideal powers pa dividing a. For a finite set M , the symbol CŒM stands for the C-vector space of all functions from M into C. A basis for this vector space is the set of all functions ex (x 2 M ) such that ex .y/ equals 1 or 0 accordingly x D y or not. Let H be a subgroup of finite index in the group G, and let be a complexvalued function on G which takes the samePvalue on each coset of G=H . We use P g2G=H .g/ as a short-hand notation for g2R .g/, where R is a complete set of representatives for G=H . In the sequel theorems are numbered independently, whereas the numbering of lemmas, propositions, examples, and corollaries share the same numbering sequence.

xix

Chapter 1

Finite Quadratic Modules

Let K be a number field of degree n over Q. We shall use O, and d for the ring of integers and for the different of K, respectively. In this chapter we shall develop a theory of finite quadratic modules over number fields, i.e. a theory of finite O-modules equipped with a quadratic form O ! K=d1 . A special emphasis is on cyclic finite quadratic modules. The main results of this chapter are Theorems 1.1 and 1.2 which give normal forms for cyclic finite quadratic modules and describe explicitly their isotropic submodules and the corresponding quotients. These results will be used in the next chapter when we decompose the spaces of cyclic Weil representations. In Sect. 1.1, we shall give the definition of finite quadratic O-modules, and we discuss their basic properties. In Sect. 1.2, we shall specialize to cyclic O-modules, we shall prove the two mentioned theorems and we study the orthogonal groups of cyclic finite quadratic modules which will be very crucial for the splittings of the spaces of cyclic Weil representations. Finally, in Sect. 1.3, we shall provide some lemmas concerning the quotient rings O=a of O modulo an integral O-ideal a which we shall need in Sect. 2.6 of Chap. 2.

1.1 Finite Quadratic O-Modules In this section we shall develop a basic theory of finite quadratic O-modules. We shall follow closely [Sko10, Sect. 1.1], where such a theory was developed for K D Q. Definition 1.1 A finite quadratic O-module, in short O-FQM, is a pair .M; Q/, where M is a finite O-module, and where Q is a non-degenerate quadratic form on M , i.e. where Q W M ! K=d1 is a map which satisfies the following properties: © Springer International Publishing Switzerland 2015 H. Boylan, Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields, Lecture Notes in Mathematics 2130, DOI 10.1007/978-3-319-12916-7_1

1

2

1 Finite Quadratic Modules

(i) For all a 2 O and x 2 M one has Q.ax/ D a2 Q.x/. (ii) The map B W M M ! K=d1 defined by B.x; y/ WD Q.x C y/ Q.x/ Q.y/ is O-bilinear and symmetric. (iii) B is non-degenerate, i.e. B.x; M / D f0g if and only if x D 0. Let M D .M; Q/ and N D .N; R/ be O-FQM. We say that there is an isomorphism between M and N , in symbols M ' N , if there exists an O-module isomorphism ' W M ! N such that R ı ' D Q. Two O-FQM are called isomorphic if there is an isomorphism between them. The automorphisms of a finite quadratic module, i.e. the isomorphisms M ! M , form a group with respect to the composition of maps, which we denote by O.M / and which we call the orthogonal group of M . In the sequel, when we write x 2 M , we mean that x is an element of M , and we write U M if U is a subset of M . Moreover, we refer to an O-submodule U of M simply as a submodule of M . Example 1.2 Let L D .L; ˇ/ be an even O-lattice, i.e. let L be a finitely generated torsion-free O-module and let ˇ be a symmetric non-degenerate O-bilinear form on L taking values in d1 such that ˇ.x; x/ 2 2d1 for all x 2 L (see Sect. 3.1 for a short resumé of the notion of O-lattices). The discriminant module of L is the O-FQM 1 # 1 : DL D L =L; x C L 7! ˇ.x; x/ C d 2 Here L# stands for the dual lattice of L, i.e. the set of all y 2 K ˝O L such that ˇ.y; L/ d1 . We have a map Tr W K=d1 ! Q=Z, a C d1 7! tr a C Z. It is easy to see that this map is well-defined. Indeed, if b 2 a C d1 , say, b D a C t for some t 2 d1 , then Tr.t/ is in Z, hence, we have tr.a/ tr.b/ mod Z. Proposition 1.3 Let M D .M; Q/ be an O-FQM. The tuple Tr.M / WD .M; Tr ıQ/ defines a finite quadratic Z-module. Proof The form Tr ıQ is obviously a quadratic form onM , viewed as a Z-module. We need to show that it is non-degenerate. Suppose Tr B.x; M / D f0g for some x 2 M . Since, for all a 2 O, we have aM M , we then have Tr aB.x; M / D f0g for all a. It is easy to see from the very definition of the different that this implies that B.x; M / D f0g. Since M is a non-degenerate O-FQM, we conclude x D 0. t u Definition 1.4 Let M D .M; Q/ be an O-FQM. We call Level.M / WD fa 2 O W aQ D 0g Ann.M / WD fa 2 O W aM D 0g the level and the annihilator of M , respectively.

1.1 Finite Quadratic O-Modules

3

Note that Level.M / and Ann.M / are integral O-ideals of K. An O-FQM M which is annihilated by a power of a prime ideal p is called, by abuse of language, a p-module. If K equals the field of rational numbers, we also call the positive integer generating Level.M / the level of M . Moreover, in this case the positive integer generating the annihilator of M is the usual exponent of the abelian group M . Proposition 1.5 Let M D .M; Q/ be an O-FQM. The following holds true: Level.M / Ann.M / 1=2 Level.M /: Proof Let B be associated bilinear form of M . We prove the first inclusion. Let u be in Level.M /. So, we have uQ D 0. This implies that uB D 0, i.e B.ux; y/ D 0 for all x; y 2 M . From the non-degeneracy of M we conclude that ux D 0. Hence, u 2 Ann.M /. Therefore, Level.M / Ann.M /. Now we prove the second inclusion. Let a 2 Ann.M /. Since B is O-bilinear, B.aM; y/ D f0g. In particular, aB.x; x/ D 0 holds true for all x in M , hence we have 2aQ.x/ D 0. So 2a 2 Level.M /. Therefore, Ann.M / 1=2 Level.M /. t u There are three operations which we can perform in the category of O-FQM: twisting, taking direct sums and quotients. Twisting is the operation which maps M D .M; Q/ to M a WD .M; aQ/, where a 2 O and a − Level.M /. The latter ensures that M a is still non-degenerate. Let .M; Q/ and .N; R/ be O-FQM with associated bilinear forms B and B 0 , respectively. We define their direct sum as M C N WD .M ˚ N; .x; y/ 7! Q.x/ C R.y// ; where M ˚ N is the direct sum of the abelian groups M and N . It is clear that the map x ˚ y 7! Q.x/ C R.y/ defines a non-degenerate quadratic form, so that M C N is indeed an O-FQM. Note that the bilinear form associated to M C N is given by the map .k ˚ l; x ˚ y/ 7! B.k; x/ C B 0 .l; y/. Similarly, we can define the direct sum of an arbitrary finite number of O-FQM. An important application of taking direct sums is the decomposition of a OFQM into local parts. For explaining this let M D .M; Q/ be a O-FQM with associated bilinear form B. We use M .p/ for the O-submodule of M which contains all elements of M that are annihilated by a power of a prime ideal p. The O-FQM M .p/ WD M.p/; QjM.p/ is called the p-part of M . Note that it is a p-module. The non-degeneracy of the quadratic from QjM.p/ follows from the following proposition. Proposition 1.6 Let M D .M; Q/ be an O-FQM. The quadratic form QjM.p/ is non-degenerate for every prime ideal p. Moreover, we have M '

a pj Ann.M /

M .p/:

4


Proof We define 'W

a

M .p/ ! M ;

fxp gpj Ann.M / 7!

pj Ann.M /

X

xp :

pj Ann.M /

Q We show first of all that ' is surjective. Set Ip WD qn k Ann.M /; q¤p qn . Note that the ideals P Ip (pj Ann.M /) are relatively prime, i.e. there exists numbers ˛p in Ip such that pj Ann.M / ˛p D 1. Let x 2 M be arbitrary. Then, ˛p x is an element of P P M .p/ and we have ' f˛p xgp D p ˛p x D . p ˛p /x D x. We show that the quadratic form QjM.p/ on M .p/ is non-degenerate for any prime ideal p dividing Ann.M /. First we shall show that for any x in M .p/ and y in M .q/, B.x; y/ D 0, where the p and the q are different prime ideals. Fix pn k Ann.M / and qm k Ann.M /. Since p ¤ q, we have pn C qm D O, i.e. there exists a 2 pn , b 2 qm such that a C b D 1. If x 2 M .p/ and y 2 M .q/. we have ax D by D 0. Hence, B.x; y/ D .a C b/B.x; y/ D B.ax; y/ C B.x; by/ D 0. Suppose that we have B.x; M.p// D f0g. Using the above fact, we have ` B x; qj Ann.M / M .q/ D B.x; M / D f0g. So x D 0, since the quadratic form Q is non-degenerate. Therefore the proposition follows. t u For explaining the third operation, namely, taking quotients, we need some preparations. Let M D .M; Q/ be an O-FQM with associated bilinear form B and U be an O-submodule of M . The dual group of U is defined as: U # WD fy 2 M W B.U; y/ D 0g:

(1.1)

Note that U # is also an O-submodule of M . Proposition 1.7 Let M D .M; Q/ be an O-FQM with associated bilinear form B and U be an O-submodule of M . The application x 7! B.x; / defines an exact sequence of O-modules: 0 ! U # ! M ! Hom.U; K=d1 / ! 0: Here Hom.U; K=d1 / denotes the group of O-module homomorphisms of U into K=d1 . In particular, one has jU j jU # j D jM j and .U # /# D U . Proof The sequence is exact at M , since U # is by definition the kernel of the map M ! Hom.U; K=d1 /, x 7! B.x; /. The surjectivity of this map can be seen as follows: every element in Hom.U; K=d1/ can be extended to an element of Hom.M; K=d1 / [Ser73, Chap. VI,Sect. 1, Proposition 1]. The latter group has order jM j [Ser73, Chap. VI,Sect. 1, Proposition 2], and it is injective (since nondegenerate) the map x 7! B.x; / from M into Hom.M; K=d1 / is therefore also surjective. The exactness of the sequence implies that Hom.U; K=d1/ ' M=U # . Because the groups are finite and j Hom.U; K=d1 /j D jU j, we obtain jU j jU # j D jM j. We have trivially U .U # /# . Applying the equality for group

1.1 Finite Quadratic O-Modules

5

orders to U # instead of U , we obtain jU jjU # j D jM j D jU # jj.U # /# j, hence U D .U # /# . t u If U is contained in U # , then B induces a well-defined bilinear form on U # =U as follows: B W U # =U U # =U ! K=d1 ;

.x C U; y C U / 7! B.x; y/:

The bilinear form B is non-degenerate. Indeed, let x C U be in U # =U and suppose B.x C U; y C U / D 0 for all y 2 U # , i.e. x 2 .U # /# . Proposition 1.7 implies then x 2 U . Although the application .x C U; y C U / 7! B.x; y/ defines a bilinear form on U # =U , the application x C U 7! Q.x/ is not well-defined unless Q vanishes on U . We call an element x of the O-FQM M isotropic, if Q.x/ D 0. An O-submodule U of M is called isotropic, if Q vanishes on U . If U is isotropic then U is contained in U # and the considerations above show that the application Q W x C U 7! Q.x/, which is now well-defined, defines a non-degenerate quadratic form on U # =U . We set M =U WD U # =U; Q and call M =U the quotient of M by the isotropic submodule U . Definition 1.8 Let M D .M; Q/ be an O-FQM. We define 1 X .M / WD p e fQ.x/g : jM j x2M This value is called the -invariant of M . Remark Let M and N be O-FQM. It is easy to see directly from the definition that .M C N / D .M /.N / and .M 1 / D .M /. Proposition 1.9 Let M D .M; Q/ be an O-FQM and U be an isotropic submodule of M . Then we have .M / D .M =U /: Proof Let B be associated bilinear form of M . Let R denote a system of representatives for the cosets in M =U . We write .M / D

XX

e fQ.x C y/g :

x2R y2U

Since U is isotropic we have Q.x C y/ D Q.x/ C B.x; y/. The inner sum becomes 0 unless x 2 U # , when it equals jU j. The result is now obvious. t u

6


Proposition 1.10 Let M D .M; Q/ be an O-FQM. Then .M / has absolute value 1. Proof Let B be associated bilinear form on M . We write j.M /j2 D

1 X e fQ.x/ C Q.y/g : jM j x;y2M

After doing the substitution y 7! y C x in the above sum, we obtain j.M /j2 D

X 1 X e fQ.y/g e fB.x; y/g : jM j y2M x2M

But the inner sum equals zero, unless y D 0, when it equals jM j (see the subsequent proposition). Hence, j.M /j2 D 1 as we claimed. t u Proposition 1.11 Let M D .M; Q/ be an O-FQM with associated bilinear form B. For y 2 M , the following holds true: Sy WD

X z2M

( e fB.z; y/g D

jM j if y D 0 0

otherwise:

Proof If y equals 0, the formula is obvious. Otherwise, there exists y0 such that B.y0 ; y/ ¤ 0, since B is non-degenerate. Substituting z 7! z C y0 we obtain that Sy equals e fB.y0 ; y/g Sy . Hence, Sy D 0. t u

1.2 Cyclic Finite Quadratic O-Modules In this section we shall give a full classification of cyclic finite quadratic O-modules, their isotropic submodule, their quotients, and their orthogonal groups. Definition 1.12 A finite quadratic O-module .M; Q/ is called a cyclic finite quadratic O-module, if the O-module M is cyclic, i.e. there exists an x 2 M such that M D Ox. Henceforth, a cyclic finite quadratic O-module is called O-CM. Proposition 1.13 Let M D .M; Q/ be an O-CM with level l. Then .2; l/2 divides l. In particular, vp .l/ > vp .2O/ for every even prime ideal dividing l. For the annihilator of M we have the formula Ann.M / D l .2; l/1 : Remark In the following we shall often tacitly use that, for integral ideals l and a D l.2; l/1 , the statement .2; l/2 jl is equivalent to lja2 .

1.2 Cyclic Finite Quadratic O-Modules

For the proof of the proposition we need a lemma. Lemma 1.14 Let a be a fractional O-ideal. The ideal b WD

7

P a2a

Oa2 equals a2 .

Proof Multiplying by a suitable integer we can assume without loss of generality that a is an integral O-ideal. Let a 2 a. We have a2 2 a2 and hence a2 jb. Vice versa, let n D vp .b/ for a prime ideal p dividing b. (Recall that vp .b/ denotes the valuation of b at p, see the section Notations.) There exists a 2 a such that n D vp .a2 /. Then n D 2k for some integer k. Hence pk ka. We also have that p2k ja2 for all a 2 a. Hence pk ja for all a 2 a. We therefore obtain pk ja, thus, pn ja2 . This proves the lemma. t u Proof of Proposition 1.13 Let B denote the bilinear form of M . Write M D O . We put Q. / D ! C d1 . Then we have Q.a / D a2 ! C d1 for all a 2 O. First of all, we show that l equals the denominator of !d. The level of M is by definition the largest O-ideal l such that lQ D 0. i.e. la2 ! 2 d1 , or, equivalently, such that l!d is an integral O-ideal. Hence l equals the denominator of !d. Next we prove a WD Ann.M / D l .2; l/1 . By the non-degeneracy of B, the annihilator of M consists of all a 2 O such that B.a; a0 / D 0 for all a0 2 O. But B.a; a0 / D 2aa0 ! C d1 . Hence the annihilator of M consists of all a 2 O such that 2a!d is integral, which is equivalent to l.2; l/1 ja. This proves the claim. By the remark it remains to show that lja2 . If a 2 a, then 0 D Q.a / D 2 a ! C d1 . This implies that a2 ! 2 d1 , i.e. the ideal a2 !d is integral. Since l is the denominator of !d, we have that lja2 for all a 2 a. Since a2 equals the ideal generated by the squares of elements in a (see Lemma 1.14) we conclude lja2 . Finally, let pl be the exact power of an even prime dividing l. If pl divided 2, then l.2; l/1 would not be divisible by p, in contradiction to .2; l/2 jl. t u By the proposition the ideal l.2; l/2 , where l is the level of an O-CM, is integral. Since it will show up in many subsequent formulas we introduce a name for this quantity. Definition 1.15 Let M be an O-CM with level l. We call Mod.M / WD l.2; l/2 the modified level of M . Theorem 1.1 (i) Let ! 2 K and let l be the denominator of !d. Assume .2; l/2 jl. Then the pair M .!/ WD O=a; x C a 7! !x 2 C d1 ; where a D l.2; l/1 , defines an O-FQM with annihilator a and level l. In fact, M .!/ is an O-CM with generator 1 C a. (ii) Every O-CM is isomorphic to an O-FQM of the form M .!/.

8


(iii) Two O-CM M .!1 / and M .!2 / are isomorphic if and only if there exists an a in O, relatively prime to l, such that !1 !2 a2 mod d1 . Here l stands for the denominator of !2 d. Proof First we prove (i). Note that the assumption .2; l/2 jl is equivalent to the statement lja2 . We show that the map Q W x C a 7! !x 2 C d1 is well-defined and that it is non-degenerate. First note that !da2 is integral (by the assumption lja2 ). For the well-definedness, we need to have that if y 2 x C a, then !x 2 !y 2 2 d1 . Write y D x C k (k 2 a). Then !x 2 !y 2 D 2!xk !k 2 2 !.2a C a2 / lies in d1 , since l divides 2a by definition of a and since l divides a2 by assumption. For the non-degeneracy of the quadratic form Q, we need to have that 2!xO d1 (x 2 O) if and only if x 2 a. Indeed, 2!xd is integral if and only if the denominator l of !d divides 2x, i.e. if and only if a D l.2; l/1 divides x. It is obvious from the construction that M .!/ has annihilator a and level l. The O-FQM M .!/ is an O-CM, since it is generated by the multiplicative neutral element 1 C a of the ring O=a. Secondly we prove (ii). Let M D .M; Q/ be a cyclic finite quadratic O-module with annihilator a. Write M D O , set ! D Q. / and let l denote the denominator of !d. Then l equals the level of M . Indeed, since M D O the ideal generated by 2 the Q.x/ (x 2 M ) equals then Q./O. By Proposition 1.13 we2have1 .2; l/ jl and 1 a D l.2; l/ . Hence M Q. / D O=a; x C a 7! Q. /x C d . It is obvious that the map x C a 7! x defines an isomorphism M Q. / ! M . Lastly, we prove (iii). Assume that there exists an isomorphism ' from M .!1 / to M .!2 /. Then the levels of both modules coincide, hence are equal to l. The annihilator of both O-FQM is a WD l.2; l/1 . Since 1 C a is a generator of the Omodule O=a there exist an a in O such that '.1 C a/ D a.1 C a/. Since ' is an isomorphism there exist also an a0 such that ' 1 .1 C a/ D a0 .1 C a/. We conclude that a0 a 1 mod a, i.e. that a is relatively prime to a. Since, a and l have the same prime divisors (see Proposition 1.13), we see that a is also relatively prime to l. Finally, since '.1 C a/ D a C a and since ' preserves the quadratic forms, we find !2 a2 !1 mod d1 . If, vice versa, !2 a2 !1 mod d1 for some a, relatively prime to the denominator l of !2 d, then !1 d has also denominator l. Indeed, write !2 a2 D !1 Ct with t in d1 . Then !2 a2 d !1 d C O, and therefore l1 !2 a2 d !1 dl1 C l1 , where l1 denotes the denominator of !1 d, from which we deduce that l1 !2 a2 d is integral. Hence l divides l1 a2 , and since a and l are coprime, we find ljl1 . Changing the role of l and l1 in the preceding argument we find also l1 jl. It is then clear that the map x C a 7! ax C a defines an isomorphism of M .!1 / ! M .!2 /. t u Corollary 1.16 The number of isomorphism classes of O-CM with a given level l equals the number of elements in .O=l/ Œ2 , where .O=l/ Œ2 denotes the kernel of the squaring map of .O=l/ . Remark Applying the Chinese remainder theorem [Neu99, I. 3, Theorem (3.6)] our theorem can be restated in the form that Q the number of isomorphism classes of OCM with a given level l equals therefore pn kl a.pn /, where a.pn / is the number of


9

solutions of x 2 D 1 in .O=pn/ . For odd p, there are exactly two solutions of x 2 D 1 in .O=pn / , i.e. a.pn / D 2. In general, a.pn / D 2e , where e denotes the number of even elementary divisors of .O=pn / . For even p, the number of solutions depends very much on the arithmetic of the number field. Proof If l is given, then we can always form an O-CM. Indeed, let b be an integral O-ideal which lie in the inverse ideal class of ld which satisfies b C l D O. Then 1 1 K such that b.ld/ D O!. Denote a WD l.2; l/ . Hence, there exists some !2in O=a; x C a 7! !x defines an O-CM (see Theorem 1.1 (i)). Let M be a cyclic O-module of level l. By Theorem 1.1 M is isomorphic to some M .!/ where l!d is an integral O-ideal relatively prime to l, and, vice versa, for every such !, the O-FQM M .!/ has level l. Moreover, M .!/ depends obviously only on ! modulo d1 . We shall prove in a moment that l!d is an integral O-ideal relatively prime to l if and only if !Cd1 generates the O-module l1 d1 =d1 . Thus if we use Il for the set of isomorphism classes of cyclic O-modules of level l, the application ! 7! M.!/ induces a surjective map from the set G of the generators l1 d1 =d1 onto Il . By Theorem 1.1 (iii) this map induces a bijection G

ı 2 .O=l/ ! Il :

It is easy to see that there is an O-module isomorphism O=l ! l1 d1 =d1 (see Lemma 1.17). ı So, the number that we are looking for equals the number of elements in .O=l/ ..O=l/ /2 . Since we have an exact sequence Sq

1 ! Ker.Sq/ ! .O=l/ ! ..O=l/ /2 ! 1; where Sq ı is the squaring map, we conclude that the number of elements in the group .O=l/ ..O=l/ /2 equals the number of elements in the kernel of Sq. This proves the corollary. It remains to determine the ! such that l!d is an integral ideal relatively prime to l. Write such an ! in the form !d D bl1 with an integral b coprime to l. Then ! D bl1 d1 Ol1 d1 D l1 d1 . Hence, ! 2 l1 d1 , and, by the assumption on !, it is so that !dl C l D O, i.e. so that O.! C d1 / D l1 d1 =d1 . It is easy to see that these reasoning can be reversed. t u Our next goal is a description of all isotropic submodules of a given O-CM and its quotients. For this we need a lemma. Lemma 1.17 Let a and b be fractional O-ideals such that a b. The quotient b=a is O-CM. Its generators are the elements Ca, where is in b such that b D O Ca. Proof Multiplying by a suitable integer, we can assume without loss of generality that a and b are integral O-ideals. Let c be a fractional O-ideal in the ideal class of b1 which is relatively prime to the (integral) O-ideal ab1 . We thus have cCab1 D O and bc D O, for some 2 b. Consequently, we have O C a D b. This implies

10


b=a D O. C a/. Indeed, let b C a 2 b=a. Hence, b D d C c, for some d 2 O, c 2 a. Then b C a D d C a D d. C a/. This proves the lemma. u t Theorem 1.2 Let M D .M; Q/ be an O-CM with level l, modified level m D l .2; l/2 and annihilator a (recall from Theorem 1.1 that a D m .2; l/). (i) The isotropic submodules of M are of the form ab1 M , where b is an integral O-ideal such that b2 jm. In particular, the sum of any two isotropic submodules is again isotropic. ı submodule of M .!/ (so that b2 jm), (ii) If M D M .!/, and ab1 a is an ı isotropic 1 then the quotient module M .!/ .ab =a/ is isomorphic to the O-FQM M .! 2 / D O=ab2 ; x C ab2 7! ! 2 x 2 C d1 ; where 2 b is such that b D O C ab1 (see Lemma 1.17). (iii) In particular, the class of O-CM is closed under taking quotients. Remark If the set of isotropic submodules of an O-FQM is closed under taking sums then it possesses only one maximal isotropic submodule, namely the sum of all isotropic submodules. Vice versa, if it possesses only one maximal isotropic submodule then the sum of any two isotropic submodules is contained in the unique maximal one, and hence also isotropic. Thus, by part (i), an O-CM possesses only one maximal isotropic submodule. Example 1.18 Note that there are also O-FQM which have this property, but are not cyclic. The O-FQM .Z=4Z Z=4Z; Q/ with Q.x C 4Z; y C 4Z/ D .x 2 C xy C y 2 /=4 C Z is such an example (for K D Q). It possesses exactly five isotropic submodules, namely, the submodules 0, h.0; Œ2 /i, h.Œ2 ; Œ2 /i, h.Œ2 ; 0/i and h.0; Œ2 /i ˚ h.Œ2 ; 0/i, where the last one is the unique maximal one. (Here we use Œx D x C 4Z.) Proof of Theorem 1.2 First we prove (i). We have that M is isomorphic to an OFQM M .!/ for some nonzero ! 2 K as given in Theorem 1.1. Clearly, any Osubmodule of M .!/ is of the form c=a for some integral O-ideal c such that a c. Let c=a be an isotropic submodule of M .!/. For all x 2 c, we have !x 2 2 d1 i.e. the ideal !dc2 is integral (see Lemma 1.14). Hence, l divides c2 . Therefore, any isotropic submodule of M .!/ is of the form c=a with some integral O-ideal c such that ljc2 ja2 . It is then clear that the isotropic submodules of M are of the form cM , where c runs through the set of integral O-ideals which satisfy ljc2 ja2 . However, it is easily checked that the following map is an isomorphism: fc O W ljc2 ja2 g ! fb O W b2 jmg;

c 7! ac1 DW b:

Hence, the first statement of (i) is proved. Let U and V be two isotropic submodules of M , say, U D ab1 1 M and V D 1 ab2 M . Then 1 1 1 1 1 U C V D ab1 1 M C ab2 M D ab1 C ab2 M D a.b1 ; b2 /b1 b2 M;


11

and since b21 and b22 divide m it is clear that the square of the least common multiple b1 b2 .b1 ; b2 /1 also divides m, i.e. that U C V is isotropic. We prove the statement (ii). Let ab1ı=a be an isotropic submodule of M .!/. To determine the quotient module M .!/ .ab1 =a/, we need to determine the dual module .ab1 =a/# . Using the Definition (1.1), we obtain .ab1 =a/# D fx C a 2 O=a W 2!xab1 d1 g D fx C a 2 O=a W bjxg D b=a: For the second equality we write !d D b0 l1 , where b0 in an integral O-ideal such that .b0 ; l/ D 1. Let x C a 2 O=a. Then we have that 2!dxab1 is integral if and only if bjx. Indeed, since a D l.2; l/1 (see Proposition 1.13), we have 2!dxab1 D 2.2; l/1 b0 b1 x. But b is relatively prime to 2.2; l/1 b0 , since b is a divisor of l.2; l/1 . Therefore, we have ı ı .ab1 =a/# .ab1 =a/ D .b=a/ .ab1 =a/ ' b=ab1 ; and hence ı M .!/ .ab1 =a/ ' b=ab1 ; x C ab1 7! !x 2 C d1 DW N : Note that the annihilator of the O-module b=ab1 equals the ideal ab2 (which is integral, since b2 divides m and m divides a). By Lemma 1.17 we know that b=ab1 is an O-CM, i.e. there exists 2 b such that b D O C ab1 . The application x C ab2 7! x C ab1 defines therefore an isomorphism

' ! N; O=ab2 ; x C ab2 7! ! 2 x 2 C d1

which proves (ii). Lastly, the statement (iii) is an immediate consequence of (i) and (ii).

t u

Corollary 1.19 Let M D .M; Q/ an O-CM, and let a, l, m denote its annihilator, level and modified level. Then the annihilator, the level and the modified level of the quotient module M =ab1 M equals ab2 , lb2 and mb2 , respectively. Proof Set U D ab1 M . By Theorem 1.2 (ii), M =U is isomorphic to some O-FQM M .! 2 / with b D O C ab1 ( 2 b). Clearly, the latter has annihilator ab2 . It is enough to show that the O-FQM M .! 2 / has level lb2 . Because, then, it is clear that the modified level of M =U equals mb2 . Write O D bb0 with some integral O-ideal b0 . The level of M .! 2 / equals the denominator of !d 2 . To show that M .! 2 / has level lb2 , it is enough to show that lb2 is relatively prime to b0 , since the denominator of !d equals l. The identity b D O C ab1 implies that .b0 ; ab2 / D 1. Since b2 jm, we have that .2; l/jab2 . Here we used the fact that a D m.2; l/ (see Proposition 1.13). Hence, .2; l/ is also relatively prime to b0 . Since

12


we have lb2 D ab2 .2; l/, obviously b0 is relatively prime to lb2 . This proves the corollary. t u We finally describe the orthogonal groups ı of O-CM. It will turn out that O.M / is isomorphic to a certain subgroup of O a , for which we introduce a special name. Definition 1.20 Let M D .M; Q/ be an O-CM with level l and annihilator a. We define: ı E.M / WD f" C a 2 O a W "2 1 mod lg: Remark Note that E.M / is well-defined. Namely, assume " "0 mod a. Since .2; l/2 jl D a.2; l/ (Proposition 1.13) we deduce .2; l/ja, hence " "0 mod .2; l/, 0 0 2 02 and ıthen a.2; l/j." " /." C " /, i.e. " " mod l. In fact, E.M / is a subgroup of O a . Proposition 1.21 Let M D .M; Q/ be an O-CM. The application g 7! mg , '

where mg denotes multiplication by g, defines an isomorphism of groups E.M / ! O.M /. Proof For the well-definedness we need to show that multiplication by an element g D " C a 2 E.M / defines an orthogonal transformation of M . Since "2 1 mod l for any x 2 M , we have Q."x/ Q.x/ D Q.x/."2 1/ D 0, i.e. we have Q."x/ D Q.x/. The injectivity is obvious. For the surjectivity we need to show that any ˛ in O.M / is given by ˛.x/ D gx for some g 2 E.M /. Write M D O for some 2 M and ˛. / D " for some " 2 O and " … a. Write x D a with a 2 O and a … a. Since ˛ is an O-module homomorphism, we have ˛.x/ D a˛. / D a" D a " D x": This proves the proposition.

t u

Proposition 1.22 Let M be an O-CM with annihilator a and modified level m. a Then ` the map " C a 7! f" C p gp defines an isomorphism of groups E.M / ' pa ka E.M .p//. Via this isomorphism the factor E.M .p// corresponds to the subgroup h"p C ai of E.M /, where "p denotes an integer in O such that "p 1 mod pa and "p C1 mod apa . If p − m, then E.M .p// is the trivial subgroup of E.M /, otherwise E.M .p// has order 2. (Recall by Proposition 1.13 that m divides a.) Proof The isomorphism follows from the Chinese remainder theorem (see for example [Neu99, I. 3, Theorem (3.6)]). It is obvious that the subgroup h"p C ai contains at most two elements.


13

If p − m, then pa k.2; l/, where l is the level of M . Here note by Proposition 1.13 that a D m.2; l/. But this implies that C1 and 1 are equivalent modulo pa , i.e. "p C1 mod a. Hence, E.M .p// is the trivial subgroup of E.M /. If pjm, then vp .2; l/ a 1. But Proposition 1.13 implies that vp .2; l/ D vp .2O/. Hence, pa − 2, i.e. C1 and 1 are inequivalent modulo pa , and thus they are inequivalent modulo a, which implies finally that E.M .p// has order 2. t u Proposition 1.23 Let M be an O-CM with level l, modified level m and annihilator a. The linear characters of E.M / are parameterized by the square-free divisors of m. More precisely, the linear characters of E.M / are of the form: f

W E.M / ! f˙1g;

f ."

C a/ D f; ." C 1/.2; l/1 ;

where f runs through the square-free divisors of m. (Here is the Möbius -function of K (see section Notations) and it is applied to the ideal f; p/.) Remark Let pa ka and pjm. If " D "p , where "p is as in Proposition 1.22, then we have f ."p C a/ D .f; p/. Indeed, since a D m.2; l/ (see Proposition 1.13) and pjm, we have vp .2; l/ a 1. However, "p C 1 is divisible by pa . Hence, p divides ." C 1/.2; l/1 . Proof of Proposition 1.23 To begin with, we show that f is well-defined. First note that the ideal ." C1/.2; l/1 is integral. Indeed, suppose that p is an even prime ideal dividing l. Set l D vp .l/ and u D vp .2O/. By Proposition 1.13, we have u < l, and hence vp .2; l/ D u. This implies that pu divides ." 1/." C 1/. Assume for contradiction that pu does not divide " C 1. Then, say, ps divide " C 1 (s < u). Since pl divides ." 1/." C 1/, we have that " 1 is divisible by pls . But l s > u, since l s > l u and l u u (this is an easy consequence of Proposition 1.13). Hence, pu divides " 1. This is a contradiction, since " 1 " C 1 mod pu . Now we show that the map f depends only on the residue class of " modulo a. Suppose x 2 " C a. We write x D " C a, for some a 2 a. Hence, we have .x C 1/.2; l/1 D ." C 1/.2; l/1 C a.2; l/1 ." C 1/.2; l/1 C m which proves well-definedness. Here we use the fact that a D m.2; l/ (see Proposition 1.13). Next we show that f defines a group homomorphism from E.M / to f˙1g. Let " C a, "0 C a 2 E.M / and p be a prime ideal divisor of m. We need to prove the following statements. (i) If pj." C 1/.2; l/1 and pj."0 C 1/.2; l/1 , then p − .""0 C 1/.2; l/1 (ii) If pj." C 1/.2; l/1 and p − ."0 C 1/.2; l/1 , then pj.""0 C 1/.2; l/1 (iii) If p − ." C 1/.2; l/1 and p − ."0 C 1/.2; l/1 , then p − .""0 C 1/.2; l/1 . We can write ""0 C 1 ." C 1/."0 1/ " C 1 "0 1 D C .2; l/ .2; l/ .2; l/ .2; l/ D

"0 C 1 " 1 ." 1/."0 C 1/ C : .2; l/ .2; l/ .2; l/

(1.2) (1.3)

14


First we prove that pj."C1/.2; l/1 if and only if p − ."1/.2; l/1 . If p is odd, then this statement is obvious. If p were even and the contrary held true, then p would divide 2.2; l/1 . But this is a contradiction, since vp .2O/ D vp .2; l/ (see above). Now using this fact it is easy to deduce .i /, .i i / (using (1.2)) and .iii/ (using (1.3)). It remains to show that every homomorphism from E.M / to f˙1g is of this form, i.e. there exists a square-free divisor f of m such that ." C a/ D f ." C a/ for any " C a 2 E.M /. Let p be a prime dividing m and let "p be as in Proposition 1.22. By setting fD

Y

p;

pjm

."p Ca/D1

we recognize the claimed statement. t u

1.3 Some Lemmas Concerning Quotients O=a In this section a stands for a nonzero integral ideal of O. Moreover, R denotes the ring O=a and W O ! R stands for the canonical projection. In the present section we shall analyze the structure of the ring R and we shall provide some lemmas which will be needed in the next chapter. Recall the well-known fact that the integral ideals of O containing a are in one to one correspondence with the ideals of R via the map b 7! .b/. Lemma 1.24 If b is an arbitrary integral ideal in O, then one has .b/ D .bCa/. Proof It is clear that .b/ .b C a/. Vice versa, let 2 .b C a/. Then D .y C x/ (y 2 b, x 2 a). Hence, D .y C x/ C a D y C a D .y/ .b/. Therefore, the claimed identity holds true. t u Lemma 1.25 The ring R is a principal ideal ring. Proof By the Chinese remainder theorem (see e.g. [Neu99, I. 3, Theorem (3.6)]), it suffices to consider a D pn , where p is a prime ideal of O. We claim first of all that the ideals of O=pn are .1/; .p/; : : : ; .pn /. Let I be an ideal of O=pn . Then I D . 1 .I // D . 1 .I / C pn / D .pm / for some 0 m n, i.e. the claim holds true. Note that the second equality follows from Lemma 1.24. For the third equality we used the fact that 1 .I / C pn is an ideal of O containing pn . Next we prove that every ideal of O=pn is principal. Fix an m such that 0 m n. It suffices to prove that the ideal .pm / is principal. Let c 2 p and c … p2 . Then pm D c m O C pn , since the greatest common divisor of c m and pn is pm . Hence, .pm / D .c m O C pn / D .c m O/ D .c m /R, where we used Lemma 1.24. This proves the lemma. t u

1.3 Some Lemmas Concerning Quotients O=a

15

Remark Let p be a prime ideal of O. The proof of Lemma 1.25 implies that the ideals of O=pn are of the form .c m /=pn (0 m n), where c 2 p and c … p2 , i.e. there are in total n C 1 ideals of O=pn. From this we conclude that O=pn is a principal local ring whose unique maximal ideal is .c/=pn . Lemma 1.26 We have ˛R D ˇR if and only if there exists some " 2 R such that ˛ D "ˇ. Proof The statement would be trivial if R possessed no zero divisors, which, however does not hold true unless a is prime. By the Chinese remainder theorem [Neu99, I. 3, Theorem (3.6)], it is enough to consider a D pn , where p is a prime ideal of O. Let I stand for the set of ideals of O=pn . We need to show that the following map .O=pn/ n.O=pn/ ! I;

Œ˛ 7! ˛R

is an injection. For that it suffices to show that .O=pn / n.O=pn / has n C 1 elements, since I has n C 1 elements (see Lemma 1.25 and the remark afterwards). (Here we use Œ˛ for the orbit of ˛ under left multiplication by elements of .O=pn/ .) Let c 2 p and c … p2 . It remains to prove the following identity n X

jŒc m C pn j D N .pn /;

(1.4)

mD0

since then we can take the elements c m C pn (0 m n) as representatives for the orbit space, i.e. the orbit space has n C 1 elements as we claimed. We calculate the number of elements in each orbit, i.e. the number jŒc m C pn j for all 0 m n . By the so-called Orbit-Stabilizer theorem, we have jŒc m C pn j D '.pn /=j Stab.c m C pn /j, where ' is the Euler’s totient function, i.e. '.a/ D j.O=a/ j for an integral O-ideal a. But we have Stab.c m C pn / D fx C pn 2 .O=pn / W xc m c m mod pn g D fx C pn 2 .O=pn / W x 1 mod pnm g: The above identity implies that Stab.c m Cpn / equals the kernel of the reduction map .O=pn/ ! .O=pnm/ . From the first isomorphism theorem for groups, we have j Stab.c m C pn /j D '.pn /='.pnm/. Therefore, jŒc m C pn j D '.pnm /. Hence, to obtain (1.4), it is enough to show '.pn / D N.pn / N.pn1 /. But from Lemma 1.25 and the remark afterwards we have that O=pn is a local principal ideal ring with the maximal ideal .c/=pn , where c 2 p and c … p2 . Since .c/=pn has N.pn1 /-many elements, i.e. the non-units in O=pn are N.pn1 / in total, the last assertion holds true. t u Remark Let p be a prime ideal of O. From the proof of Lemma 1.26 we have that the elements c m C pn (0 m n), where c is an element in p but not in p2 , can be

16


taken as representatives for the orbit space of the left action of the group .O=pn/ on .O=pn/. Lemma 1.27 If O D xO C yO C a, then there exists x 0 ; y 0 2 O such that x 0 x mod a and y 0 y mod a with O D x 0 O C y 0 O. Proof The statement is obvious if x D y D 0. Without loss of generality we assume y ¤ 0. Let g WD xO C yO. We set y1 WD

Y

pt ;

y2 WD .yO/y1 1 :

pt kyO pjg

Obviously, y1 and y2 are relatively prime. Let t be an integral O-ideal in the inverse ideal class of y2 a which is relatively prime to xya. Then there exists a 2 O such that aO D y2 at. Set x 0 D x C a, y 0 D y. It remains to show that h WD x 0 O C yO equals O. Assume for contradiction that there exists a prime ideal p dividing h. Then p divides either y1 or y2 . If p divides y1 , then by the very definition of y1 , the ideal p divides g, and hence it divides x. But since p also divides x 0 , the ideal p divides aO D y2 at. But by the choice of t, the ideals p and t are relatively prime, hence p divides a which contradicts with the assumption. If p divides y2 , then p divides aO, and hence it divides xO. But this implies that p divides g, and hence it divides y1 which contradicts with the fact that y1 and y2 are relatively prime. This proves the lemma. t u Lemma 1.28 If R D ˛R C ˇR, then there exists x 2 ˛ and y 2 ˇ such that O D xO C yO. Proof Since is a surjection, we have .O/ D .x/.O/ C .y/.O/, where ˛ D .x/ and ˇ D .y/ with x; y 2 O. Then we have .O/ D .xO C yO/, i.e. O D xO C yO C a. Lemma 1.27 implies now the result. t u a b .b/ Lemma 1.29 The map W SL.2; O/ ! SL.2; R/, c d 7! .a/ .c/ .d / defines an epimorphism. Proof We write and R for SL.2; O/ and SL.2; It is clear that R/, respectively. .a/ .b/ the map is a group homomorphism. Let B WD .c/ .d / 2 R with a; b; c; d 2 O. Here note that since is a surjection, every element of R is of this form. To prove the lemma we need to find an element A in such that .A/ D B. Since B 2 R , we have .c/R C .d /R D R. From Lemma 1.28, there exists x in .c/ and y in .d / such that xO C yO D O. Since .x/ D .c/ and .y/ D .d /, we have .ad bc/ D .ay bx/. Hence, we can write ay bx D 1 C k.x y/ for some k 2 a. Hence, .a C k/y .b C k/x D 1. Therefore, the bCk matrix aCk is an element of and, obviously, it is mapped to B by . This x y proves the lemma. t u

1.3 Some Lemmas Concerning Quotients O=a

17

Lemma 1.30 Given ˛; ˇ 2 R. There exists 2 R and A 2 SL.2; R/ such that .0; /A D .˛; ˇ/. Here multiplication of a row vector with a matrix is done in the usual way. Proof By the Chinese remainder theorem [Neu99, I. 3, Theorem (3.6)] it is enough to consider a D pn . Let a 2 ˛ and b 2 ˇ. From the remark after Lemma 1.26, we can write a c m1 e1 mod pn and b c m2 e2 mod pn (0 m1 ; m2 n), where e1 ; e1 2 .O=pn/ . If m1 m2 , we have 1 1 D .c m1 e1 ; c m2 e2 / .a; b/ mod pn : .0; c m1 / e0 c me m 2 1e 1

2

.e1 /1 we see that the statement of the By taking D .c m / and A D .e0 1 / .c m2 m1 e / 2 lemma holds true. If m2 m1 , then using the above argument, we find A 2 SL.2; O=pn / and in O=pn such that .0; /A D .ˇ; ˛/. Since .ˇ; ˛/ D .˛; ˇ/S , where S D 10 1 0 , we obtain .0; /AS 1 D .˛; ˇ/. This proves the lemma. t u

Chapter 2

Weil Representations of Finite Quadratic Modules

We carry over the notations of the previous chapter. As before, K denotes a number field of degree n over Q, and we use O, d for the ring of integers and the different of K, respectively. Moreover, we shall use for the group SL.2; O/ and Q for a certain central extension of (see Sect. 2.2 for the definition of Q ). Occasionally, we shall denote by R for a ring R, the group SL.2; R/. In this chapter we shall associate Weil representations to finite quadratic Omodules (O-FQM) and develop a basic theory of these representations. The main result of this chapter will be Theorem 2.5, which describes the complete decomposition of cyclic Weil representations, i.e of Weil representations associated to cyclic O-FQM, and Theorem 2.6, which gives the explicit description of all onedimensional subrepresentations of cyclic Weil representations. The latter theorem will play an important role when we determine explicitly all singular Jacobi forms of certain indices, since in Chap. 4, we shall see that the singular Jacobi forms will correspond to the one-dimensional subrepresentations of certain cyclic Weil representations. In Sect. 2.1, we shall briefly recall notations and facts concerning representations of groups which will be used in the sequel. In Sect. 2.2, we shall define the Weil representations of Q associated to O-FQM. In fact, though we shall use throughout the term representations we view the Weil representations rather as modules over Q . In Sect. 2.3, we shall study decompositions of Weil representations. For arbitrary finite quadratic modules, our decompositions of the associated Weil representations are in general not complete, i.e. they are neither splittings into direct sums nor the subrepresentations which occur in the given decompositions are irreducible. However, as we shall see in Sect. 2.4, for cyclic Weil representations, these decompositions are in fact complete. The proof of this completeness relies on Theorem 2.7 of Sect. 2.6, which provides an upper bound for the number of irreducible Q -submodules of an arbitrary Weil representation. Using the dimension formulas for the irreducible Q -submodules of a cyclic Weil representation, we shall

© Springer International Publishing Switzerland 2015 H. Boylan, Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields, Lecture Notes in Mathematics 2130, DOI 10.1007/978-3-319-12916-7_2

19

20

2 Weil Representations

be able to determine in Sect. 2.5, the one-dimensional submodules of cyclic Weil representations. In Sect. 2.6, for deriving our estimate for the number of irreducible Q submodules of cyclic Weil representations, we have to develop a machinery which is also interesting for its own sake since it gives a explicit description of the Weil representations as projective representations of D SL.2; O/.

2.1 Review of Representations of Groups To fix the language, we shall briefly recall those basic notions and facts from the general theory of representations of groups which we shall need in the sequel. In particular, we shall introduce and discuss the notion of a projective action of a group on a vector space. This notion will be useful for us in later sections. In the following, G denotes a multiplicative group with identity element 1. For the convenience of the reader we shall give proofs of some basic propositions for which we could not find suitable references. Definition 2.1 Let G be a group acting from left on a set X . We use GnX for the set of orbits of the G-action. For an element v 2 X , we use Stab.v/ for the subset of elements of G which are fixed under the G-action, i.e. Stab.v/ D fg 2 G W gvDvg. (In fact, the set Stab.v/ is a subgroup of G.) If G acts from right on X , the set of orbits of the G-action is denoted by X=G. Unless otherwise stated when we speak of a group action, we suppose that the group acts from the left. Proposition 2.2 Suppose there exits a group homomorphism from G onto H . If H acts on a set X , then G also acts on X , via gv WD .g/v. If is surjective then the number of elements of GnX equals the number of elements of H nX . t u

Proof The proof is obvious. 0

Definition 2.3 Let f be a group homomorphism from G to G and be a surjective group homomorphism from G onto H . We say that f factors through (or sometimes, if is clear from the context, that f factors through H ), if there exits a group homomorphism f from H to G 0 such that f ı D f . The following proposition is a standard proposition from basic algebra. Proposition 2.4 Let G, G 0 , H , and f be as in Definition 2.3. Then f factors through if and only if Ker./ Ker.f /. Proof If f exists such that f ı D f then obviously Ker./ Ker.f /. Assume vice versa Ker./ Ker.f /. Let h 2 H be given. Since is a surjection, there exists g 2 G such that .g/ D h. We define f W H ! G 0 by f .h/ D f .g/. By the assumption this map is well-defined, i.e. does not depend on the particular choice of

2.1 Review of Representations of Groups

21

the preimage g of h. By definition f ..g// D f .g/. Finally it is obvious that f is a group homomorphism. t u Definition 2.5 A G-module is a vector space V together with an operation from G V to V of G on V such that, for each g in G, the map v 7! gv is linear. If the action is clear from the context then we simply say that V is a G-module. The group homomorphism W G ! GL.V /;

.g/.v/ D gv

is called the representation afforded by V . If G acts from the right on V , then we say that V is a right G-module. (In this case satisfies .gh/ D .h/.g/ for all g, h in G.) Remark If there exists a representation of G on V , i.e. a group homomorphism W G ! GL.V /, then V becomes a G-module via the action .g; v/ 7! gv WD .g/.v/ and is the representation afforded by this V -module. Definition 2.6 Let V be a G-module and be the representation afforded by V . Suppose that there exists a surjective group homomorphism from G onto H . If factors through , then we say that the representation of G factors through a representation of H and that the G-action on V factors through an action of H . Definition 2.7 Let V be a finite dimensional complex Hilbert space with inner product h ; i. We call a representation W G ! GL.V / unitary if .g/ is unitary for every g in G (i.e. if h.g/v; .g/wi D hv; wi for all v, w in V ). We say that G acts unitarily on V , if the afforded representation is unitary. Definition 2.8 Let V be G-module and be the representation afforded by V . A subspace W of V is called a G-invariant subspace of V or a G-submodule of V , if gW W (g 2 G). The G-module V or the representation is called irreducible, if there is no proper nonzero G-invariant subspace of V . Definition 2.9 Let V be a G-module and be the representation afforded by V . For g 2 G, we use tr.g; V / for the trace of the matrix determined by the automorphism .g/ of V . The map V W G ! C defined by V .g/ D tr.g; V / is called the character of the G-module V and the character or trace of . An irreducible character of G is a character of an irreducible G-module. A linear character of G is a character of a one-dimensional G-module. Remark Every linear character of a finite group G gives rise to a group homomorphism from G to S1 , and vice versa. Ln Proposition 2.10 If V D i D1 Vi is a decomposition of V into G-invariant subspaces, then the character of V equals the sum of the characters of Vi , i.e. we have:

V .g/ D

n X i D1

Vi .g/

.g 2 G/:

22


Proof We proceed by induction onL n. For n D 1, the result is immediate. We prove the result for n D 2. Write V D A B, where A and B are G-invariant subspaces of V . Let fa1 ; : : : ; as g be a basis for A and fb1 ; : : : ; bt g be a basis for B, and let g in G. Since A andP B are G-submodules of P V , we have gai 2 A and gbj 2 B, and s t therefore gaP gb D i D j kD1 cki aj and lD1 dlj bl with P P cki ; dljP2 C. But then tr.g; A/ D k ckk , tr.g; B/ D l dl l and tr.g; V / D k ckk C l dl l , which is the claimed formula. V/ D n 2. Using the previous result, wehave L Suppose Ltr.g; n1 . By induction hypothesis, we have tr g; tr.g; Vn / C tr g; in1 V V i D1 i D1 i D Ln1 tr.g; V /. This proves the proposition. t u i i D1 Definition 2.11 Let V and W be G-modules. A C-linear map ' W V ! W is called G-linear, if '.gv/ D g'.v/ (g 2 G, v 2 V ). Definition 2.12 Let V and W be G-modules. Let and stand for the representations afforded by V and W , respectively. We say that V and W are isomorphic as G-modules, and that and are equivalent, if there exists a G-linear isomorphism W V ! W (or, equivalently, if there exists an isomorphism of vector spaces W V ! W such that ı .g/ D .g/ ı for all g 2 G). Proposition 2.13 ([FH91, Cor. 2.13, Cor. 2.14]) Let G be a finite group. The set of irreducible characters of G is finite. Two G-modules are isomorphic as G-modules if their characters coincide. Definition 2.14 For a G-module V we use V G for the subspace of G-invariants of V , i.e. the subspace of all v in V such tat gv D v for all g in G. Proposition 2.15 Let G be a finite group. If V is a G-module, then the application 1 P G v 7! jGj g2G gv defines a surjective map ' W V ! V . Proof It is clear that the map is well defined, i.e. '.V / V G . Suppose v 2 V G . Then we have gv D v for all g. Hence '.v/ D v, which implies that the map is surjective. t u Lr Proposition 2.16 Let V be a G-module. Suppose V D i D1 Vi with irreducible G-submodules Vi . If W is a nonzero irreducible G-submodule of V , then W is Glinearly isomorphic to Vi for some i . P Proof Let Pi denote the projection of V onto Vi . Then riD1 Pi D 1. Since W ¤ 0, there exists some i such that Pi .W / ¤ 0. Since Pi .W / is a G-submodule of Vi , and Vi is irreducible, we have Pi .W / D Vi . So, the map Pi jW is surjective. But the kernel of Pi jW is a G-submodule of W , so it must be zero, since W is irreducible and Pi jW ¤ 0. This proves the proposition. t u Proposition 2.17 ([FH91, Prop. 1.8]) Let G be finite, let GO denote the set of O let V denote the sum of those Girreducible characters of G. For in G, submodules of V which afford the character . Then V is the largest G-submodule of V whose character is a multiple of . Moreover, one has


23

V D

M

V :

(2.1)

2GO

Lm If V D j D1 Wj is a decomposition of V into G-submodules Wj such that the character of Wj is a multiple of an irreducible character L

j and such that j 6D k for j 6D k, then Wj D V j , and the splitting V D Wj coincides with the decomposition (2.1) (after deleting zero spaces up to permutation of the summands). Definition 2.18 The decomposition (2.1) is called the canonical decomposition of the G-module V . Remark If G is abelian, then GO is a group with respect to the usual multiplication of functions. Namely, GO coincides with the group of linear P characters of G, called O we have V D the dual group of G. In this case, for 2 G, v Cv, where the sum is over all v 2 V which satisfy gv D .g/v for all g 2 G. In other words, we have V D fv 2 V W gv D .g/v; 8g 2 Gg: Proposition 2.19 We carry over the notations of Proposition 2.17. Let Vi be an irreducible G-submodule of V . Then, dim V D

dim Vi X

V .g/ V .g/: jGj g2G i

P Proof From [FH91, Eq. (2.32)], we know that W dim Vi =jGj g2G Vi .g/g defines a projection from V onto V . Let v1 ; : : : ; vm be a basis for V and vmC1 ; : : : ; vn be a basis for the kernel of . Hence, v1 ; : : : ; vn becomes a basis for V . We then have .vi / D vi for 1 i m and .vi / D 0 otherwise. Hence dim V equals the trace of the map . This proves the proposition. t u Corollary 2.20 Let G be a finite group and V be a G-module. We have dim V G D

1 X

V .g/: jGj g2G

Proof This follows immediately from Proposition 2.19 in the case of being the trivial character. u t Let V and W be G-modules. The spaces V ˚ W and V ˝ W are also G-modules via .g; v ˚ w/ 7! .gv ˚ gw/;

.g; v ˝ w/ 7! .gv ˝ gw/;

(2.2)

respectively. The space of all C-linear maps from V to W is denoted by Hom.V; W /. In particular, we set V WD Hom.V; C/. Moreover, the space of all G-linear maps

24


from V to W is denoted by HomG .V; W /. In particular, the space HomG .V; V / is called the intertwining algebra of V . It is not difficult to see that Hom.V; W / is a G-module via the following G-action .g; / 7! g ;

g

.v/ WD g.g 1 v/:

(2.3)

In fact, (2.3) defines also a G-module structure on Hom.V; W / if V D W and if V is not a G-module but only a projective G-module (see Definition 2.23 below), provided the projective representation afforded by V satisfies .g1 /.g/ D 1. We denote by V the G-module whose underlying vector space is the dual space V of V and where the G-action is given by: .g; / 7! g

where g .v/ D .g 1 v/:

(2.4)

The spaces V ˝ W and Hom.V; W / can be identified via the following Gmodule isomorphism V ˝ W ! Hom.V; W /;

˝ ! 7! “v 7! .v/! 00 :

(2.5)

Proposition 2.21 Let V , W be G-modules. The following holds true: HomG .V; W / D Hom V; W /G : Proof From (2.3), for any g 2 G and any v 2 V , we have g .v/ D g.g 1 v/. So, if is G-linear (i.e. 2 HomG .V; W /), then clearly g D (i.e. 2 Hom V; W /G ). On the other hand, if g D , then clearly g 1 .v/ D .g 1 v/ which implies that is G-linear. This proves the proposition. t u Proposition 2.22 Let G be a finite group, and V be a G-module. The number of irreducible G-submodules of V is less than or equal to the dimension of the space HomG .V; V /. Proof We denote by j (j D 1 : : : ; m) the characters of the distinct irreducible Gsubmodules of V (see Proposition 2.13 for finiteness of the number of irreducible

j characters). Let V D ˚m be the canonical decomposition of V (see j D1 V

j Proposition 2.17), where V is the sum of those G-submodules of V which have character j . It is enough to prove

j HomG .V; V / ' ˚m ; V j /: j D1 HomG .V

(2.6)

Ldj Namely, let V j D kD1 Vj;k be a decomposition into irreducible submodules. Then, from the fact dim HomG .V j ; V j / D dj2 which we shall prove in a moment, we obtain dim HomG .V; V / m (since dj2 1 for all j ).


25

First we prove dim HomG .V j ; V j / D dj2 . From Proposition 2.21, we have HomG .V j ; V j / D Hom.V j ; V j /G . Using Corollary 2.20, we then have dim HomG .V j ; V j / D

1 X

j

j .g/:

jGj g2G Hom.V ;V /

But via the identification in (2.5) and [FH91, Prop. 2.1], we have

Hom.V j ;V j / D V j V j D dj2 j j : Using [FH91, Eq. (2.10)], we now recognize the claimed identity. Finally, we prove (2.6). Let L 2 HomG .V; V /. If we can show that for each j , L.V j / is a subset of V j , then obviously the map L 7! .LjV j /j defines an Pdj isomorphism. Since L is a linear map, we can write L.V j / D kD1 L.Vj;k /. The kernel of LjVj;k is either 0 or Vj;k , since Vj;k is an irreducible G-module. If the kernel of LjVj;k is Vj;k , there is nothing to prove. Suppose the kernel of LjVj;k is 0. Then L.Vj;k / ' Vj;k , and hence L.Vj;k / V j . But this implies that L.V j / V j . t u Definition 2.23 A projective action of G on a vector space V is a map G V ! V , .g; v/ 7! gv such that, for all g; h 2 G, there exists a constant .g; h/ 2 C such that (i) g.hv/ D .g; h/.gh/v (ii) 1v D v .v 2 V / .

.v 2 V; g; h 2 G/,

The space V is then called a projective G-module. The map W G ! GL.V /, .g/ D gv is called the projective representation afforded by the projective Gmodule V . The map W G G ! C is called the multiplier system of the projective G-module V . Remark Note that the projective representation afforded by the G-module V satisfies .g/.h/ D .g; h/.gh/ for all g, h in G. If vice versa is a projective representation of G, i.e. if is a map W G ! GL.V / such that for all g; h 2 G there exists a constant .g; h/ 2 C which satisfies .g/.h/ D .g; h/.gh/

.g; h 2 G/;

then the map .g; v/ 7! .g/.v/ defines a projective G-module structure on V and is the projective representation afforded by V . Remark Note that the multiplier system of a projective G-module satisfies .1; g/ D .g; 1/ D 1

(2.7)

.g 0 ; g 00 /.g; g 0 g 00 / D .g; g0 /.gg0 ; g 00 /;

(2.8)

26


as follows immediately from the axioms (i) and (ii). Indeed, for proving (2.8), we write, for v 2 V g g 0 .g 00 v/ D .g 0 ; g 00 /g .g 0 g 00 /v D .g 0 ; g 00 /.g; g 0 g 00 /.gg0 g 00 /v g g 0 .g 00 v/ D .g; g0 /.gg0 /.g 00 v/ D .g; g0 /.gg0 ; g 00 /.gg0 g 00 /v; and comparing yields the claimed cocycle relation. Definition 2.24 Let V be a projective G-module with multiplier system . We define GV to be the set GV D f.g; / W g 2 G; 2 C g; where C is the subgroup of C generated by the .g; h/ (g; h 2 G) together with the multiplication .g; / .g 0 ; 0 / WD gg0 ; 0 .g; g0 / :

(2.9)

Proposition 2.25 The multiplication (2.9) defines the structure of a group on GV . The sequence

1!C ! GV ! G ! 1; where ./ D .1; / and .g; / D g, and the subgroup .C / lies in the center of GV . In short, GV is a central extension of G by the abelian group C . Proof First we show that GV becomes a group with the operation given in (2.9). The neutral element from (2.7). The inverse of an arbitrary of GV is .1; 1/ as follows element .g; / is g 1 ; 1 .g; g1 /1 . It remains to prove the associativity of the multiplication, i.e. we need to prove .g; / .g 0 ; 0 / .g 00 ; 00 / D .g; / .g 0 ; 0 / .g 00 ; 00 /: On the left we have 0 00 gg g ; 0 00 .g 0 ; g 00 /.g; g 0 g 00 / : On the right we have

gg0 g 00 ; 0 00 .g; g0 /.gg0 ; g 00 / :

Applying (2.8) shows that both sides coincide. The exactness of the given sequence is obvious. That .C / lies in the center of GV follows again from (2.7). t u

2.2 The Weil Representation W .M /

27

Proposition 2.26 Let V be a projective G-module. The space V becomes a GV module via the following GV -action: .g; /; v 7! .g; /v WD .gv/: Proof Clearly .1; 1/ acts as identity. Let .g; /; .g0 ; 0 / 2 GV . For checking the second axiom for a G-action we calculate .g; / .g 0 ; 0 / v D gg0 ; 0 .g; g0 / v D 0 .g; g0 /gg0 v D 0 g.g 0 v/ D g. 0 g 0 v/ D .g; / .g 0 ; 0 /v : t u

This proves the proposition.

2.2 The Weil Representation W.M / Theorem 2.1 The group D SL.2; O/ is generated by S D 1 b (b 2 O). 0 1

0 1 1 0

and Tb D

Proof A theoremof Vaserstein [Vas72, First Thm.] states that is generated by the matrices T c WD 1c 01 and Tb (b; c 2 O). However, we have the following easily deduced identity: T b D S Tb S 1 : t u Let M D .M; Q/ be an O-FQM with associated bilinear form B. We use CŒM for the complex vector space of maps M ! C. Recall from the section Notations that the functions ex (x 2 M ) form a basis of CŒM . To the generators of from Theorem 2.1 we assign linear operators U.S / and U.Tb / on CŒM by setting for the basis elements ex : U.Tb /ex D e fbQ.x/g ex

.b 2 O/ X e fB.y; x/g ey :

1 U.S /ex D .M / p jM j y2M

(2.10)

1 P Recall from Definition 1.8 that we have .M / D pjM x2M e fQ.x/g. j As it will turn out later we can extend the map A 7! U.A/ (A one of our generators) to a projective representation of (Theorem 2.8). Hence we can find a central extension M of which acts on CŒM such that the action of suitable preimages of S and Tb in the extension is given by the operators U.S / and U.Tb / (see Proposition 2.26). However, this extension would a priori depend

28


on the particular underlying O-FQM. Since, on the one hand side, we need such an extension (not necessarily central) which does not depend on the underlying OFQM, and since we do not want to analyze these extensions M more closely here, we adopt the following strategy. For the rest of this chapter we fix once and for all a group Q such that the following four conditions are satisfied: (i) Q acts on W .M / for all O-FQM M (ii) There exists Tb (b 2 O), S 2 Q such that, for every O-FQM M , we have, for all x 2 M , the identities Tb ex D U.Tb /ex ;

S ex D U.S /ex ;

(2.11)

where U.Tb / and U.S / are given by (2.10). (iii) Q is generated by S and Tb (b 2 O). (iv) There is an epimorphism from Q to which maps S to S and Tb to Tb for b in O. In fact, such group exist. For example, we can take for Q the free group generated by elements S and Tb (b 2 O). In general, every group Q satisfying (2.11) will be a quotient of this free group. If K is totally real and if we restrict our theory to O-FQM which are discriminant modules of a totally positive definite even O-lattice (see Definition 3.3), then by Theorem 3.4 we can take for Q the metaplectic cover of SL.2; O/ defined in the next chapter in Sect. 3.3. Definition 2.27 Let M be an O-FQM. We write W .M / for the Q -module CŒM with the Q -action (2.11). By slight abuse of language, we shall refer to W .M / as the Weil representation associated to M . The Weil representation associated to an O-CM is called a cyclic Weil representation. As we stated in the beginning the goal of this chapter will be to decompose W .M / into smaller Q -submodules. For this the particular choice of Q is not important as it is explained by the following proposition, whose obvious proof we leave to the reader. Proposition 2.28 Suppose Q1 L is a group satisfying (i), (ii), (iii) and (iv) (with Q replaced by Q1 ). If W .M / D j Mj , where the Mj are -submodules, then the Mj are also Q1 -submodules. If Mj W .M / is irreducible with respect to Q , then it is also irreducible with respect to Q1 . It is, of course, still an interesting question to describe the smallest quotient of the free group generated by elements S and Tb (b 2 O) which we can take for Q . In fact, the precise answer can be given. Since we do not need it here we confine ourselves to describe it at the end of this section. Before doing so we note an important property of the Weil representations, which will be needed later.

2.2 The Weil Representation W .M /

29

˝ ˛ By _j_ , we denote the Hermitian scalar product on W .M / which is anti-linear in the second argument and which satisfies: ˝

˛

(

ex j ey D

1 if x D y

(2.12)

0 otherwise:

Proposition 2.29 Let M D .M; Q/ be an O-FQM. The operators U.Tb / (b 2 O) and U.S / are unitary with respect to the scalar product in (2.12). In particular, the action of Q is unitary with respect to this scalar product. Proof It suffices to show that the operators U.Tb / and U.S / are unitary. For the former ones this is obvious. For proving that U.S / is unitary P let B be the 0 bilinear form of M and let v; v 2 W .M /, so that v D x2M v.x/ex and P v 0 D x 0 2M v 0 .x 0 /ex 0 . By (2.10) we have X .M / X U.S /v D p v.x/ e fB.y; x/g ey ; jM j x2M y2M and similarly for v 0 . Hence we have ˝

X ˚ ˛ j.M /j2 X U.S /vjU.S /v 0 D v.x/v 0 .x 0 / e B.y; x C x 0 / : jM j 0 y2M x;x 2M

The inner sum equals zero if x 0 ¤ x (see Proposition 1.11), and otherwise it equals jM j. From Proposition 1.10, we know j.M /j D 1. It follows that U.S / is unitary. t u We describe the smallest group Q satisfying (2.11). For this we describe, first of all, the smallest central extensions of SL.2; O/ which the projective representations (2.10) can be lifted to. We use p for the Kubota symbol at the prime p, i.e. for the SL.2; Kp / 2-cocycle p .A; B/ D

x.A/ x.B/ ; x.AB/ x.AB/

: p

Here ._; _/p denotes the Hilbert symbol of the completion Kp of the field K at the prime p, and where, for A D ac db , we use x.A/ D c if c 6D 0 and x.A/ D d otherwise. The Kubota symbol defines a 2-cocycle of SL.2; Kp / [Kub67, Thm.]. For a finite set S of primes of K, we use Mp.2; O; S / for the central extension of SL.2; O/ by the group f˙1g which consists of all pairs .A; t/ with A on SL.2; O/ and t D ˙1, equipped with the multiplication

Y A; t B; t 0 D AB; t t 0 p .A; B/ : p2S

30


For a given prime ideal p and p-module M , we define a function on the group of units O of O by setting "M .a/ D .M a /=.M 1 /: One can prove that either "M .a/ is a character of O for odd p, whereas this is not always true if p is even (see [BS14a, Prop. 5.1]). For a given O-FQM let ˚ SM WD p W "M .p/ is not a character : Thus SM is a subset of the prime ideals of K dividing 2. Theorem 2.2 ([BS14a, Thm. 6.2]) For a given O-FQM M D .M; Q/, the applications .Tb ; t/ 7! tU.Tb / and .S; t/ 7! tU.S / can be extended to a representation U W Mp.2; O; SM / ! GL.CŒM /. Finally, the smallest Q satisfying (2.11) is the projective limit of the projective system built from the finitely many groups Mp.2; O; SM / together with the obvious 0 0 homomorphisms from Mp.2; O; SM / onto Mp.2; O; SM / for SM SM . More Q precisely, one can describe as the group of all elements .A; t/, where t is an element of f˙1gn , and where the multiplication is given by

A; ftp gpj2 B; ftp0 gpj2 D AB; ftp tp0 p .A; b/gpj2 :

Each Mp.2; O; SM / is a quotient of Q , namely, as homomorphic image under the map Q ! Mp.2; O; SM /;

Y 7 A; A; ftp gpj2 ! tp : p2SM

2.3 Decomposition of Weil Representations The purpose of this section will be to determine subrepresentations of the Q modules W .M /, which were introduced in the preceding section. We shall not derive a complete decomposition in general. However our results will suffice to give a complete decomposition of W .M / in the important case of a cyclic M . The main results are Theorems 2.3 and 2.4. For our language concerning representations of groups the reader is referred to Sect. 2.1. For the notions concerning finite quadratic O-modules that we use in the sequel, e.g. orthogonal groups, isotropic modules an so on, we refer to reader to Sect. 1.1. We shall first explain the main results of this section and state three auxiliary propositions which are needed for the understanding of the main results. The

2.3 Decomposition of Weil Representations

31

rest of this section is dedicated to the proofs of these auxiliary propositions (Propositions 2.30, 2.34, and 2.35) and the proofs of the main results. The decomposition of the Q -modules is based on two principles. The first one is that the Weil representation of a quotient of an O-FQM M embeds naturally into W .M / as Q -submodule. Proposition 2.30 Let M D .M; Q/ be an O-FQM and U be an isotropic submodule of M . The linear map U W W .M =U / ,! W .M /;

eX 7!

X

ey

y2X

defines a Q -linear embedding (i.e. an injective Q -module homomorphism). Definition 2.31 Let M D .M; Q/ be an O-FQM. We define the new part W .M /new of W .M / as the orthogonal complement of the subspace X

U W .M =U /

U M U isotropic U ¤0

with respect to the scalar product in (2.12). Here the sum is over all isotropic submodules U 6D 0 of M . Remark By Proposition 2.30 the spaces U W .M =U / are Q -invariant, and hence their sum is so too. Since Q acts unitarily (see Proposition 2.29) the space W .M /new is in fact a Q -submodule of W .M /. Theorem 2.3 Let M D .M; Q/ be an O-FQM. We have the following decomposition of W .M / into Q -submodules: W .M / D W .M /new ˚

X

U W .M =U /new :

(2.13)

U M U isotropic U ¤0

If M contains only one maximal isotropic submodule, then the second sum in (2.13) is an orthogonal sum with respect to the scalar product (2.12). Remark Note that the decomposition (2.13) is a direct sum decomposition for OCM, since a cyclic M contains only one maximal isotropic submodule (see the remark after Theorem 1.2). Recall also that there exist also O-FQM which are not cyclic but contain only one maximal isotropic submodule. The condition that there exists only one maximal isotropic submodule is not necessary for the decomposition in (2.13) to be direct as the subsequent Example 2.32 shows. However, this condition is also not superfluous as we shall show in the Example 2.33 below.

32


Example 2.32 We show that the sum (2.13) applied to the finite quadratic Z-module N WD .Z=2Z Z=2Z; Q/, where Q.x C 2Z; y C 2Z/ D xy=2 C Z, is direct. The nonzero isotropic submodules of N are U1 D h.Œ0 ; Œ1 /i, U2 D h.Œ1 ; Œ0 /i. (Here we use Œx D x C 2Z.) Since jUi# j jUi j D 4 (Proposition 1.7) the quotient modules N =Ui are trivial, in particular, W .N =Ui / D W .N =Ui /new . They are spanned by the vectors e.Œ0 ;Œ0 / C e.Œ0 ;Œ1 / and e.Œ0 ;Œ0 / C e.Œ1 ;Œ0 / , respectively, which are obviously linearly independent. We thus have W .N / D W .N /new ˚ U1 W .N =U1 /new ˚ U2 W .N =U2 /new . Example 2.33 Let N 0 WD .Z=2Z Z=2Z; Q0 /, where Q0 denotes the quadratic form Q0 .x C 2Z; y C 2Z/ D .x 2 C xy C y 2 /=2 C Z. We show that the sum (2.13) applied to M WD N 0 ˚N , where N is as in ˝ Example 2.32, is not ˛ direct. The ˝ nonzero isotropic˛ submodules of M are U D .Œ0 ; Œ0 / ˚ .Œ0 ; Œ1 / , U D 1 2 ˝ ˛ ˝ ˛ .Œ0 ; Œ0 / ˚ .Œ1 ; Œ0 / , U D .Œ1 ; Œ1 / ˚ .Œ1 ; Œ1 / , U D .Œ0 ; Œ1 / ˚ .Œ1 ; Œ1 / and U5 D 3 4 ˝ ˛ .Œ1 ; Œ0 /˚.Œ1 ; Œ1 / . Note that, for all i , Ui is maximal. The order of M =Ui equals 4 (Proposition 1.7). Since the Ui are maximal, the Z-FQM M =Ui are anisotropic, i.e have no nonzero isotropic submodules. (In fact, one can show that M =Ui is isomorphic to N 0 .) Hence we have Ui W .M =Ui / D Ui W .M =Ui /new . Since W .M / has dimension 16 the sum of the five four-dimensional spaces Ui W .M =Ui /new cannot be direct. The second principle for decomposing a Weil representation W .M / is the natural action of the orthogonal group O.M / on W .M / coming from the permutation representations given by its action on M . The main observation is that this action intertwines with the action of Q . Proposition 2.34 The group O.M / acts on the space W .M / via linear continuation of the map: '; ex 7! ' ex WD e'.x/ :

(2.14)

This action is unitary with respect to the scalar product 2.12. The action of O.M / and the action of Q on W .M / commute. In fact, the action of the orthogonal groups enables us to further decompose the spaces W .M //new into Q -submodules as is explained by the next proposition. Proposition 2.35 Let M D .M; Q/ be an O-FQM. The space W .M /new is O.M /invariant. Theorem 2.4 Let M D .M; Q/ be an O-FQM. For each irreducible character of O.M /, the sum W .M /new; of those O.M /-submodules of W .M /new which afford the character , is invariant under Q . In particular, we have the decomposition of W .M /new into Q -submodules W .M /new D

M

b

2O.M /

W .M /new; :

(2.15)


33

1

(Recall that O.M / denotes the set of irreducible characters of the orthogonal group O.M /.) Remark Note that the components of the decomposition (2.15) are in general not irreducible Q -modules. However, for O-CM, they turn out be irreducible (see Sect. 2.4). Proof of Proposition 2.30 Let B stand for the bilinear form associated of M . It is enough to prove the result for the elements Tb (b 2 O), S . Let b 2 O. Using the Tb -action in (2.11), the result holds true for Tb , since we have the following identity for any x 2 U # : X U .Tb exCU / D e fbQ.x/g U .exCU / D e fbQ.x/g ey D

X

X

e fbQ.y/g ey D

y2xCU

y2xCU

Tb ey D Tb U .exCU /:

y2xCU

The third identity follows from the very definition of isotropic submodules. 1 and CM WD .M / pjM . To prove the We set CM =U WD .M =U / p 1# j jU =U j

claimed identity for S , first we determine U .S exCU /. Later we shall compare this with S U .exCU /. For any x 2 U # , using the S -action in (2.11), we have X e fB.y C U; x C U /g U eyCU U .S exCU / D CM =U yCU 2M =U

X

D CM =U

yCU 2M =U

X

D CM =U

X

e fB.y C U; x C U /g X

ey 0

y 0 2yCU

e fB.y C U; x C U /g ey 0

yCU 2M =U y 0 2yCU

D CM =U

X

e fB.y; x/g ey :

y2U #

On the other hand, again from the S -action in (2.11), we have X X X ˚ S U .exCU / D S ey 0 D CM e B.y; y 0 / ey y 0 2xCU

D CM

X

y2M

D CM

X y2M

D jU jCM

X

ey

y 0 2xCU y2M

˚ e B.y 0 ; y/

y 0 2xCU

e fB.y; xg ey X

y2U #

X

e fB.y; u/g

u2U

e fB.y; x/g ey :

34


For the last identity we used the fact that the inner sum in the previous sum is zero unless y 2 U # , when it equals jU j. To obtain the claimed identity for S , it remains to prove the identity jU jCM D CM =U . But from Proposition 1.9, we have p p t .M / D .M =U / and from Proposition 1.7, we have jM j D jU j jU # =U j. u Remark For every isotropic submodule U of M , the orthogonal projection of W .M / onto U W .M =U / is given by the formula: M PU .v/

X 1 Dp jU j X 2M =U

X

! v.x/

x2X

X

ex :

x2X

P This follows from the fact that the vectors p1jU j x2X ex (X 2 M =U ) form an orthonormal basis of the space U W .M =U /. Note that v is in W .M /new if and only M if PU .v/ D 0 for all U ¤ 0. For the proof of Theorem 2.3, we need two lemmas. Lemma 2.36 Let M be an O-FQM and U V be isotropic submodules of M . The following diagram is commutative

where ' is induced by the canonical isomorphism .x C U / C V =U 7! x C V . Proof First note that V U # , since V isotropic (i.e. V V # ) and V # U # (see the assumption). Note also that V =U is an isotropic submodule of M =U and .V =U /# equals V # =U . This shows that the map ' is well-defined. The following identity proves the lemma: V ı '.e.xCU /CV =U / D V .exCV / D

X y2xCV

X

D

ey D

X

X

ey 0

yCU 2.xCU /CV =U y 0 2yCU

U .eyCU / D U ı V =U .e.xCU /CV =U /:

yCU 2.xCU /CV =U

t u Lemma 2.37 Let U , V be isotropic submodules of the O-FQM M such that U CV is isotropic. Then we have M

PU V D

p M =V jU \ V jV P.U CV /=V :

(2.16)


35

Proof By the remark after Theorem 2.3, we have 1 M PU .ez / D p jU j

X

ey

.z 2 M /:

y2U # yz mod U

First we evaluate the left hand side of (2.16) at ex0 CV (x0 2 V # ). We have X 1 M PU V .ex0 CV / D p jU j y2x0 CV

X z2U #

1 X ez D p jU j v2V

zy mod U

1 X Dp jU j v2V

X

X

ez

z2U #

zx0 Cv mod U

ex0 CvCu :

u2U x0 CvCu2U #

The map '

f.u; v/ 2 U C V W x0 C v C u 2 U # g ! fz 2 U # \ V # W z x0 mod U C V g given by '.u; v/ D x0 C v C u is obviously a surjective map. The well-definedness of ' follows from the fact that U V # and V U # , since U C V is isotropic. We claim that each fiber has jU \ V j-many elements. Let z be an element of the right hand side. Since ' is surjective, there exist .u; v/ such that '.u; v/ D z. The number of elements in the fiber of z equals the number of elements in f.u; v/ 2 U C V W u C v 0 mod U C V g. But this set has clearly jU \ V j-many elements. Therefore, we have 1 M PU V .ex0 CV / D jU \ V j p jU j

X

ez :

(2.17)

z2U # \V # zx0 mod U CV

# Since .U C V /=V D U # \ V # =V , we have 1 M =V V P.U CV /=V .ex0 CV / D q j U C V =V j 1 D q j U C V =V j

X

V e Y

Y 2.U # \V # /=V Y x0 CV mod .U CV /=V

X

X

y2Y Y 2.U # \V # /=V Y x0 CV mod .U CV /=V

ey :

36


By doing the substitution Y 7! .y/ (where Y D y C V and is the canonical projection from U # \ V # onto U # \ V # =V ), we obtain M =V

V P.U CV /=V .ex0 CV / D

1 1 q jV j jU C V =V j

X

X

y2U # \V #

y 0 y mod V

yx0 mod U CV

ey 0 :

We study the map ' 0 from f.y; y 0 / 2 .U # \ V # /2 W y x0 mod U C V; y 0 y mod V g to fz 2 U # \ V # W z x0 mod U C V g which is defined by .y; y 0 / 7! y 0 . The map ' 0 is surjective. Indeed, let z be an element of the latter set. Then clearly ' 0 .z; z/ D z. We claim that each fiber has jV j-many elements. The fiber of the element z equals fy 2 U # \ V # W y x0 mod U C V; y z mod V g. But since y z 2 V implies that y x0 mod U C V , we observe that the fiber of z has jV j-many elements. Therefore, we have 1 M =V V P.U CV /=V .ex0 CV / D q j U C V =V j

X

ez :

(2.18)

z2U # \V # zx0 mod U CV

In view of the identities (2.17) and (2.18), to show that (2.16) holds true, it remains to prove the following identity: p 1 1 jU \ V j p : D jU \ V j q jU j j U C V =V j However, theorem for modules we have the isomor fromthe secondisomorphism phism U C V =V ' U= U \ V , and hence the claimed identity holds true. u t Proof of Theorem 2.3 We proceed by induction on the order of M . If M does not possess isotropic submodules, then there is nothing to prove. Otherwise, by the definition of W .M /new , we have W .M / D W .M /new ˚

X

U W .M =U /:

U M U is isotropic U ¤0

By induction hypothesis for U ¤ 0, we can write W .M =U / D

X V =U M =U V =U isotropic

new V =U W .M =U /=.V =U / :


37

Inserting this into the first identity, we obtain W .M / D W .M /new ˚

X

X

new U V =U W .M =U /=.V =U / :

U M V =U M =U U is isotropic V =U isotropic U ¤0

The claimed decomposition follows now by the identity new U V =U W .M =U /=.V =U / D V W .M =V /new which is obvious from Lemma 2.36. For proving the second statement of the theorem, assume that there is only one maximal isotropic submodule in M , or, equivalently, that the set of isotropic submodules of M is closed under addition. Let U and V be isotropic submodules, U ¤ V . It suffices to show that V W .M =V /new is orthogonal to U W .M =U /. By Lemma 2.37, we have p M M =V PU V W .M =V /new D jU \ V jV P.U CV /=V W .M =V /new : Since U ¤ V , we have U C V =V ¤ 0. Hence the right hand side of the last identity is zero (see the second remark after the proof of Proposition 2.30). But this M means V W .M =V /new is in the kernel of the orthogonal projection PU , hence it M is perpendicular to the image of PU , which equals U W .M =U /. This proves the theorem. t u Proof of Proposition 2.34 It is clear that the map in the statement of the proposition defines indeed an action. The action is unitary with respect to (2.12), since the elements of the orthogonal group are in fact automorphisms on M . We show in the following that the actions of O.M / and W .M / commute. Let B be associated bilinear form of M . Let ' 2 O.M / and b 2 O, x 2 M . The action of Tb (see (2.11)) and O.M / commute, since: 'Tb ex D e fbQ.x/g 'ex D e fbQ.x/g e'.x/ D e fbQ.'.x//g e'.x/ D Tb e'.x/ D Tb 'ex : Similarly, the action of S (see (2.11)) and O.M / commute, since we have: .M / X .M / X e fB.y; x/g 'ey D p e fB.y; x/g e'.y/ 'S ex D p jM j y2M jM j y2M .M / X ˚ .M / X D p e B.' 1 .y/; x/ ey D p e fB.y; '.x//g ey jM j y2M jM j y2M D S e'.x/ D S 'ex :

38


To obtain the third identity, we did the substitution '.y/ 7! y in the previous sum. The forth identity holds true by the very definition of the orthogonal group. t u For the proof of Proposition 2.35 we need a lemma. Lemma 2.38 Let M D .M; Q/ be an O-FQM and U be an isotropic submodule of M . If U is fixed by O.M /, then for ' 2 O.M /, we have ' U D U U .'/;

(2.19)

where U W O.M / ! O.M =U / is defined by U .'/.x C U / D '.x/ C U . Proof Let B be associated bilinear form on M . We need to show first of all that the map U is well-defined. Let ' 2 O.M /. First we show that for any x 2 U # , '.x/ is an element of U # , i.e. B.'.x/; U / D 0. But this follows from the very definition of the orthogonal group. By the same reasoning, we also have U .'/ 2 O.M =U /. For the well-definedness it remains to show that U does not depend on the choice of the representatives of U # =U . Let x 0 2 x C U . Write x 0 D x C u (u 2 U ). But we have '.x 0 / '.x/ D '.x C u/ '.x/ D '.u/ 2 U . The last identity follows from the assumption that '.U / D U . The statement of the lemma holds true, since we have: X X X ' U .exCU / D 'ey D e'.y/ D ey y2xCU

D

X

y2xCU

' 1 .y/2xCU

ey D U .e'.x/CU / D U .U .'/exCU /:

y2'.x/CU

For the third identity we did the substitution '.y/ 7! y in the previous sum. To obtain the forth identity we used the assumption '.U / D U . u t Proof of Proposition 2.35 This follows immediately from Proposition 2.34 and Lemma 2.38. t u For the proof of Theorem 2.4, we need a lemma.

1

Lemma 2.39 Let M be an O-FQM. The space W .M /new; ( 2 O.M /) is a Q submodule of W .M /new . P Proof Write W .M /new; D i 2I Wi , where fWi gi 2I is the set of all O.M /submodules of W .M /new affording the character . It suffices to show that the spaces ˛Wi for ˛ 2 Q , is again an O.M /-submodule of W .M /new affording the character . But this follows immediately from the fact that x 7! ˛x defines an O.M /-module isomorphism of Wi and ˛Wi since the actions of Q and of O.M / commute as Proposition 2.34 shows. t u Proof of Theorem 2.4 Proposition 2.35 implies that the space W .M /new is O.M /invariant. Hence, by Proposition 2.17, we have the decomposition as stated in

2.4 Complete Decomposition

39

the theorem. Finally by Lemma 2.39, we know that the components of the decomposition are Q -submodules of W .M /new . t u

2.4 Complete Decomposition of Cyclic Representations In this section we shall show that, for a cyclic O-module M , the decomposition of W .M / resulting from the combination of Theorems 2.3 and 2.4 is complete, i.e. the components occurring in the decomposition are all irreducible. In addition, we shall derive dimension formulas for these irreducible components. Recall from Sect. 1.2 that for an O-FQM M D .M; Q/ the level l, the modified level m and the annihilator a satisfy m D l.2; l/2 and a D l.2; l/1 . Recall also, that, for cyclic M , the isotropic submodules are all of the form ab1 M , where b runs through the square divisors of m. Finally recall, that, for a cyclic M , the elements of the orthogonal group O.M / are given by multiplication by the elements g in the subgroup E.M / .O=a/ of all " C a such that "2 1 mod l (see Proposition 1.21). Via this identification of O.M / with E.M / we shall henceforth consider W .M / as an E.M /-module via the action .g; v/ 7! gv, where .gv/.x/ D v."x/ if g D " C a. In the following we consider W .M / as an E.M /-module. Definition 2.40 Let M D .M; Q/ be an O-CM with level l and modified level m. For a square-free divisor f of m, we set W .M /f WD fv 2 W .M / W v.gx/ D

f .g/ v.x/for

all g 2 E.M /, x 2 M g:

Here f denotes the linear character f ." C a/ D f; ." C 1/.2; l/1 of E.M / (see Proposition 1.23). Moreover, we define W .M /new;f WD W .M /new \ W .M /f : Remark Note that for an O-CM M , the spaces W .M /new;f coincide with the 0 spaces W .M /new; f occurring in Theorem 2.4, where f0 is the character of O.M / corresponding to f under the isomorphism E.M / ' O.M / of Proposition 1.21. Theorem 2.5 Let M D .M; Q/ be an O-CM with level l, modified level m and annihilator a. (i) We have the following decomposition of W .M / into Q -submodules: W .M / D

M

ı ab1 M W .M ab1 M /new :

b2 jm

Here the sum is over all integral O-ideals b whose square divides m.

(2.20)

40


(ii) For W .M /new we have the decomposition M

W .M /new D

W .M /new;f

(2.21)

fjm f squaref ree

into Q -submodules. The W .M /new;f are irreducible Q -submodules. (iii) For any square-free divisor f of m, we have dim W .M /

new;f

Y1

.f; p/ Y 1 1 1C 1 : D N.a/ 2 N.p/ 2 N.p2 / 2 p jm

pkm

(2.22) We subdivide the proof of the theorem into three parts. Proof of Theorem 2.5 (i) The decomposition given in (i) is the decomposition of Theorem 2.3 specialized here to the case of cyclic finite quadratic O-modules. The directness of the sum comes from the fact that a cyclic O-FQM fulfills the assumption stated in Theorem 2.3, namely that it possesses only one maximal isotropic submodule (see the remark after Theorem 1.2). t u Next we shall prove (iii). For that we need a lemma. Lemma 2.41 Suppose M D .M; Q/ be an O-CM with annihilator a and modified level m. The character W .M /new of the E.M /-module W .M /new satisfies

W .M /new ." C a/ D

X

.b/ N " 1; ab2 :

b2 jm

Proof Since the space W .M / is O.M /-invariant (see (2.14)), using the decomposition in part (i) of Theorem 2.5 and also Proposition 2.10, we have ı X tr " C a; W .M / D tr " C ab2 ; W .M ab1 M /new :

(2.23)

b2 jm

Here we also used the identity ı ı tr " C a; ab1 M W .M ab1 M /new D tr " C ab2 ; W .M ab1 M /new which is a consequence of Lemma 2.38 (when weıapply this lemma we used Corollary 1.19, which says that the annihilator of M ab1 M equals ab2 ). If we can show that the following identity holds true X ı tr " C a; W .M /new D

.b/ tr " C ab2 ; W .M ab1 M / ; b2 jm

(2.24)

2.4 Complete Decomposition

41

then the claimed formula holds true. Indeed, since W .M / is a permutation representation with respect to the action of the group O.M / (see (2.14)), we then obviously have ı tr " C ab2 ; W .M ab1 M / D N " 1; ab2 : Now we prove (2.24). First we calculate the right hand ı side of (2.24). By inserting the value in (2.23) specialized to tr " C ab2 ; W .M ab1 M // and ı using Corollary 1.19 (which says that the annihilator and the modified level of M ab1 M equal ab2 and mb2 , respectively), we obtain X b2 jm

.b/

X

tr " C ab2 b02 ; W .M 0 /new

b02 jmb2

X

D

X tr " C ab002 ; W .M 0 /new

.b/: bjb00

b002 jm

ı Here M 0 D M =ab1 M .ab2 b01 .ab1 M /# =ab1 M /. For the above identity we used the fact that the underlying module of M 0 is isomorphic to O=ab002 (which follows from Theorem 1.1 (ii) and Theorem 1.2 (ii)). But the inner sum in thesecond identity equals zero unless b00 D O. Therefore, new , i.e. (2.24) holds true. t u the above identity equals tr " C a; W .M / Proof of Theorem 2.5 (iii) By Proposition 2.19, the dimension of the E.M /-module W .M /new;f is given by dim W .M /new;f D

X 1 j E.M /j

f .g/ W .M /new .g/;

g2E.M /

where we used that f .g/, which is explained in Proposition 1.23, is real. We write the formula for W .M /new .g/ from Lemma 2.41 in the form

W .M /new ." C a/ D ( I."; p/ D

Y

I."; p/;

pjm

N." 1; pa /

if pkm

N." 1; pa / N." 1; pa2 /

if p2 jm;

where pa is the exact power of p dividing a. Inserting this quantity into the dimension formula we obtain dim W .M /new;f D

1 j E.M /j

X "Ca2E.M /

f ."

C a/

Y pjm

I."; p/:

42


Using the decomposition of E.M / into p-parts as given in Proposition 1.22 we can write dim W .M /new;f D

Y1 pjm

2

X

f ."

C a/ I."; p/;

"Ca2h"p Cai

where "p 1 mod pa , "p C1 mod apa . (We used here that I."; p/ depends only on " modulo pa , and that the order of E.M / equals 2r , where r is the number of different prime factors of m.) We denote the factor corresponding to p by S.p/. Recall that f ."p C a/ D 1 if pjf and f ."p C a/ D C1 otherwise (see Proposition 1.23 and the remark afterwards). In other words, f ."p C a/ D .f; p/. Inserting this and the formulas for I."; p/ into the sum S.p/ we obtain 1 S.p/ D 2

(

N.pa / C .f; p/ N.2; pa / N.p / N.p a

a2

/ C .f; p/ N.2; p / N.2; p a

a2

/

if pkm if p2 jm:

It remains to prove that N.2; pa / D N.pa1 / if pkm, and N.2; pa / D N.2; pa2 / if p2 jm. This is obvious if p is odd. If p is even and pkm, then pa ka D m.2; l/ implies pa1 k.2; l/; but by Proposition 1.13 we have vp .2; l/ D vp .2/. Similarly, if p is even and p2 jm, then pa ka D m.2; l/ implies that a 2 vp .2; l/, hence a 2 vp .2/. This proves the claimed formula. t u For the proof of the remaining part (ii) we need a lemma. Lemma 2.42 Let m be an integral O-ideal. The number of pairs .b; f/ of integral O-ideals such that b2 jm and f is a square-free divisor of mb2 equals 0 .m/, i.e. the number of integral O-ideal divisors of m. Proof Denote the number of pairs .b; f/ in question by I.m/. It is easy to see that the function I from the set of integral O-ideals into N is multiplicative, i.e. it satisfies I.gh/ D I.g/I.h/ for O-ideals g and h with .g; h/ D 1. Hence, we have that Q I.m/ equals pn km I.pn/. Since 0 .m/ is also multiplicative, it suffices to show that, for each prime ideal power pn , we have I.pn / D 0 .pn /. Indeed, I.pn / equals the number of pairs .pk ; O/ with 0 2k n plus the number of pairs .pk ; p/ with 0 2k < n. Hence ( I.p / D n

2.1 C b n2 c/

if n is odd

1C

if n is even:

2b n2 c

We observe that in any case I.pn / D n C 1, for each p, which equals 0 .pn /. This proves the lemma. t u Proof of Theorem 2.5 (ii) The decomposition (2.21) given in (ii) is the decomposition of Theorem 2.4, which we specialize here to cyclic finite quadratic O-modules.

2.5 The One Dimensional Subrepresentations

43

Note that by the remark after Definition 2.40, the components in (ii) coincide with the ones given in Theorem 2.4. It remains to prove that the components in (2.21) are irreducible. We shall prove in the next section (see Corollary 2.47) that the number of irreducible Q submodules of W .M / is less than or equal to the number 0 .m/. If we insert the decompositions (2.21) for M =ab1 M with b running through the square divisors of m into the decomposition (2.20), we have split W .M / into as many Q -submodules as there are pairs of integral O-ideals .b; f/ with b2 jm and f a squarefree divisor of mb2 . By Lemma 2.42 these are exactly 0 .m/-many Q -submodules in the decomposition of W .M /, i.e. as many as our upper bound for irreducible submodules in W .M /. From the dimension formulas (2.22) it is clear that none of the components in this splitting W .M / can be zero. Therefore the Q -submodules in this splitting cannot split further and must hence be irreducible. This proves the theorem. t u

2.5 The One Dimensional Subrepresentations In the present section, we shall prove that, for a cyclic M , the space W .M / contains one-dimensional Q -submodules if and only if the level of M is a character ideal (see the subsequent definition) times a square dividing the modified level of M . Moreover, we shall also determine basis elements for the one-dimensional submodules of cyclic Weil representations. Recall that if M is an O-FQM with level l, the modified level of M equals l.2; l/2 , and the annihilator of M equals l.2; l/1 . Definition Q 2.43 Q A character ideal is an integral O-ideal c, which is of the form c D siD1 pi tj D1 q3j , where the pi are pairwise different prime ideals of degree one dividing 3, and where the qj are pairwise different prime ideals of degree and ramification index one dividing 2. Remark Note that s or t might be equal to zero. If t D 0, then c is called an odd character ideal. Definition 2.44 For a prime ideal p of degree one over 3, we use p for the nontrivial Dirichlet character modulo p. For a prime ideal q of degree one over 2, we use q2 for the nontrivial Dirichlet character modulo q2 . For square-free products g and h of prime ideals of degree one over 3 and 2, respectively, we set

gh2 WD

Y pjg

p

Y

q2 ;

qjh

and call gh2 the totally odd character modulo gh2 .

(2.25)

44


Remark Note that, for primes p and q as in the definition, the groups of units .O=p/ and .O=q2 / have both order 2, so that there is indeed for each group a unique nontrivial character. We state the main result of this section. Theorem 2.6 Let M D .M; Q/ be an O-CM with level l, annihilator a and modified level m. (i) The space W .M / contains one-dimensional Q -submodules if and only if l is a character ideal times a square dividing the modified level of M . (ii) The space W .M / contains at most one one-dimensional Q -submodule. (iii) Suppose that W .M / contains a one-dimensional Q -submodule. If we write l D gh3 b2 , where gh3 is the character ideal dividing l and b2 a divisor of the modified level of M , then we have a D gh2 b2 and m D ghb2 . The onedimensional Q -submodule equals U W .M =U /new;gh , where U D ab1 M D gh2 bM . It is spanned by U

X s2O=gh2

gh2 .s/ egs D

X

gh2 .s/ ex :

(2.26)

x2M;s2O=gh2 xs mod U

Here g D C U is a generator of M =U , and gh2 denotes the totally odd Dirichlet character modulo gh2 . The rest of this section is devoted to the proof of the theorem. For this it is convenient to introduce a name for the prime ideals occurring in character ideals. Definition 2.45 A prime ideal of degree 1 and ramification index 1 above 2 is called a .2; 1; 1/-ideal. A prime ideal of degree 1 above 3 is called a .3; 1/-ideal. Thus, a character ideal c is a product of different .3; 1/-ideals and cubes of different .2; 1; 1/-ideals. For proving the theorem we first consider the new parts of the spaces W .M /. Lemma 2.46 Let M be an O-CM with level l. The space W .M /new contains onedimensional Q -submodules if and only if l is a character ideal. If l is a character ideal, say, l D gh3 , then W .M /new contains exactly one one-dimensional Q submodule, namely W .M /new;gh . Proof First suppose that l is a character ideal such that l D gh3 . Then the modified level of M equals gh. Hence, by Theorem 2.5 (iii) we have that W .M /new;gh is one dimensional, i.e. W .M / contains one-dimensional Q -submodules. Next suppose that W .M /new contains one-dimensional Q -submodules. Let m be the modified level of M . By Theorem 2.5 (ii) (and by Proposition 2.16) there exists a square-free divisor f of m such that W .M /new;f is one dimensional. If we can show that l D gh3 and f D gh for a product g of different .3; 1/-ideals and a product h of different .2; 1; 1/-ideals, then this proves the lemma.

2.5 The One Dimensional Subrepresentations

45

We write dim W .M /new;f D P1 P2 , where P1 and P2 are the contributions from odd and even prime ideals, respectively. By the assumption that W .M /new;f is one dimensional, we have P1 D 1 and P2 D 1 (see also Proposition 1.6). Using (2.22) and also the fact that a D m.2; l/, we can write 1 D P1 D N.m1 /

Y pkm1

N.p/1

Y N.p/ C .f; p/ 2

pkm1

Y p2 jm

N.p/2

Y N.p/2 1 ; 2 2

(2.27)

p jm1

1

where m1 stands for the odd part of m. Since the second and the forth products and also N.m1 / times the first and the third products in (2.27) are all integers, obviously we need to have first of all that m is square-free. Moreover, for all pkm1 , we need to have N.p/ C .f; p/ D 1: 2 But this implies that N .p/ D 3 and .f; p/ D 1 for each pkm1 . Therefore, we have that m1 D g, and the odd part of f equals g, where g is a product of different .3; 1/-ideals. Now we consider the even part. Using (2.22), we have Y Y 1 D P2 D 2s N m2 .2; l/ N.p/ C .f; p/ N.p/1 pkm2

pkm2

Y

p2 jm2

N.p/2

Y

N.p/2 1 ;

(2.28)

p2 jm2

where m2 denotes the even part of m, and s denotes the number of distinct prime ideal divisors of m2 . First note that the second and the forth products in (2.28) are integers. Also 2s N.m2 .2; l// times the first and the third products in (2.28) are integers. Indeed, this follows from the fact that every prime ideal dividing m2 occurs in the prime ideal decomposition of .2; l/, since m D l.2; l/2 . Therefore, we need to have first of all that m2 is square-free. Furthermore, we need to have N.2; l/ D 2s ;

N.p/ C .f; p/ D 1

for all pkm2 . But the first identity implies that m2 D .2; l/ D h, where h is a product of different .2; 1; 1/-ideals. The second identity implies that N.p/ D 2 and

.f; p/ D 1 for each pkm2 . Therefore, we have that m2 D h and that the even part of f equals h.

46


As a consequence, we obtain m D m1 m2 D gh, f D gh, and hence we have that l D m.2; l/2 D ghh2 D gh3 , which proves the lemma. t u Remark Note that if M is anisotropic, i.e. M does not contain isotropic submodules, then W .M /, which equals W .M /new , contains one-dimensional Q -submodules if and only if the level of M is a character ideal (see Lemma 2.46). Proof of Theorem 2.6 Proof of part (i). Suppose that the space W .M / contains one-dimensional Q -submodules. By Theorem 2.5 (and Proposition ı 2.16), there exists an integral O-ideal b with b2 jm such that the space W .M ab1 M /new;f is one dimensional. Lemma 2.46 implies that the level of M =ab1 M is a character ideal. But the level of M =ab1 M equals lb2 (Corollary 1.19). Hence l is of the claimed form. Suppose that l is as given in the statement of the theorem, i.e l D cb2 , where b2 jm and c is a character ideal. Set U WD ab1 M . Since the level of M =U equals lb2 D c (see Corollary 1.19), which is a character ideal, we deduce from Lemma 2.46 that the space W .M =U /new, and hence the space U W .M =U /new W .M / contains one-dimensional Q -submodules. Proof of part (ii). First we show that amongst the Q -submodules in the decomposition of W .M / obtained on combining (2.20) and (2.21), there is at most one one-dimensional Q -submodule. Suppose W .M / contains two one-dimensional new;fi Q -submodules in the decomposition, say W .M =ab1 (i D 1; 2). Then, by i M/ 1 Lemma 2.46, the level li of M =abi M is equal to a character ideal, say gi h3i , and 3 2 fi D gi hi . From Corollary 1.19, we know that li D lb2 i . Hence, l D gi hi bi . But this implies that g1 D g2 , h1 D h2 and b1 D b2 (use that g1 h1 D g2 h2 is the squarefree part of the unique factorization of l into a product of a square-free ideal and a square), i.e. the claimed result holds true. Now suppose that W is a one-dimensional Q -submodule of W .M /. Then by Proposition 2.16, we have W ' W .M =ab1 M /new;f , for some b and f as in Eqs. (2.20) and (2.21). If W .M / does not contain any one-dimensional Q -submodule in the decomposition, there is nothing to prove. If there is a onedimensional Q -submodule in the decomposition of W .M /, it is unique by the new;f1 above argument. We call it W .M =ab1 . Hence, all the other Q -submodules 1 M/ 1 new;f W .M =ab M / with b ¤ b1 or f ¤ f1 have dimension bigger than one. Our aim new;f1 is to show that W D W .M =ab1 . We denote by Pb;f the projection from 1 M/ 1 new;f W onto the space W .M =ab P / . It suffices to prove for all w P 2 W , the identity Pb1 ;f1 .w/ D w. The identity b;f Pb;f D 1 implies that w D b;f Pb;f .w/. But Pb;f .w/ D 0 for all .b; f/ ¤ .b1 ; f1 /. Indeed, the kernel of the map Pb;f jW must be equal to W , since otherwise the map Pb;f jW would be a Q -linear isomorphism from W onto a one-dimensional Q -submodule of W .M =ab1 M /new;f , whereas the latter is irreducible and has dimension bigger than one. Hence, we have w D Pb1 ;f1 .w/, which proves (ii). Proof of part (iii). Suppose W .M / contains a one-dimensional Q -submodule, say W . As we saw in the proof of part (ii), we then have l D gh3 b2 with b2 jm, and W D U W .M =U /new;gh , where U D ab1 M .

2.6 The Number of Irreducible Components

47

For proving the claimed identities for a and m it suffices to show that h D .2; l/ (since, for any O-CM, we have a D l.2; l/1 and m D l.2; l/2 ). For this we write m D b2 t. Since m D l.2; l/2 we have .2; l/2 D gh3 t1 , and since g and h are square-free and relatively prime, therefore .2; l/jh. But h divides 2 and it divides l, hence .2; l/ D h. P Finally, let I WD s2O=gh2 gh2 .s/ egs , where g D C U is a generator of the O-CM M 0 D M =U ' O=gh2 . Since I is clearly different from 0, it remains to show that I is in W .M 0 /new;gh . First of all, note that W .M 0 /new D W .M 0 / since M 0 has modified level gh, and hence has no isotropic submodules different from zero (see Theorem 1.2). In other words, we only have to show that hI D gh .h/I for all h in E.M 0 /. But this follows immediately from the fact that E.M 0 / D .O=gh2 / and gh D gh2 . This proves the theorem. t u

2.6 The Number of Irreducible Components In the present section we shall find an estimate for the number of irreducible subrepresentations of Weil representations. For cyclic Weil representations this number can be made even more explicit. Namely, we have: Theorem 2.7 Let M D .M; Q/ be an O-FQM with level l. Then the number of irreducible Q -submodules of less than or equal to the number of elements W .M /is 0ı 0 of M M O=l . Here M M is the set of all v in M M such that v is trivial (for v , we refer to Lemma 2.58 below). Corollary 2.47 Let M D .M; Q/ be an O-CM with modified level m. The number of irreducible Q -submodules of W .M / is less than or equal to the number of integral O-ideal divisors of m, i.e. 0 .m/. We prove the above theorem with two different methods. This section is divided accordingly into two subsections. Both approaches calculate the dimensions of the intertwining algebras of Weil representations. In fact, the number of irreducible G-submodules of a G-module V is bounded by the dimension of the intertwining algebra of V (see Proposition 2.22). As a side result of our second approach we also obtain the following theorem: Theorem 2.8 Let M D .M; Q/ be an O-FQM with level l and associated bilinear form B. There exists a projective representation of which satisfies, for any z 2 M , the following formulas (b 2 O) (i) .Tb /ez D e fbQ.z/g eP z 1 0 (ii) .S /ez D .M / pjM 0 2M e fB.z ; z/g ez0 . z j

48


2.6.1 The First Approach Before we can give the proofs of Theorem 2.7 and Corollary 2.47, we need several lemmas. Lemma 2.48 Let M D .M; Q/ be an O-FQM. The bilinear map

X X X Œ_; _ W W .M 1 / W .M / ! C; _ v.x/ex ; v 0 .x 0 /ex 0 WD v.x/v 0 .x/ x2M

x 0 2M

x2M

is Q -invariant. Proof Let B be the bilinear form of M . It is enough to prove the lemma for the standard generators Tb (b 2 O) and S . We shall prove only P the invariance under 0 S , since the invariance under T is obvious. Write v D x2M v.x/ex and v D b P 0 0 x 0 2M v .x /ex 0 . From the S -action in (2.11), we have X .M 1 / X S v D p v.x/ e fB.y; x/g ey jM j x2M y2M and X ˚ .M / X 0 v .x/ e B.y 0 ; x 0 / ey 0 : S v0 D p jM j x 0 2M y 0 2M Hence, we have ŒS v; S v 0 D

X ˚ .M 1 /.M / X v.x/v 0 .x 0 / e B.y; x x 0 / : jM j 0 y2M x;x 2M

From Proposition 1.11, the inner sum is zero unless x D x 0 , otherwise it equals jM j. We now recognize ŒS v; S v 0 D Œv; v 0 , since Proposition 1.10 implies that sigmainvariant .M / has absolute value one. t u Lemma 2.49 Let M D .M; Q/ be an O-FQM. Then the linear map W .M 1 / ! W .M / ;

v 7! “v 0 7! Œv; v 0 00

defines a Q -module isomorphism. Proof We denote the above map by '. First we show that ' is Q -linear. For that, using (2.4), it is enough to show that for any v 2 W .M 1 /, v 0 2 W .M / and ˛ in Q , the identity Œ˛ 1 v; v 0 D Œv; ˛v 0 holds true. But if do the substitution v 7! ˛v, Lemma 2.48 implies the result.


49

To show that ' is an isomorphism, it suffices to show that ' is an injection, since the spaces have the same dimension. Let v be an element of the kernel of P 0 0 0 ', i.e. Œv; v D 0 for all v 2 W .M / . We write v D v.x/e x and v D x2M P 0 0 x 0 2M v .x /ex 0 . Then we have Œv; v 0 D

X

v.x/v 0 .x/ D 0:

x2M

If we take v 0 D ex0 for some x0 2 M , then the above identity implies that v.x0 / D 0. Repeating the same argument by choosing other elements of M , we observe that v D 0, i.e. ' is an injection. t u Lemma 2.50 Let M D .M; Q/, N D .N; R/ be O-FQM. Then the linear map W .M C N / ! W .M / ˝ W .N /;

ex˚y 7! ex ˝ ey

defines a Q -module isomorphism. Proof We denote the map in the lemma by '. Clearly ' is an isomorphism of vector spaces CŒM ˚ N and CŒM ˝ CŒN . Hence it remains to show that ' is Q -linear. It is enough to prove this fact for Tb (b 2 O) and S . Let QR denote the quadratic form on M C N . Recall that QR.x ˚ y/ D Q.x/ C R.y/. Let b 2 O, x 2 M and y 2 N . The Tb -action in (2.11) and ' commute, since '.Tb ex˚y / D e fbQR.x ˚ y/g ex ˝ ey D e fb.Q.x/ C R.y//g ex ˝ ey D e fbQ.x/g ex ˝ e fbR.y/g ey D Tb ex ˝ Tb ey D Tb .ex ˝ ey / D Tb '.ex˚y /: Let B, C and BC stand for associated bilinear forms of M , N and M C N , respectively. Recall BC.x 0 ˚ y 0 ; x ˚ y/ D B.x 0 ; x/ C C.y 0 ; y/. Similarly, the identity '.S ex˚y / D D

.M C N / jM C N j

X

˚ e BC.x 0 ˚ y 0 ; x ˚ y/ ex ˝ ey

x 0 ˚y 0 2M ˚N

X ˚ .M C N / X ˚ e B.x 0 ; x/ ex ˝ e C.y 0 ; y/ ey jM C N j 0 0 x 2M

y 2N

D S '.ex˚y / proves that the S -action in (2.11) and ' commute. For the last identity we used the remark after Definition 1.8, which states that .M C N / D .M /.N /, and we also used jM C N j D jM jjN j. t u

50


Lemma 2.51 Let M D .M; Q/ be an O-FQM. Then Q HomQ W .M /; W .M / ' W M 1 C M : Proof From the mapgiven in (2.5), it is easy to see that W .M / ˝ W .M / and Hom W .M /; W .M / are isomorphic as Q -modules. Lemma 2.49 implies that the spaces W .M / and W .M 1 / are Q -module isomorphic. Using Lemma 2.50 we obtain that W .M 1 /˝W .M / is Q -module isomorphic to W M 1 CM . Therefore, we have that Hom W .M /; W .M / is Q -module isomorphic to W M 1 C M . Therefore Proposition 2.21 implies the result. t u Lemma 2.52 Let M D .M; Q/ be an O-FQM with annihilator a. The group O=a acts on the right of M M via bCa 7! .x; y/A; .x; y/; A D aCa cCa d Ca

.x; y/A WD .ax C cy; bx C dy/:

Proof First we show that the above multiplication is well-defined. Let a0 2 a C a. We have a0 D aCt for some t 2 a. But a0 x D .aCt/x D ax, since tx D 0. Let v D bCa a0 Ca b 0 Ca .x; y/ 2 M M and A, B 2 O=a . Write A D aCa cCa d Ca and B D c 0 Ca d 0 Ca . Since it is obvious that v1 D v, the following identity proves the lemma: B.vA/ D a0 ax C a0 cy C c 0 bx C c 0 dy; b 0 ax C b 0 cy C d 0 bx C d 0 dy D ABv: t u Remark Let M D .M; Q/ be an O-FQM with level l and annihilator a. By Proposition 1.5, we have that l a, so there is a reduction map from O=l onto O=a. Hence, using Proposition 2.2 and Lemma 2.52, we obtain that O=l also acts on M M . 2.53 Let M D .M; Q/ be an O-FQM with level l. For fixed A D Lemma aCl bCl in O=l , the map cCl d Cl fA W M M ! C ;

˚ .x; y/ 7! e abQ.x/ C bcB.x; y/ C cdQ.y/

satisfies the following identity fAB .v/ D fA .v/fB .vA/: Here .v; A/ 7! vA is the action in Lemma 2.52 (see also the remark afterwards). Proof First note that fA .v/ depends only on the coset of a. Let a0 2 a C l. We have a0 D a C l for some l 2 l. But a0 Q.x/ D .a C l/Q.x/ D aQ.x/ for


51

0 Cl b 0 Cl be in O=l and v D .x; y/ be in M M . any x 2 M . Let B D ac 0 Cl 0 d Cl Let B stand for associated bilinear form of M . The following proves the claimed identity: ˚ fA .v/fB .vA/ D e abQ.x/ C bcB.x; y/ C cdQ.y/ ˚ e a0 b 0 Q.ax C cy/ C b 0 c 0 B.ax C cy; bx C dy/ C c 0 d 0 Q.bx C dy/ ˚ D e ab C a2 ab0 C 2aba0 c 0 C c 0 d 0 b 2 Q.x/ ˚ e cd C a0 b 0 c 2 C 2a0 c 0 cd C c 0 d 0 d 2 Q.y/ ˚ e bc C a0 b 0 ac C a0 c 0 .ad C bc/ C c 0 d 0 bd B.x; y/ ˚ D e .aa0 C bc0 /.ab0 C bd0 /Q.x/ ˚ ˚ e .ab0 C bd0 /.ca0 C dc0 /B.x; y/ e .ca0 C dc0 /.cb0 C dd0 /Q.y/ D fAB .v/: For the third identity we used a0 d 0 b 0 c 0 1 mod l and that ad bc 1 mod l, and, in addition, the fact that lQ.x/ D 0 for any x 2 M , l 2 l. t u Lemma 2.54 Let M D .M; Q/ be an O-FQM with level l. The group O=l acts on CŒM M via .A; ev / 7! Aev ;

Aev WD fA1 .v/evA1 ;

where fA is as in Lemma 2.53. Proof Let A, B be in O=l and v be in M M . Since it is obvious that 1ev D ev , the following identity proves the lemma: A.Bev / D fB 1 .v/fA1 ..vB/1 /e.vB/1 A1 D f.AB/1 .v/ev.AB/1 D ABev : For the second identity we used Lemma 2.53.

t u

Definition 2.55 Let M D .M; Q/ be an O-FQM level l. In the following the O=lmodule CŒM M described in Lemma 2.54 is denoted by P .M /. Remark Let a be a nonzero integral O-ideal. There is an epimorphism from Q onto O=a which maps Tb (b 2 O) and S to Tb and S reduced modulo a, respectively. This epimorphism obtained by composing the epimorphism from Q onto (see Sect. 2.2) and the epimorphism from onto O=a (see Lemma 1.29). Remark The above remark and Proposition 2.2 imply that P .M / can be viewed as a Q -module.

52


Lemma 2.56 Let M D .M; Q/ be an O-FQM with bilinear form B and level l. The linear map W .M 1 C M / ! P .M /;

W ex˚y 7!

X

e fB.z; y/g e.yx;z/

z2M

defines a Q -module isomorphism. Proof First note by the second remark after Definition 2.55 that P .M / is a Q module. It is clear that the map is an isomorphism. It remains to show that is Q linear, i.e. ˛ex˚y D .˛/ex˚y for ˛ D Tb (b 2 O) or S . Here stands for the epimorphism from Q onto O=l explained in the first remark after Definition 2.55. Let b 2 O. For Tb the claimed identity holds true, since for any x; y 2 M , we have .Tb /ex˚y D

X

e fB.z; y/g f.Tb /1 .y x; z/e.yx;z/.Tb /1

z2M

D e fbQ.y x/g

X

e fB.z; y/g e.yx;b.xy/Cz/

z2M

D e fbQ.y x/ C bB.y; y x/g

X

e fB.z; y/g e.yx;z/

z2M

˚ X D e b Q.y/ Q.x/ e fB.z; y/g e.yx;z/ D Tb ex˚y : z2M

To obtain the third identity we did the substitution z 7! z b.y x/ in the previous sum. We refer to the Tb -action in (2.11) to see that the last identity holds true. We have X e fB.z; y/g f.S /1 .y x; z/e.yx;z/.S /1 .S /ex˚y D z2M

D

X

e fB.z; y/g e fB.y x; z/g e.z;yx/

z2M

D

X

e fB.z; x/g e.z;yx/ D

z2M

X

e fB.z; x/g e.z;yx/ :

z2M

To obtain the last identity we did the substitution z 7! z in the previous sum. Now we apply the S -action in (2.11). The claimed identity holds true for S also, since for any x; y 2 M , similarly we have S ex˚y D

X ˚ .M 1 C M / X ˚ e B.y 0 ; z y/ e B.x 0 ; x/ e.y 0 x 0 ;z/ jM j 0 0 z;y 2M

x 2M


D D

53

X X ˚ 1 X ˚ e B.x 0 ; x/ e.x 0 ;z/ e B.y 0 ; z y C x/ jM j 0 z2M 0 X

x 2M

y 2M

˚ e B.x 0 ; x/ e.x 0 ;yx/ :

x 0 2M

To obtain the second identity above we did the substitution x 0 7! y 0 x 0 and changed the order of summation in the previous sum. Moreover, we also used the fact that .M 1 C M / D 1 which follows from the remark after Definition 1.8 (which says that .M 1 C M / D .M 1 /.M / D .M /.M /) and Proposition 1.10 (which says that .M / has absolute value one). The last identity follows from the fact that the inner sum in the previous identity is 0 unless z D y x, when it equals jM j (see Proposition 1.11). t u Lemma 2.57 Let M be an O-FQM. Then we have Q HomQ W .M /; W .M / ' P .M / : Proof This is immediate from Lemmas 2.51 and 2.56.

t u

Lemma 2.58 Let M D .M; Q/ be an O-FQM. For fixed v 2 M M , the map

v W Stab.v/ ! 1 , A 7! fA .v/ defines a group homomorphism. t u

Proof This is immediate from Lemma 2.53.

Proof of Theorem 2.7 The number of irreducible Q -submodules of the space W .M / is bounded by the dimension of the space HomQ W .M /; W .M / (see Proposition 2.22), which equals by Lemma 2.57 the dimension of P .M /Q . But we have P .M /Q D P .M /O=l (see the first remark after Definition 2.55 and Proposition 2.2). It enough to show that the dimension of the space P .M /O=l equals the upper bound given in the statement of thetheorem.ı Let vi (1 i m) be a set of representatives for M M O=l. We claim that the space P .M /O=l has as basis the elements Li WD

1

X

jO=lj A2

fA1 .vi /evi A1

.1 i m/

O=l

unless they are zero. The space P .M /O=l is spanned by the operators Li (see Proposition 2.15), and obviously these elements are linearly independent whenever they are nonzero. First we show that whenever vi and vj lie in the same orbit, Li and Lj differ by a constant. Write vj D vi B 1 for some B 2 O=l. But the claim holds true, since we have

54


Lj D

X

1

jO=lj A2

fA1 .vi B 1 /e.vi B 1 /A1

O=l

D

1

X

fB 1 .vi /jO=lj A2O=l

f.AB/1 .vi /evi .AB/1 D

1 fB 1 .vi /

Li :

For the second identity we used Lemma 2.53, and for the last identity we did the substitution A 7! AB1 in the previous sum. Next we determine when the operators Li are equal to zero. We write Li D D

1 jO=lj 1 jO=lj

X

X

f.AB/1 .vi /evi .AB/1

A Stab.vi /2O=l = Stab.vi / B2Stab.vi /

X

evi A1

A Stab.vi /2O=l = Stab.vi /

X

f.AB/1 .vi /:

B2Stab.vi /

For the second identity we used vi B 1 D vi which follows from the fact that B is an element of Stab.vi /. Since the elements evi A1 (A 2 O=l ) are linearly independent, the operators Li are equal to zero if for all A 2 O=l, we have X

f.AB/1 .vi / D 0:

B2Stab.vi /

However, from Lemma 2.53, and the fact that vi B 1 D vi , we have the identity f.AB/1 .vi / D fB 1 .vi /fA1 .vi /. Since fA1 .vi / being a root of unity can never be zero, hence we have X fB 1 .vi / D 0: B2Stab.vi /

But if Li D 0, then vi must be nontrivial (see Lemma 2.58 for vi ). As a consequence, we obtain that the space P .M /O=l has basis the operators Li for which the characters vi are trivial. Therefore, the dimension of the 0 ı t u space P .M /O=l equals the number of elements of M M O=l. For the proof of Corollary 2.47 we need a lemma. Recall that the annihilator and the modified level of an O-CM M of level l equals l.2; l/1 and l.2; l/2 , respectively. Lemma 2.59 Let a be a nonzero integral O-ideal, let R WD O=a and let I stand for the set of integral ideals of R. We define ı I W R R R ! I;

Œ˛; ˇ 7! ˛R C ˇR:


55

Here R acts on R R via formal multiplication of row vectors in R R with matrices in R . Moreover, Œ˛; ˇ stands for the orbit of .˛; ˇ/ of under this action. Then the map I defines a bijection. Proof First we show that I is well-defined. Let v D .˛; ˇ/ and w D .˛ 0 ; ˇ 0 / in R R. We need to show that if v, w lie in the same orbit, then D I.Œw /. Suppose I.Œv / v and w lie in the same orbit i.e. w D vA for some A D ı 2 R . Then we have I.Œw / D I.ŒvA / D .˛ C ˇ /R C .˛ C ˇı/R D . C /˛R C . C ı/ˇR: Hence, I.Œw / I.Œv /. On the other hand, we have I.Œv / D I.ŒwA1 / D .˛ 0 ı ˇ 0 /R C .˛ 0 C ˇ 0 /R D .ı /˛ 0 R C . /ˇ 0 R: Similarly, we have I.Œv / I.Œw / which proves the well-definedness. The surjectivity of I follows from Lemma 1.25. Next we prove the injectivity. Suppose I Œv D I Œw . From Lemma 1.30 we have that every orbit contains an element whose first entry equals zero. Suppose .0; 1 / is contained in Œv , and .0; 2 / is contained in Œw . Hence, we have 1 R D 2 R (by the assumption). By applying 1 Lemma 1.26, we obtain 1 D "2 for some " 2 R . Hence, .0; 1 / D .0; 2 / " 0 0" , i.e Œv D Œw , which proves that I is an injection. t u Proof of Corollary 2.47 Let l and a be the level and annihilator of M , respectively. We set R WD O=a. From Theorem 2.7 we know that the number of irreducible Q -submodules of W .M / is less than or equal to the number of elements of R 0 ı 0 0 R =O=l. But we have R R O=l D R R =R DW U (see Proposition 2.2 and the first remark after Definition 2.55). It is enough to prove the identity jU j D 0 .m/. Let I be the bijection in Lemma 2.59, and let I be the set of integral O-ideals of R. If can show that I.U / D f.x C a/R I W .2; l/jxg;

(2.29)

then the claimed identity holds true. Indeed, since .x C a/R being an ideal of R must contain a, i.e. we have xja, and since a.2; l/1 D m. By Theorem 2.7 we have ı U D fŒv 2 R R R W v D 1g; where v is as in Lemma 2.58. Let Œv 2 U . From Lemma 1.30 we know that Œv contains an element of the form .0; x C a/ for some x 2 O. Then, since v D 1, we have fA .0; x C a/ D 1, for all A 2 Stab.0; x C a/. By a direct computation, we obtain Stab.0; x C a/ D

˚ aCa

bCa cCa d Ca

2 R W ajcx; ajdx x

56


˚ and fA .0; x C a/ D e cd!x 2 . To prove (2.29), we need to show that the following holds true: 8A D

aCa

bCa cCa d Ca

2 Stab.0; x C a/;

ljcdx2

if and only if .2; l/jx:

(2.30)

bCa Suppose first of all that .2; l/jx and A D aCa cCa d Ca in Stab.0; x C a/. Hence, we have ajcdx. Therefore, l D a.2; l/jcdx2 . (Recall here that a D l.2; l/1 .) Suppose now that the left hand side of (2.30) holds true. Let c be an integral Oideal which lies in the inverse ideal class of a which is relatively prime to l. Then O D ca for some 2 K. We take c D x 1 , d D 1 C x 1 . Then we have cdx2 D x 1 .1 C x 1 /x 2 D 2 C x. By the assumption ljcdx2 , we have that l divides c2 a2 C cax, i.e. la1 divides c2 a C cx. But c is chosen so that it is relatively prime to l, hence la1 divides ca C x. But this implies that la1 , which equals .2; l/, divides x, since .2; l/ also divides ca. Therefore the identity (2.30) holds true, which proves finally the corollary. t u

2.6.2 The Second Approach Let M be an O-FQM with level l. In this subsection we use some tools which we already introduced in the previous subsection. Namely, we use the action of O=l on M M (as given in Lemma 2.52) and the remark afterwards, and we also use the function fA .v/ (v 2 M M ) attached to an element A of O=l (as given in Lemma 2.53). In this subsection we give another proof of Theorem 2.7, and we shall prove Theorem 2.8. For the proofs of these theorems we need again several lemmas. Lemma 2.60 Let M D .M; Q/ be an O-FQM with bilinear form B. The space CŒM is a projective M M -module via .x; y/; ez 7! .x; y/ez WD e fB.z; y/g exCz : More precisely, one has v.wez / D .v; w/.v C w/ez , where ˚ .v; w/ D e B.x 0 ; y/

v D .x; y/; w D .x 0 ; y 0 / :

(2.31)

Proof Let v D .x; y/ and w D .x 0 ; y 0 / be in M M and z 2 M . Then we have ˚ ˚ v.wez / D e B.z; y 0 / e B.x 0 C z; y/ exCx 0 Cz D .v; w/.v C w/ez ; where .v; w/ D e fB.x 0 ; y/g.

t u


57

Definition 2.61 Let M D .M; Q/ be an O-FQM with associated bilinear form B. Let l be the level of the finite quadratic Z-module Tr.M / (see Proposition 1.3). We define H.M / WD f.v; / W v 2 M M; 2 l g with the operation .v; / .w; 0 / D v C w; 0 .v; w/ ; where .v; w/ denotes the cocycle (2.31). This group is called the Heisenberg group associated to M . In the sequel, we write .x; y; / instead of ..x; y/; / for the elements of H.M /. Remark From Lemma 2.60 and Proposition 2.25 we see that H.M / is indeed a group, more precisely, a central extension of M M by l . Lemma 2.62 Let M D .M; Q/ be an O-FQM. The space CŒM is an H.M /module via .v; /; ez 7! .v; /ez WD vez : For the action of M M on CŒM , we refer the reader to Lemma 2.60. Proof By Proposition 2.26 and Lemma 2.60, it follows that CŒM is an H.M /module. u t Lemma 2.63 Let M D .M; Q/ be an O-FQM. The character CŒM of the H.M /module CŒM satisfies (

CŒM .v; / D

0

if v ¤ 0

jM j otherwise:

Proof From Lemma 2.62,CŒM is an H.M /-module. Let B be the bilinear form of M and v D .x; y/; 2 H.M /. The following identity proves the claimed identity

tr .v; /; CŒM D

(

0

if x ¤ 0

P z2M

e fB.z; y/g

otherwise;

since the sum above is zero unless y D 0, when it equals jM j (see Proposition 1.11). t u Lemma 2.64 Let M D .M; Q/ be an O-FQM. The space CŒM is an irreducible H.M /-module.

58


Proof Using Lemma 2.63, we have 1 H.M /

X

j CŒM .v; /j2 D

.v;/2H.M /

1 X 1 jM j2 jj2 D jM j2 l D 1 ljM j2 ljM j2 2 l

t u

which proves the lemma (using [FH91, Cor. 2.15]).

Lemma 2.65 Let M D .M; Q/ be an O-FQM with level l. The group O=l acts from the right on H.M / via .v; /; A 7! .v; /A WD vA; fA .v/ ; where fA .v/ is as in Lemma 2.53. Remark Note that the above map commutes with the embedding and the canonical projection given in the following exact sequence

H.M / ! M M ! 1: 1 ! l ! Proof of Lemma 2.65 It is enough to show that for fixed A in O=l, the map h 7! hA defines a group homomorphism of H.M /, since Lemma 2.53 and the remark after Lemma 2.52 ensure the fact that the map in the statement of the lemma satisfies the axioms of anaction. bCl 0 0 Let A D aCl cCl d Cl be in O=l and let h D .v; / and h D .w; / be in H.M /. We have hA h0A D vA C wA; 0 fA .v/fA .w/.vA; wA/ : On the other hand, we have .h h0 /A D vA C wA; 0 .v; w/fA .v C w/ : Hence, it remains to show that the following identity holds true .v; w/fA .v C w/ D .vA; wA/fA .v/fA .w/: Calculating both sides separately and inserting the following values ˚ .vA; wA/ D e B.ax0 C cy0 ; bx C dy/ ;

˚ .v; w/ D e B.x 0 ; y/ ;

we see that the following identity proves the assertion: .vA; wA/fA .v/fA .w/ ˚ ˚ D e B.ax0 C cy0 ; bx C dy/ e abQ.x/ C bcB.x; y/ C cdQ.y/ ˚ e abQ.x 0 / C bcB.x 0 ; y 0 / C cdQ.y 0 /


59

˚ D e abQ.x C x 0 / C cdQ.y C y 0 / ˚ e bcB.x; y C y 0 / C bcB.x 0 ; y C y 0 / C B.x 0 ; y/ ˚ D e abQ.x C x 0 / C bcB.x C x 0 ; y C y 0 / C cdQ.y C y 0 / ˚ e B.x 0 ; y/ D .v; w/fA .v C w/: For the third identity we used ad bc 1 mod l, and the fact that lB D 0 for any l 2 l. t u Definition 2.66 Let M be an O-FQM with level l. Using the action of O=l on H.M / from Lemma 2.65, we define J.M / WD O=l Ë H.M /: We call J.M / as the Jacobi group associated to M . Remark The group operation in J.M / is given by .A; h/ .B; h0 / WD .AB; hB h0 /: The fact that J.M / becomes a group with this operation is a well-known fact in basic algebra. More explicitly, for h D .v; / and h0 D .w; 0 /, the above operation is given by A; .v; / B; .w; 0 / D AB; ..vB/ C w; 0 fB .v/..vB/; w// : Remark Henceforth, for the elements of J.M /, we use .A; v; / instead of .A; .v; //. We view O=l (where l is the level of M ) and H.M / as subgroups of J.M / via the maps A 7! .A; 1/ and h 7! .1; h/, respectively. Moreover, via the map ˛ 7! ..˛/; 1/, the group Q can be viewed as a subgroup of J.M /, where is the epimorphism from Q onto O=l explained in the first remark after Definition 2.55. Lemma 2.67 Let M D .M; Q/ be an O-FQM with level l. For h D .v; / 2 H.M / and A 2 O=l , we have AhA1 D 1; vA1 ; fA1 .v/ : Proof The following identity proves the claimed identity: AhA1 D .A; 0; 1/.1; v; /.A1 ; 0; 1/ D .A; 0; 1/ A1 ; vA1 ; fA1 .v/ D 1; vA1 ; fA1 .v/ : t u

60


Lemma 2.68 Let M D .M; Q/ be an O-FQM with level l. For fixed A 2 O=l , we define A W H.M / ! H.M /;

A .h/ D AhA1 :

If is the representation afforded by the H.M /-module CŒM (see Lemma 2.62), then the representations and ı A of H.M / are equivalent. Proof First note from Lemma 2.67 that AhA1 lies in H.M / for any h in H.M /. It is easy to see that ı A defines a representation of H.M /. Using Proposition 2.13, it suffices to show that the of and ı A are equal. Let B be the bilinear traces bCl and let h D v D .x; y/; be in H.M /. The form of M . Write A D aCl cCl d Cl trace of ı A becomes ( 0 1 tr AhA ; CŒM D P fA1 .v/ z2M e fB.z; bx C ay/g

if dx cy ¤ 0 otherwise:

Here we used Lemma 2.67 and the action in Lemma 2.62, also the identity vA1 D .dx cy; bx C ay/ (see Lemma 2.52 and the remark afterwards). We use Proposition 1.11 to evaluate the above sum. We obtain that it is zero unless ay D bx, when it equals jM j. Therefore, we recognize that this value coincides with the trace of in Lemma 2.63. t u Lemma 2.69 Let M D .M; Q/ be an O-FQM with level l and be the representation afforded by the H.M /-module CŒM (see Lemma 2.62). For each A 2 O=l, there exists an (up to multiplication by a constant) unique ı.A/ 2 GL.CŒM / such that the following holds true: ı.A/.h/ı.A/1 D .AhA1 /

.h 2 H.M //:

(2.32)

Proof Let A 2 O=l. By Lemma 2.68 we know that ı A and are equivalent to each other. Hence, there exists an element ı.A/ of GL.CŒM / such that (2.32) holds true. It remains to show that ı.A/ is unique up to multiplication by a constant. Assume there exists .A/ 2 GL.CŒM / which satisfies also (2.32). Let h 2 H.M /. We then have 1 ı.A/.h/. 1 ı/1 .A/ D .h/: We denote 1 .A/ı.A/ by '.A/. Using the above identity we obviously have '.A/.hv/ D h'.A/.v/ for any v 2 CŒM . But this implies that '.A/ defines an H.M /-linear map on CŒM . Since from Lemma 2.64 we know that CŒM is an irreducible H.M /-module, the result follows from Schur’s Lemma (see e.g [FH91, Lem. 1.7]). t u


61

Lemma 2.70 Let M D .M; Q/ be an O-FQM with level l. For A an element of SL.2; O/, let ı.A/ be an element of GL.CŒM / satisfying (2.32). The map A 7! ı.A/ defines a projective representation of O=l . Proof Let A and B be in O=l . By assumption, ı.AB/ satisfies (2.32). It is enough to show that ı.A/ı.B/ also satisfies the same identity, since then by proceeding as in the proof of Lemma 2.69, the statement of the lemma holds true. But we have ı.A/ı.B/.h/.ı.A/ı.B//1 D ı.A/ BhB1 ı.A/1 D ABhB1 A1 D .ABh.AB/1 /: Here we used the assumption that ı.A/ and ı.B/ satisfy (2.32).

t u

Proof of Theorem 2.8 For z 2 M and b 2 O, we define L.Tb /ez WD e fbQ.z/g ez : Let be the epimorphism in Lemma 1.29 from onto O=l. If we can show that the operators L.Tb / and ı..Tb // differ by a constant, then multiplying ı..Tb // with a suitable constant so that it satisfies (i) and using Lemma 2.70, we see that part (i) of theorem holds true. To show that these operators differ by a constant it is enough to show that L.Tb / also satisfies (2.32) (see the proof of Lemma 2.69). But for any h D .x; y; / 2 H.M /, we have L.Tb /.h/L.Tb /1 ez D e fbQ.x/ C B.z; bx y/g ezCx D fT 1 .v/e fB.bx C y; z/g ezCx b D .Tb /h.Tb /1 ez : To obtain the first identity we used L.Tb /1 ez D e fbQ.z/g ez and the action in Lemma 2.62. For the second identity we used Lemmas 2.67 and 2.62. For z 2 M , we set 1 X ˚ L.S /ez WD .M / p e B.z0 ; z/ ez0 : jM j z0 2M We show that the operators L.S / and ı..S // differ by a constant. Then, proceeding as in the previous case, we obtain that part (ii) of theorem also holds true. But for any h D .x; y; / 2 H.M /, we have L.S /.h/L.S /1 ez D

1 X ˚ e B.x; z00 / ez00 jM j 00 z 2M

X ˚ e B.z0 ; y z C z00 / z0 2M

62


D e fB.x; y z/g ezy D fS 1 .v/e fB.z; x/g ezy D ..S /h.S /1 /ez : 1 P 0 For the first identity we used L.S /1 ez D .M / pjM z0 2M e fB.z ; z/g ez0 and j the action in Lemma 2.62. Moreover, we also used the fact that .M / has absolute value one (see Proposition 1.10). The second identity follows from the fact that the sum in the previous identity is zero unless z00 D y z, when it equals jM j (see Proposition 1.11). For the third identity we used Lemmas 2.67 and 2.62. This proves the theorem. t u

Remark Let M D .M; Q/ be an O-FQM with level l. By (2.3) and also Lemma 2.70 we have that O=l acts on Hom.CŒM ; CŒM / via .A; / ! A ;

A

.v/ D ı.A/ ı.A/1 .v/ :

Here ı.A/ is any element of GL.CŒM / which satisfies (2.32). Using the first remark after Definition 2.55 and Proposition 2.2 we have that Q also acts on the space Hom.CŒM ; CŒM /. On the other hand, since W .M / is a Q -module (see Sect. 2.2), the group Q acts on Hom.W .M /; W .M // via (see (2.3)) .˛; / ! ˛ ;

˛

.v/ D .˛/ .˛/1 .v/ ;

where is the representation afforded by Q -module W .M /. But from Theorem 2.8 we know that and ı differ only by a constant. Therefore, the action of Q on the spaces Hom.CŒM ; CŒM / and Hom.W .M /; W .M // coincide. We can now give the second proof of Theorem 2.7. Proof of Theorem 2.7 The dimension of HomQ W .M /; W .M / provides a bound for the number of irreducible Q -submodules of W .M / (see Proposition 2.22). By Q Proposition 2.21 HomQ W .M /; W .M / in fact equals Hom W .M /; W .M / . From the previous remark the latter space equals Hom CŒM ; CŒM O=l . Therefore, it is enough to show that the dimension of this latter space equals the upper bound given in the statement of the theorem. Since CŒM is an irreducible H.M /-module (see Lemma 2.64),the elements .v; 1/ (v 2 M M ) form a basis for the space Hom CŒM ; CŒM (see [Ser77, Prop. 10]). Here is the representation afforded by the H.M /-module CŒM . By Proposition 2.15 and the action given in the previous remark we have that Hom CŒM ; CŒM O=l is spanned by the operators L.v/ WD

1

X

jO=lj A2

O=l

ı.A/.v; 1/ı.A/1

.v 2 M M /:


63

ı We claim that the nonzero L.v/, where the v are representatives for M M O=l, form a basis for the space Hom CŒM ; CŒM O=l . For that first we need to show that if v and w lie in the same orbit (see Lemma 2.52 and the remark afterwards), then L.v/ and L.w/ differ by a constant. We write v D wB1 for some B 2 O=l. Then we have L.v/ D

1

X

jO=lj A2

ı.A/.wB1 ; 1/ı.A/1 :

O=l

Since ı.A/ satisfies (2.32), we have ı.A/.wB1 ; 1/ı.A/1 D A.wB1 ; 1/A1 : We know A.wB1 ; 1/A1 D 1; wB1 A1 ; fA1 .wB1 / from Lemma 2.67 and also fA1 .wB1 / D f.AB/1 .w/=fB 1 .w/ from Lemma 2.53. Now inserting these values to the above identity gives ı.A/.wB1 ; 1/ı.A/1 D

1 fB 1 .w/

ı.AB/.w; 1/ı.AB/1 :

Inserting this quantity to the last sum and doing the substitution A 7! AB1 proves the assertion. Next we determine when L.v/ D 0. We write L.v/ D

X

1 jO=lj

X

ı.AB/.v; 1/ı.AB/1 :

B Stab.v/2O=l = Stab.v/ A2Stab.v/

We have the following identity ı.AB/.v; 1/ı.AB/1 D AB.v; 1/.AB/1 D 1; .vB/; fA .v/fB .v/ D ..vB//fB .v/fA .v/: The first identity follows since ı.AB/ satisfies (2.32). The second identity follows from Lemmas 2.67, 2.53 and the fact that A 2 Stab.v/. The last identity is implied by the action in Lemma 2.62. Therefore, we obtain L.v/ D

1 jO=lj

X B Stab.v/2O=l = Stab.v/

..vB//fB .v/

X

fA .v/:

A2Stab.v/

Clearly, L.v/ D 0 if and only if the inner sum equals zero. But the inner sum equals zero if and only if v is nontrivial (see Lemma 2.58 for v ).

64


Consequently, the space Hom CŒM ; CŒM O=l has as basis the operators L.v/ for which the characters v are trivial. Therefore the dimension of the space 0 ı Hom CŒM ; CŒM O=l equals the number of elements in M M O=l. t u

Chapter 3

Jacobi Forms over Totally Real Number Fields

From this chapter on, the number field K is assumed to be totally real. This restriction is necessary for guaranteeing the holomorphicity of Jacobi forms. As before, we shall simply write O, d for the ring of integers and different of K, respectively. Furthermore, we shall use D SL.2; O/, and we shall write Q for the metaplectic cover of SL.2; O/ which will be defined in Sect. 3.3. In addition, for a subring R of K, we shall denote by R the group SL.2; R/ and by QR the metaplectic cover of R . In the present chapter we shall develop a theory for Jacobi forms over number fields. In particular, we shall see that there is a one-to-one correspondence between the spaces of Jacobi forms and certain spaces of vector-valued Hilbert modular forms (see Theorem 3.5). As an immediate corollary, we shall deduce that the spaces of Jacobi forms are finite dimensional (see Corollary 3.53). Certain spaces of functions, the Jacobi theta functions which can be viewed as modules over Q (see Theorem 3.1), will play an important role in this context. In the next chapter, we shall define a Q -module isomorphism between the spaces of Weil representations associated to certain discriminant modules and the spaces of these theta functions. This will be a key step for the explicit description of the singular Jacobi forms whose index is a rank one O-lattice. We shall also calculate the matrix coefficients of the action of Q on the spaces of Jacobi theta functions (see Theorem 3.1). In Sect. 3.1, we shall recall or develop those basic facts about integral lattices over number fields (the O-lattices) which are crucial for the definition of Jacobi forms. In Sect. 3.2, we shall introduce some basic notations which will help to avoid clumsy notations when dealing with Jacobi forms and Hilbert modular forms. In Sect. 3.3, we shall define the metaplectic cover of SL.2; O/, which will be necessary to include Jacobi forms of half integral weight. In Sect. 3.4, we shall introduce the notions of Heisenberg groups and the Jacobi groups associated to O-lattices, and we list several results concerning the actions of these groups on the spaces of holomorphic functions, which will be helpful for defining Jacobi forms. In Sect. 3.5,


65

66

3 Jacobi Forms

we shall introduce the spaces of Jacobi theta functions associated to O-lattices. Later in the same section, we shall study the spaces of Jacobi theta functions as Q -modules, and moreover we shall determine the matrix coefficients of the Q action. In Sect. 3.6, we shall finally define Jacobi forms, and we study their Fourier developments and theta expansions. In Sect. 3.7, we shall show that the spaces of Jacobi forms are isomorphic to spaces of vector-valued Hilbert modular forms. In particular, we shall be able to prove that a Köcher principle holds true for Jacobi forms and that the spaces of Jacobi forms are finite dimensional.

3.1 O-Lattices Definition 3.1 An integral lattice over O is a pair L D .L; ˇ/, where L denotes a finitely generated torsion-free O-module, and where ˇ W L L ! d1 is a map which satisfies the following properties: (i) The map ˇ is O-bilinear and symmetric. (ii) The map ˇ is non-degenerate (i.e. ˇ.x; L/ D f0g if and only if x D 0). For simplicity, the integral lattices over O will be named O-lattices. We sometimes write shortly x 2 L for x 2 L. Proposition 3.2 Let L D .L; ˇ/ be an O-lattice. Then Tr.L/ WD L; trK=Q ıˇ defines a Z-lattice. Proof The bilinear form trK=Q ıˇ is non-degenerate, since the Q-bilinear map a; b 7! trK=Q .a; b/ is non-degenerate (a; b 2 K). Clearly, trK=Q ıˇ is Z-bilinear and symmetric, which proves the proposition. t u Let L D .L; ˇ/ be an O-lattice. The O-lattice L0 D .L0 ; ˇ 0 / is called an Osublattice of L, if L0 is an O-submodule of L, and ˇ 0 is the restriction of ˇ to L0 L0 . For x 2 L# , here and in the following, we set ˇ.x/ WD 12 ˇ.x; x/. If ˇ.x/ 2 d1 , then L is called an even O-lattice, otherwise it is called an odd O-lattice. Every odd O-lattice contains an even O-sublattice. Indeed, the map x 7! ˇ.x; x/ C 2d1 defines a group homomorphism from L to d1 =2d1 . If L is even, then it is the trivial homomorphism. If L is odd, then the kernel of this homomorphism is an even O-sublattice of L. For 1 j n D ŒK W Q , let j be the embeddings of K into R. If j ı ˇ.x; x/ > 0 for all j and all nonzero x 2 L, then L is called totally positive definite. Note that the notion of totally positive definite O-lattices is a generalization to number fields of positive and integral Z-lattices. We say that there is a homomorphism from L to L0 , if there is an O-module homomorphism ' W L ! L0 which is isometric, i.e. such that ˇ 0 .'.x/; '.y// D ˇ.x; y/ (x; y 2 L). Note that every homomorphism ' between totally positive definite O-lattices is injective (indeed, if '.x/ D 0, then 0 D ˇ 0 '.x/; '.y/ D

3.1 O-Lattices

67

ˇ.x; y/ for all y, which implies x D 0 since ˇ is non-degenerate). The O-lattices L and L0 are called isomorphic, and we write L ' L0 , if there is an isomorphism between them. Recall that every torsion-free finitely generated O-module L is isomorphic as an O-module to an O-module of the form O r1 ˚ a for some positive integer r and a fractional O-ideal a [FT93, § II.4, Thm. 13(b)]. Moreover, the integer r and the ideal class of a are uniquely determined by L [FT93, § II.4, Thm 13(c)]. The integer r is called the rank of L. The ideal class of a is called the Steinitz-invariant of L. Clearly, r equals the dimension of the K-vector space K ˝O L, i.e. r D dimK K ˝O L. If A is a ring extension of O, we denote the A-bilinear extension of ˇ, namely the bilinear map A ˝O L A ˝O L ! A ˝O d1 , also by ˇ. We use ˇ.x/ D 12 ˇ.x; x/ if 2 is invertible in A ˝O d1 . If d1 is contained in A, we identify the A-module A ˝O d1 with A (via the O-linear map induced from the O-bilinear map .a; d / 7! ad). The dual of L is defined as L# WD fx 2 K ˝O L W ˇ.x; L/ 2 d1 g: Note that L# is again a finitely generated torsion-free O-module, and that we # have .L# / D L [O’M00, §82F] (loc. cit. the dual of a lattice is defined slightly differently than ours, but it is easy to modify the arguments given loc. cit. so that they extend also to our situation). Definition 3.3 Let L D .L; ˇ/ be an even O-lattice. We define the discriminant module of L as the O-FQM: DL WD L# =L; x C L 7! ˇ.x/ C d1 : Remark Note that, for the well-definedness of the quadratic form Q W x C L 7! ˇ.x/ C d1 of DL the evenness of L is crucial. The non-degeneracy of Q comes # from the fact that .L# / D L. From [Ebe02, § 1.1] we know that L# =L is finite. By the level, modified level and the annihilator of an O-lattice L we mean the level, modified level and the annihilator of the O-FQM DL . The reader is referred to Sect. 1.1 for basic notions about finite quadratic O-modules. For an isotropic submodule U Definition 3.4 Let L D .L; ˇ/ be an even O-lattice. of L# =L, we define L=U WD 1 .U /; ˇ , where is the canonical projection from L# onto L# =L. Remark Note that L=U is again an even O-lattice (the non-degeneracy of ˇ on 1 .U / 1 .U / follows from the easily proven fact that x 7! ˇ.x; / defines an isomorphism of the K-vector space K ˝O L with its dual and that 1 .U / contains a basis of this vector space.)

68

3 Jacobi Forms

Proposition 3.5 Let L D .L; ˇ/ be an even O-lattice and be the canonical projection from L# to L# =L. The map x C 1 .U / 7! .x/ C U defines an isomorphism from DL=U to DL =U . Proof The statement is an obvious consequence of the very definition of L=U . u t The remaining statements of this section concern O-lattices of rank one and will not be used before the next chapter. Definition 3.6 Let c be a nonzero fractional O-ideal, and ! be a nonzero element of K such that ! 0 and !c2 d1 . We set .c; !/ WD .c; .x; y/ 7! !xy/ :

(3.1)

Note that, .c; !/ defines a totally positive definite O-lattice of rank one. Proposition 3.7 Let .c; !/ be as in the above definition, and assume that .c; !/ is even. Then the discriminant module of .c; !/ is an O-CM. Moreover, the annihilator, level and the modified level of .c; !/ equal c2 !d, 2c2 !d and c2 !d=2, respectively. The annihilator and the level of L are divisible by 2 and 4, respectively. Proof Let L D .L; ˇ/ D .c; !/. Since L# D fy 2 K W !yc d1 g D .c!d/1 , we have DL D .c!d/1 =c; x C c 7! !x 2 =2 C d1 : Here note that c .c!d/1 , since the O-lattice .c; !/ is integral. Lemma 1.17 implies then that DL is an O-CM. It is easy to see that the annihilator, level and the modified level of L are of the claimed form. Since L is even, !dc2 is divisible by 2. Therefore, the last statement also holds true. t u Proposition 3.8 Every homomorphism between O-lattices of the form (3.1) is given by a multiplication of some nonzero element in K. Proof Let .c; !/ and .c0 ; ! 0 / be as in Definition 3.6. Let ' W .c; !/ ! .c0 ; ! 0 / be a homomorphism. We can find a positive integer N such that N c is integral. Since ' is an O-module homomorphism, we have N'.x/ D '.Nx/ D Nx'.1/ for all x 2 c. Hence, '.x/ D x'.1/ for all x 2 c. In addition, since ' is isometric, we have !xy D ! 0 '.x/'.y/ D ! 0 xy'.1/2 for all x; y 2 c. This implies that for nonzero x and y, we have ! D ! 0 '.1/2 . Since, ! and ! 0 are nonzero elements of K, we obtain that '.1/ ¤ 0, i.e. ' is injective. t u Proposition 3.9 Let .c; !/ and .c0 ; ! 0 / be as in Definition 3.6. Then the lattices .c; !/ and .c0 ; ! 0 / are isomorphic if and only if ! 0 D a2 ! and c0 D a1 c for some a 2 K . Proof Suppose that ' is an isomorphism from .c; !/ to .c0 ; ! 0 /. From Proposition 3.8, we have '.x/ D xa (x 2 c) for some nonzero a 2 K. Since ' is a surjection, we have c0 D ca. Moreover, since ' is isometric, we have !xy D

3.2 Algebraic Prerequisites

69

! 0 '.x/'.y/ D ! 0 xya2 for all x; y 2 c. Then, by taking nonzero x and y, we have ! 0 D a2 !. The other inclusion is obvious. t u Proposition 3.10 Let L D .L; ˇ/ be a totally positive definite O-lattice of rank one. Then L is isomorphic to an O-lattice of the form .c; !/ as in Definition 3.6. Proof Since L has rank one, using [FT93, § II.4, Thm. 13(b)], we obtain that L is isomorphic to a fractional O-ideal, say c. Let ' be an isomorphism from c onto L. Note that there exists a positive integer N 0 such that N 0 c is integral. Then for any x 2 c, we have N 0 '.x/ D '.N 0 x/ D N 0 x'.1/, i.e. '.x/ D x'.1/. Note that since ' is an isomorphism, a WD '.1/ ¤ 0. Let c 2 c such that ˇ.ca; ca/ ¤ 0, and let x; y 2 c. We can find a positive integer N such that Nx; Ny; Nc 2 O. Then we have Nc2 ˇ.'.x/; '.y// D Nc2 ˇ.ax; ay/ D cˇ.Ncax; ay/ D Nxcˇ.ca; ay/ D xˇ.ca; Ncay/ D Nxyˇ.ca; ca/: We set ! WD ˇ.ca; ca/=c 2 . Hence, ' defines an isomorphism from the lattice L onto .c; !/. The fact that !c2 lies in d1 follows from the integrality of the Olattice L, and that ! 0 follows from the totally positive definiteness of the Olattice L. t u

3.2 Algebraic Prerequisites We set C WD C ˝Q K;

R WD R ˝Q K:

We view C as a ring with respect to the multiplication induced from the map defined by .z ˝ a; z0 ˝ a0 / 7! .zz0 / ˝ .aa0 / (z; z0 2 C, a; a0 2 K), and as algebra over C and over K via the maps satisfying .z; z0 ˝ a/ 7! .zz0 / ˝ a and .a; z ˝ a0 / 7! z ˝ .aa0 /, respectively. In particular, we identify C and K with their images in C under the embeddings z 7! z ˝ 1 and a 7! 1 ˝ a. Similar conventions are made for R, which we view as a subring of C . In particular, the group O can be identified with its image in C under the embedding b 7! 1 ˝ b (b 2 O). Hence the group becomes a subgroup of SL.2; C / of 2 2-matrices over the ring C with determinant 1. We use N and tr for the norm and trace of the C-algebra C . Thus, if c is an element of C and f .x/ is the characteristic polynomial of the endomorphism of C given by multiplication by c, then f .x/ D x m tr c x m1 C C .1/m N.c/. Likewise, if c is in C , then Y X .c/; tr c D .c/: N.c/ D 2E

2E

70

3 Jacobi Forms

Here E is the set of the C-linear continuations of all embeddings W K ,! C to C-linear maps W C ! C (we use the same letter for the embedding and its linear continuation). Thus .z ˝ a/ D z.a/. The maps 2 E are coordinate functions of Q ' ! Cn (n D ŒK W Q ), where we take coordinate the ring isomorphism 2E W C wise multiplication as multiplication on Cn . In particular, an element c of C is invertible (i.e. multiplication by c is an isomorphism of C ) if and only if N.c/ is different from 0. The Q-bilinear map .z; a/ 7! z ˝ a induces a Q-linear involution on C which we also indicate by placing a bar over the argument. We set H WD fz 2 C W = .z/ > 0; for all 2 E g: Note that H is an open subset of C . Proposition 3.11 The group R acts on H via .A; / 7! A WD .a C b/.c C d /1 :

(3.2)

Proof First of all, for c and d in R and in H the element c C d is invertible and A is in H . For proving this we use the easily proved identity .A A/.c C d /.c C d / D : The left hand side under ( 2 E ) has hence strictly positive imaginary part. This shows that N.c C d / 6D 0, hence that c C d is invertible. So, we have = .A/ D = ./ = .c C d /.c C d / > 0 ( 2 E ). 0 0 Next, it obviously holds true that 1 D . Let A D ac db , B D ac 0 db 0 be in R and 2 H . Then we have B.A/ D

a Cb 0 a0 c Cd C b

a Cb 0 c 0 c Cd C d

D

.a0 a C b 0 c/ C a0 b C db0 D .BA/: .c 0 a C cd0 / C c 0 b C dd0 t u


Under the identification of C with C from above the set H corresponds to the subset of vectors w in Cn whose components have all positive imaginary part, and the trace and the norm of w become p the sum and the product of its components, respectively.For w 2 C , we use w for the element in C which corresponds to p p the element w1 ; : : : ; wn in Cn under the above isomorphism which sends w to w1 ; : : : ; wn . For the choice of the square root of a complex number, we refer to the section “Notations”. For an O-lattice L D .L; ˇ/ of rank r, the O-module LC WD C ˝O L (similarly, LR WD R ˝O L) becomes a C -module via C -linear continuation of the following map n

.w0 ; w ˝ x/ 7! w0 w ˝ x

.w; w0 2 C ; x 2 L/;

3.3 The Metaplectic Cover QR of R

71

which contains L and K as O-submodules via the identifications x 7! 1 ˝ x and a 7! .1 ˝ a/ ˝ 1 (x 2 L, a 2 K), respectively. Moreover, LC becomes a C-vector space of dimension nr via linear continuation of the following map .z; w ˝ x/ 7! zw ˝ x

.z 2 C; w 2 C ; x 2 L/:

3.3 The Metaplectic Cover QR of R Definition 3.12 cover of R (resp. ) is the set of tuples of the metaplectic The form A D ac db ; w , where A 2 R (resp. A 2 ) and w W H ! C a holomorphic function satisfying w2 ./ D N.c C d /, with the following operation .A; w/ .B; v/ WD AB; w.B/v./ : (Here B denotes the group action (3.2).) In the following we denote the metaplectic cover of R (resp. ) by QR (resp. Q ). Remark Note that R (resp. ) is in fact a group. The group QR is a central extension of R

1 ! h.1; 1/i ! QR ! R ! 1;

(3.3)

where is the map .A; w/ 7! A and the second In the arrow p is the inclusion. following, for A D ac db 2 R , we write A WD A; N c C d . Note that the map A 7! A from R to QR does not in general define a group homomorphism. We know from Sect. 3.2 that can be embedded into R . Hence, the group Q can also be viewed as a subgroup of R . For later use we determine a set of generators for Q . Proposition 3.13 The group Q is generated by the elements Tb D .Tb ; 1/ (b 2 O), p S D S; N . / and I WD .1; 1/. If the degree n of K over Q is odd, then Q is already generated by Tb and S . Proof Let A D ac db 2 and be the projection in (3.3). We know from Theorem 2.1 that A D .A / can be written as a word in S D .S / and Tb D .Tb / (b 2 O). Hence, A can be written as a word in .Tb / , S and an element lying in the kernel of , which equals .1; ˙1/. Since every element of Q is either of the form A , or A I , the first statement holds true. If n is odd, then I D .S /4 , i.e. the second statement holds true. t u

72

3 Jacobi Forms

3.4 The Jacobi Group of an O-Lattice In the present section, we shall define the Heisenberg group and the Jacobi group associated to an O-lattice. Moreover, we shall study various actions of these groups which are important in the sequel. As explained in the section “Notations”, we shall use e fcg for exp.2 i tr.c//, where c 2 C . Definition 3.14 Let L D .L; ˇ/ be an O-lattice. The Heisenberg group associated to L is H.LR / WD f.x; y; / W x; y 2 LR ; 2 C g together with the operation ˚ .x; y; / .x 0 ; y 0 ; 0 / D x C x 0 ; y C y 0 ; 0 e ˇ.x; y 0 / ˇ.x 0 ; y/ =2 :

(3.4)

Moreover, we use H.L# / and H.L/ for the subgroups ˚ H.L# / D .x; y; / W x; y 2 L# ; 2 2l ˚ H.L/ D .x; y; e fˇ.x; y/=2g/ W x; y 2 L : Here l is the exponent of the abelian group L# =L. Remark The exact sequence 0 ! C !H.LR / ! LR LR ! 0 7!.x; y; / 7! .x; y/ shows that H.LR / is a central extension of LR LR by C . Note that the elements .x; 0; 1/ and .0; y; 1/ (x; y 2 L) generate H.L/. Note also that H.L/ is a normal subgroup of H.L# /. Proposition 3.15 The operation (3.4) defines indeed a group structure on H.LR /. Proof The neutral element is .0; 0; 1/. For an element .x; y; / of H.LR /, the inverse element equals .x; y; 1 /. The associativity follows from the following identity: .x; y; / .x 0 ; y 0 ; 0 / .x 00 ; y 00 ; 00 / ˚ D x C x 0 C x 00 ; y C y 0 C y 00 ; 0 00 e ˇ.x 0 ; y 00 / ˇ.x 00 ; y 0 / =2 ˚ e ˇ.x; y 0 C y 00 / ˇ.x 0 C x 00 ; y/ =2 D x C x 0 C x 00 ; y C y 0 C y 00 ; 0 00 ˚ e ˇ.x C x 0 ; y 00 / C ˇ.x; y 0 / ˇ.x 00 ; y C y 0 / ˇ.x 0 ; y/ =2

3.4 The Jacobi Group of an O-Lattice

73

˚ D x C x 0 ; y C y 0 ; 0 e ˇ.x; y 0 / ˇ.x 0 ; y/ =2 .x 00 ; y 00 ; 00 / D .x; y; / .x 0 ; y 0 ; 0 / .x 00 ; y 00 ; 00 /: t u For later use we note the following Proposition 3.16 Let L D .L; ˇ/ be an O-lattice and let l be the exponent of L# =L. Then H.L# /=H.L/ is a central extension of L# =L by 2l . More precisely the applications 7! .0; /H.L/ and .x; /H.L/ 7! x C L define an exact sequence 1 ! 2l ! H.L# /=H.L/ ! L# =L ! 1: The order of H.L# /=H.L/ equals, in particular, 2ljL# =Lj2 . Remark It is not hard to show that, for even L, the group H.L# /=H.L/ is also a central extension of the Heisenberg group H.DL / associated to the O-FQM DL (see Definition 2.61). Proof of Proposition 3.16 The proposition follows easily by a straightforward calculation. t u Proposition 3.17 Let L D .L; ˇ/ be an O-lattice. The group R acts on the group H.LR / from the right via .x; y; /; A ! .x; y; /A WD .x; y/A; : Here, .x; y/A stands for the formal multiplication of the row vector .x; y/ and the matrix A. More precisely, for A D ac db , we use .x; y/A D .ax C cy; bx C dy/. Proof To prove the proposition we need to show that the axioms of a group action are satisfied, and that, for fixed A 2 R , the map h 7! hA defines a group homomorphism of H.LR /. Let A; B 2 R and h 2 H.LR /. Write A D ac db , 0 0 B D ac 0 db 0 and h D .x; y; /. It is obvious that h1 D h, and, since we have ..x; y/A /B D ..xa C yc/a0 C .xb C yd/c 0 ; .xa C yc/b 0 C .xb C yd/d 0 / D ..aa0 C bc0 /x C .ca0 C dc0 /y; .ab0 C bd0 /x C .cb0 C dd0 /y/ D .x; y/AB ; the first part holds true. Next we need to show hA h0A D .h h0 /A (h0 2 H.LR /). Write h0 D .x 0 ; y 0 ; 0 /. The identity e

˚ ˇ.ax C cy; bx0 C dy0 ˇ.ax0 C cy0 ; bx C dy/ =2 ˚ D e ˇ.x; y 0 / ˇ.x 0 ; y/ =2

clearly proves the second part.

t u

74

3 Jacobi Forms

Definition 3.18 Let L D .L; ˇ/ be an O-lattice. We use J.LR / for the semidirect product of R and H.LR / with respect to the action in Proposition 3.17, in short J.LR / D R Ë H.LR /: Similarly, we use J.L# / WD Ë H.L# / and, if L is even J.L/ WD Ë H.L/. Remark Recall from the general definition of semidirect products that J.LR / consists of all pairs .A; h/ of elements A in R and h in H.LR / together with the operation .A; h/ .B; h0 / D .AB; hB h0 /: More explicitly, the operation can be written as A; .x; y; / B; .x 0 ; y 0 ; 0 / D AB; ax C cy C x 0 ; bx C dy C y 0 ; ˚ 0 e ˇ.ax C cy; y 0 / ˇ.x 0 ; bx C dy/ =2 : For the definition of J.L# / and J.L/ to make sense, we need that H.L# / and H.L/ are invariant under the action of on the Heisenberg group. For H.L# / this is always true, whereas for H.L/ this holds true only if L is even. Indeed, if h D .x; y; e fˇ.x; y/=2g/ is in H.L/, then, for A in , we have that hA lies in H.L/ .0; e fabˇ.x/ C cdˇ.y/g/. But e fabˇ.x/ C cdˇ.y/g equals 1 for all A in and all x and y in L if and only if L is even. Note that J.L# / and J.L/ are subgroups of J.L R /. We view R and H.LR / as subgroups of J.LR / via the maps A 7! A; 1 and h 7! 1; h , respectively. So, when we write Ah, we mean the element .A; 1/ .1; h/. Lemma 3.19 The map W R H ! H , defined by .A; / WD c C d

.A D

a b c d /

satisfies the following identity (cocycle identity): .A; B/.B; / D .AB; /: 0 Proof Let A D ac db , B D ac 0 true, since we have:

b0 d0

(3.5)

be elements of R . The identity (3.5) holds

.A; B/.B; / D .cB C d /.c 0 C d 0 / D ca0 C cb0 C dc0 C dd0 D .ca0 C dc0 / C cb0 C dd0 D .AB; /: t u


75

Proposition 3.20 Let L D .L; ˇ/ be an O-lattice. The group R acts on H LC via z : A; .; z/ 7! A.; z/ WD A; .A; / Moreover, H.LR / also acts on H LC via .x; y; /; .; z/ 7! .x; y; /.; z/ WD .; z C x C y/: 0 0 Proof Let A D ac db , B D ac 0 db 0 be elements of R and .; z/ 2 H LC . Since obviously we have 1.; z/ D .; z/, the following identity proves the first statement: B.A.; z// D B.A/;

z .A; /

.B; A/

!

D BA;

z .BA; /

D BA.; z/:

Here to obtain the second identity we used (3.5). For proving the second statement, let h D .x; y; / and h0 D .x 0 ; y 0 ; 0 / be elements of H.LR /. Obviously, we have 1.; z/ D .; z/. Furthermore, we calculate h0 h.; z/ D .x 0 ; y 0 ; 0 / .x; y; /.; z/ D ; z C .x C x 0 / C .y C y 0 / ˚ D x 0 C x; y 0 C y; 0 e ˇ.x 0 ; y/ ˇ.x; y 0 / =2 .; z/ D .h0 h/.; z/: t u


Lemma 3.21 Let L D .L; ˇ/ be an O-lattice. For any y 2 H LC , h 2 H.LR / and A 2 R , we have h.Ay/ D A.hA y/: Proof Write y D .; z/, h D .x; y; / and A D ac db . The claimed identity holds true, since we have z C .ax C cy/ C bx C dy z C x.A/ C y D A; h.Ay/ D A; .A; / .A; / D A.; z C .ax C cy/ C bx C dy/ D A.hA y/: t u Proposition 3.22 Let L D .L; ˇ/ be an O-lattice. The group J.LR / acts on H LC via

.A; h/; .; z/ ! 7 .A; h/.; z/ WD A.h.; z//:

76

3 Jacobi Forms

0 Proof Let .A; h/, .B; h / be in J.LR / and u 2 H LC . Since obviously we have 1; 1 u D u, the following identity proves the proposition

.B; h0 / .A; h/u D .B; h0 / A.hu/ D B h0 A.hu/ ; D B.A.h0A .hu/// D .BA/..h0A h/.u// D .BA; h0A h/.u/ D .B; h0 / .A; h/ .u/: Here we used Lemma 3.21 to obtain the third identity, and for the forth identity we used the second part of Proposition 3.20. t u Proposition 3.23 Let k space Hol.H / via

2 Z. The group R acts from the right on the

k .; A/ 7! jk A ./ WD N .A; / .A/: (See (3.19) for the function .A; /). Proof Let A; B 2 R , 2 Hol.H / and 2 Hol.H /. The following identity proves the proposition, since we obviously have jk 1 D : k k jk A jk B ./ D N .B; / N .A; B/ .A.B// k D N .AB; / .AB/ D jk AB ./: t u

The second identity follows from (3.5) and Proposition 3.11.

Proposition 3.24 Let k 2 Z and L D .L; ˇ/ be an O-lattice. The group R acts from the right on Hol.H LC / via k .; A/ 7! jk;L A .; z/ WD N .A; / e

where A D

a b c d

cˇ.z/ z A; ; .A; / .A; /

. (Recall that ˇ.z/ D 12 ˇ.z; z/.)

Proof Let 2 Hol.H LC / and B 2 R . Write B D the following identity implies the proposition

a0

b0 c0 d 0

. Since jk 1 D ,

jk;L A jk;L B .; z/

D N .B; /

k

AB;

k N .A; B/ e

z .AB; /

0 c ˇ.z/ cˇ.z/ e .B; /2 .A; B/ .B; /


D N..AB; //

k

77

z .ca0 C dc 0 /ˇ.z/ AB; e .AB; / .AB; /

D jk;L AB .; z/:

To obtain the second identity we used (3.5) and Proposition 3.11, also the identity c0 c ca0 C dc 0 D C ; .AB; / .B; / .A; B/.B; /2 where we used a0 d 0 b 0 c 0 D 1 and (3.5).

t u

Proposition 3.25 Let k 2 Z and let L D .L; ˇ/ be an O-lattice. The group H.LR / acts from the right on Hol.H LC / via ; .x; y; / 7! jk;L .x; y; / .; z/ WD e fˇ.x/ C ˇ.x; z/ C ˇ.x; y/=2g .; z C x C y/: Proof Let 2 Hol.H LC / and h; h0 2 H.LR /. Write h D .x; y; / and h0 D .x 0 ; y 0 ; 0 /. Since we obviously have jk;L 1 D , the following identity proves that we have indeed an action: ˚ jk;L h jk;L h0 .; z/ D 0 e ˇ.x C x 0 / C ˇ.x C x 0 ; z/ C ˇ.x; y 0 / ˚ e ˇ.x 0 ; y 0 /=2 C ˇ.x; y/=2 .; z C .x C x 0 / C y C y 0 / ˚ D 0 e ˇ.x; y 0 / ˇ.x 0 ; y/ =2 ˚ e ˇ.x C x 0 / C ˇ.x C x 0 ; z/ C ˇ.x C x 0 ; y C y 0 /=2 .; z C .x C x 0 / C y C y 0 / D jk;L .h h0 / .; z/: t u Lemma 3.26 Let k 2 Z and let L D .L; ˇ/ be an O-lattice. For any element in Hol.H LC /, A 2 R , h 2 H.LR /, we have jk;L h jk;L A D jk;L A jk;L hA : Proof Let y 2 H LC . Write y D .; z/, h D .x; y; / and A D on the left

a b c d

. We have

78

3 Jacobi Forms

k cˇ.z/ jk;L h jk;L A .; z/ D N .A; / e .A; / z z e Aˇ.x/ C ˇ x; C xA C y : C ˇ.x; y/=2 A; .A; / .A; /

Since hA D .xa C yc; xb C yd; /, on the right we have

jk;L A jk;L hA .; z/ D e fˇ.xa C yc/ C ˇ.xa C yc; z/ C ˇ.xa C yc; xb C yd/=2g k cˇ.z C .xa C yc/ C xb C yd/ N .A; / e .A; / z C .xa C yc/ C xb C yd A; : .A; /

The claimed identity follows now from the following identities: z C .xa C yc/ C xb C yd z C xA C y D ; .A; / .A; / cˇ.z C .xa C yc/ C xb C yd/ C .c C d /ˇ.xa C yc/C c C d ˇ.xa C yc; xb C yd/ 2 c C d D cˇ.z/ C .a C b/ˇ.x/ C ˇ.x; z/ C ˇ.x; y/: 2 .c C d /ˇ.xa C yc; z/ C

The first one is obvious. The second one follows using ad bc D 1.

t u

Proposition 3.27 Let k 2 Z and let L D .L; ˇ/ be an O-lattice. The group J.LR / acts from the right on Hol.H LC / via ; .A; h/ 7! jk;L .A; h/ WD jk;L A jk;L h: Proof Let 2 Hol.H LC /, .B; h0 / 2 J.LR /. From Propositions 3.24 and 3.25, we have jk;L .1; 1/ D . Moreover, we have (writing j for jk;L )

j.A; h/ j.B; h0 / D jA jh jB jh0 D jA jB jhB jh0 D jAB j.hB h0 / D j.AB; hB h0 / D j .A; h/.B; h0 / :

The first and fourth identities follow from the very definition of the J.LR /-action. For the second identity we used Lemma 3.26, for the third identity we used Propositions 3.24 and 3.25. t u


79

If we replace the integer k in Proposition 3.23 with a half integer, then the action does not anymore define an action because of the ambiguity of the square root of N.c C d /. To solve the problem of this square root, we have to pass to the metaplectic cover QR of R (recall Sect. 3.3 for its definition). For a number k in 12 Z, we define the action .A; w/; 7! jk ..A; w/ of QR on Hol.H / and Hol.H LC / as in the Propositions 3.23 and 3.24, respectively, but with the factor k replaced by w./2k . It is clear that this defines indeed an action. N c C d Thus we can state Proposition 3.28 Let L D .L; ˇ/ be an O-lattice and k 2 12 Z. The group QR acts on the right of the space Hol.H LC / via ; .A; w/ 7! jk;L .A; w/.; z/ WD w./

2k

z cˇ.z/ A; : e .A; / .A; /

(3.6)

Definition 3.29 Let L D .L; ˇ/ be an O-lattice. The semidirect product of QR and H.LR / with respect to the action

.x; y; /; .A; w/ ! .x; y; /.A;w/ WD .x; y/A;

(3.7)

is denoted by JQ .LR /, and is called the Jacobi group associated to L. We set also JQ .L# / WD Q Ë H.L# / and, if L is even, JQ .L/ WD Q Ë H.L/. Remark We view QR and H.LR / as subgroups of JQ .LR / via the maps ˛ 7! ˛; 1 and h 7! .1; h/, respectively. So, when we write ˛h, we mean the element .˛; h/. Remark If we combine the action in (3.6) with the action of H.LR / on the space Hol.H LC / (see Proposition 3.25), we obtain a right action of JQ .LR / on Hol.H LC /. We shall now define certain differential operators on the space of smooth complex valued functions C1 .H LC /. For this end let E denote the set of all C-linear extensions of the embeddings from K into R to C-linear maps from C into C (as already introduced in Sect. 3.2). Note that tr ıˇ W LC LC ! C is the C-bilinear continuation of the non-degenerate Z-linear form .x; y/ 7! tr ıˇ.x; y/ from L L ! Z (see Proposition 3.2) we conclude that tr ıˇ is non-degenerate on LC LC . It is then easy to prove that there is a basis of the C-vector space LC with coordinate functions z;j (j D 1; : : : ; r, 2 E ) such that, for any in E and all z1 and z2 in LC , we have ı ˇ.z1 ; z2 / D

r X j D1

z;j .z1 /z;j .z2 /:

(3.8)

80

3 Jacobi Forms

We view z;j also as functions on H LC by setting z;j .; z/ D z;j .z/ for in H and z 2 LC . Furthermore, we use for the function on H LC such that .; z/ D ./. Note that ˚ ˚ 2E z;j 2E ;1j r W H LC ! Hn Cnr defines a biholomorphic map. Here H denotes the usual upper half plane in C and r and n are the rank of L and the degree of K over Q, respectively. For in E we set WD

@2 @2 C : : : C ; @z2;r @z2;1

H WD

@ 1 ; @ 4 i

(3.9) (3.10)

and call these operators the -Laplace operator and -Heat operator on H LC , respectively. Lemma 3.30 For 2 E , 2 H and s in LC , we have the following formulas: H e fˇ.s/ C ˇ.s; z/g D 0 H e fˇ.z/=g D

r e fˇ.z/=g : 2

(3.11) (3.12)

Here the expressions on the left after the differential operators are considered as functions in .; z/ on H LC . P P Proof As immediate consequences of (3.8) and tr ˇ.z/= D 2E 21 rj D1 z2;j one obtains e fˇ.s; z/g D 2.2 i /2 ˇ.s/ e fˇ.s; z/g ; @ e fˇ.s/g D 2 i ˇ.s/ e fˇ.s/g ; @ 4 i ˇ.z/ 2 i r e fˇ.z/=g ; e fˇ.z/=g D @ 2 i ˇ.z/ e fˇ.z/=g D e fˇ.z/=g : @ 2 The claimed identities of the lemma are now obvious.

t u

Proposition 3.31 Let L D .L; ˇ/ be an O-lattice of rank r, and let in E . Then for any 2 Hol.H LC /, and ˛ D .A; w/ 2 Q , we have 2 H jr=2;L ˛ D .A; / H jr=2;L ˛:

3.5 The Jacobi Theta Functions

81

Proof It suffices to prove the claimed identity for the standard generators Tb (b in O), I and S of Q . Except for S the claimed identity is then obvious. For proving the identity for S we write first of all p r jr=2;L S .; z/ D .1=; z=/ e fˇ.z/=g N : Thus jr=2;L S is a product of three functions, which we denote by f1 , f2 and f3 , respectively. Applying now the heat operator H , yields accordingly H jk;L S .; z/ D .H f1 / f2 f3 C f1 .H f2 / f3

where r D

˚

@ @z ;j

j

@ 1 .r f1 / .r f2 / f3 C f1 f2 f3 ; 2 i @ (3.13)

. A short calculation, using E D

P j

z ;j @z@ ;j , shows

H f1 D 2 H .1=; z=/ C 2 E .1=; z=/; H f2 D

r f2 ; 2

1 1 .r f1 / .r f2 / D 2 .E /.1=; z=/f2 ; 2 i

r @ f3 D f3 : @ 2 Here, the second identity is identical with (3.12). We observe that the second and fourth term in (3.13) cancel. Moreover, the third term and the second term in the formula for H f1 multiplied by f2f3 cancel. Finally, the remaining first term of H f1 multiplied by f2 f3 equals 2 H jr=2;L S . This proves the proposition. t u

3.5 The Jacobi Theta Functions In this section we introduce and study certain spaces of Jacobi theta functions which will be important in all remaining chapters. We shall show that these spaces of Jacobi theta functions are Q -modules (see Theorem 3.1), and we shall calculate explicitly the matrix coefficients of the associated representations. For t 2 C , we shall use q t for the function on H such that q t ./ D e ft g :

82

3 Jacobi Forms

Definition 3.32 Let L D .L; ˇ/ be a totally positive definite even O-lattice. For x 2 L# =L, we set X q ˇ.s/ e fˇ.s; z/g . 2 H ; z 2 LC /: (3.14) #L;x .; z/ WD s2L# sx mod L

We refer to these functions as the Jacobi theta functions associated to L. Moreover, we set ˚ L WD spanC #L;x W x 2 L# =L : Remark It is easily verified that the series defining #L;x are absolutely convergent and that the #L;x are holomorphic (here one needs that L is totally positive definite). Note also that #L;x depends only on the residue class of x modulo L. Proposition 3.33 For fixed , the functions z 7! #L;x .; z/ (x 2 L# =L) defined in (3.14) are linearly independent. In particular, the dimension of the C-vector space L equals jL# =Lj. Proof Fix in H , and let x (x 2 L# =L) be complex numbers. Set .z/ WD P # x2L# L=L x #L;x .; z/. It is immediate from the definition of #L;x that, for y 2 L , # we have #L;x .; z C y/ D #L;x .; z/e fˇ.y; x/g. For each x0 2 L , we therefore have X X X .z C y/e fˇ.y; x0 /g D x #L;x .; z C y/e fˇ.y; x0 g y2L# =L

y2L# =L x2L# =L

D

X

X

x #L;x .; z/e fˇ.y; x x0 g

y2L# =L x2L# =L

D x0 jL# =Lj#L;x0 .; z/: For the last identity we used Proposition 1.11. Hence, if .z/ vanishes identically, then x0 D 0 unless #L;x0 .; z/ vanishes for all z. But the latter is impossible since #L;x0 .; z/, considered as a Fourier development in z, would vanish identically only if all coefficients q ˇ.r/ were identically zero. t u The main results of this section is the following theorem. Theorem 3.1 Let L D .L; ˇ/ be a totally positive definite even O-lattice of rank r. The space L is a Q -module. More precisely, for x in L# and ˛ in Q , say, ˛ p equals ac db ; N c C d , we have #L;x jr=2;L ˛ D czL˛ .˛/

X

e

˚ ˇ.ax C cy; bx C dy/ ˇ.x; y/ =2

y2L# =L

e fˇ.bx C dy; z˛ /g #L;z˛ CaxCcy ;


83

where czL˛ .˛/ D

e fbdˇ.z˛ /g lim #L;dz˛ j˛ .i t ˝ 1; 0/: t !1 jSc =Lj

For z˛ and Sc , we refer to Lemma 3.41 below. Before we give the proof of this theorem at the end of the section we deduce various consequences. Corollary 3.34 Let n D ŒK W Q . We have (i) #L;x jr=2;L Tb D e fbˇ.x/g #L;x P (b 2 O) (ii) #L;x jr=2;L S D p 1# i nr=2 y2L# =L e fˇ.y; x/g #L;y jL =Lj

(iii) #L;x jr=2;L I D .1/r #L;x . Proof The formulas in (i), (iii) are immediate consequence of Theorem 3.1. For (ii), note that the element z˛ can be taken to be zero (since here Sc D L; see Lemma 3.41). Comparing the formula of the theorem for ˛ D S and (ii) shows that it remains to prove 1 lim #L;0 jr=2;L S .i t ˝ 1; 0/ D p i nr=2 : t jL# =Lj However, this is an immediate consequence of the transformation formula [Ebe02, Prop. 5.7]. t u Corollary 3.35 We carry over the notations of Theorem 3.1. Let l denote the level Q of L, and let Q0 .l/ be the inverse image of 0 .l/ WD Ol O O \ SL.2; O/ in . (i) There exists a linear character " of 0 .l/ such that #L;x jr=2;L ˛ D ".˛/e fabˇ.x/g #L;ax for all ˛ in Q0 .l/. (ii) There exists a quadratic Dirichlet character modulo l such that, for ˛ D . ac db ; w/, one has ".˛/q D .d /, where q D 2 if the rank r of L is odd, and q D 1 otherwise. (iii) Set ˚ L D ˛ D A; w 2 Q W A 2 .l/; ".˛/ D 1 : The projection of L on .l/ is surjective. The group L acts trivially on L . Remark Note that L is normal. Indeed, L equals the group of all ˛ is the inverse image of .l/ in Q which fix L point wise.

84

3 Jacobi Forms

Proof Since c is in L we can choose z˛ D 0. Using that, for all y in L# , we have cy in L and cˇ.y/ in d1 (since c is in l), the transformation formula of the theorem L simplifies to #L;x jr=2;L ˛ D ".˛/. e fabˇ.x/g #L;ax , where ".˛/ D c0 .˛/jL# =Lj. It is clear that ".˛/ is a linear character of 0 .l/. By [Ebe02, Prop. 5.10] we know that ".˛/q D .d / for a quadratic Dirichlet character modulo l (for deducing this from Prop. 5.10 in [Ebe02] let L2 WD .L ˚ L; ˇ2 /, where ˇ2 .x; y/ D ˇ.x/ C ˇ.y/, if r is odd, and let L2 D L otherwise, and choose loc. cit. V D K ˝ .L ˚ L/ and D L ˚ L; note that [Ebe02] only treats lattices of even rank which holds true for L2 ). For proving (iii) we note that " is trivial on the inverse image Q .l/ of .l/ if r is even. Otherwise " is quadratic on Q .l/, and ".I / D 1. But then, for A in .l/ we have ".A / D 1 or ".A I / D 1. This proves the corollary. t u For the proof of Theorem 3.1 we shall need some preparations. One of the main tools is the action of the Heisenberg group H.L# / on L , which we shall now explain. Proposition 3.36 The application #; .x; y; / 7! #jr=2;L .x; y; / (where r is the rank of L) defines a right H.L# /-module structure on the space L . More precisely, we have ˚ #L;x 0 jr=2;L .x; y; / D e ˇ.x; y/=2 C ˇ.x 0 ; y/ #L;x 0 Cx :

(3.15)

The group H.L/ acts, in particular, trivially on L . Remark The space L can thus be viewed as an H.L# /=H.L/-module. Recall that H.L# /=H.L/ is a finite group of order 2ljL# =Lj2 where l is the exponent of L# =L (see Proposition 3.16) . Proof of Proposition 3.36 Using the very definition of the jr=2;L -action of the Heisenberg group (see Proposition 3.25) we find #L;x 0 jr=2;L .x; y; /.; z/ D e fˇ.x/ C ˇ.x; z/ C ˇ.x; y/=2g

D e fˇ.x; y/=2g

X

X

q ˇ.s/ e fˇ.s; z C x C y/g

s2L# sx 0 mod L

q ˇ.sCx/ e fˇ.s C x; z/g e fˇ.s; y/g

s2L# sx 0 mod L

˚ D e fˇ.x; y/=2g e ˇ.x 0 ; y/ #L;xCx 0 .; z/: The last identity follows on noting that, for s x 0 mod L, we have that e fˇ.s; y/g equals e fˇ.x 0 ; y/g, and by substituting s x for s in the sum. This proves the formula for the action of H.L# /.


85

˚ Recall that H.L/ is generated by the elements x; y; e 12 ˇ.x; y/ (x; y 2 L). But for these elements h and by the just proved formulas we obviously have #L;x 0 jr=2;L h D #L;x 0 . This proves the second statement. t u Proposition 3.37 The character H.L# / of the H.L# /-module L satisfies (

H.L# / .x; y; / D

jL# =Lj e fˇ.x; y/=2g

if x; y 2 L

0

otherwise:

In particular, L is an irreducible H.L# /-right module. Remark Note that the formula for H.L# / implies that, for any ˛ in Q and h in H.L# /, we have H.L# / .˛ 1 h˛/ D H.L# / .h/. Proof of Proposition 3.37 By the formula (3.15) for the action of H.L# / we have ( P e fˇ.x; y/=2g x 0 2L# =L e fˇ.x 0 ; y/g tr .x; y; /; L D 0

if x 2 L otherwise:

But the sum in the above identity is zero unless y 2 L (see Proposition 1.11) which proves the first statement. For the second statement it suffices to prove that L , viewed as module over the finite group H.L# /=H.L/ is irreducible. Indeed, we have X

1 jH.L# /=H.L/j

jtr h; L j2 D

h2H.L# /=H.L/

X jL# =Lj2 1 D 1; # jH.L /=H.L/j 2 2l

t u

which implies the irreducibility [FH91, Cor. 2.15].

Lemma 3.38 Let U denote the subgroup 0 L# 1 of H.L# /. For any ˛ in Q , the 1 space L˛ U˛ of functions in L which are invariant under the subgroup ˛ 1 U˛ of H.L# / is one dimensional. Proof As already in the proof of the preceding proposition we view L as module over the finite group G WD H.L# /=H.L/. Let be the canonical projection .V / from H.L# / onto G. We then have LV D L , where V D ˛ 1 .U /˛. But then, by standard representation theory (see Corollary 2.20), we have .V /

dim L

D

X X 1 1 tr v; L D tr u; L : j.U /j j.U /j v2.V /

u2.U /

The second identity is an immediate consequence of the invariance under conjugation with ˛ of the character of H.L# / as explained in the remark after

86

3 Jacobi Forms

Proposition 3.37. Butby thesame proposition tr u; L D j.U /j if u is the neutral element of G, and tr u; L D 1 otherwise. The lemma is now obvious. t u Recall from the previous section that E denotes the set of the C-linear extensions to C D C ˝Q K of the (Q-linear) embeddings of K into the field of real numbers. Recall also that, for each in E we have associated the Heat operator H [see (3.10)]. Lemma 3.39 For any 2 E , and for any # in L , we have H # D 0: Proof Since every # in L has a Fourier development in terms of the functions e fˇ.s/ C ˇ.s; z/g (s 2 L# ), and these functions are annihilated by H (see Lemma 3.30) the lemma is obvious. t u Proposition 3.40 Let be a holomorphic function on H LC . Then following statements are equivalent: (i) 2 L (ii) There is a half integral k such that jk;L h D for all h 2 H.L/, and H D 0 for all 2 E . Proof Recall that L has basis #L;x (x 2 L# =L). .i/ H) .ii/. The invariance property follows from Proposition 3.36, which states that H.L/ acts trivially on L with k D r=2. The second property is the preceding lemma. .ii/ H) .i/. Since is fixed under the action of H.L/, we have, in particular, .; z/ D jk;L .0; y; 1/.; z/ D .; z C y/ for any y 2 L. Hence, we can write .; z/ D

X

s ./q ˇ.s/ e fˇ.s; z/g

s2L#

for suitable functions s ./ on H . By the same assumption again, for any x 2 L, we have .; z/ D jk;L .x; 0; 1/.; z/ D .; z C x/e fˇ.x/ C ˇ.x; z/g. But this implies .; z/ D

X

s ./q ˇ.s/ e fˇ.s; z C x/g e fˇ.x/ C ˇ.x; z/g

s2L#

D

X

s ./q ˇ.sCx/ e fˇ.s C x; z/g :

s2L#

Since this implies that the functions s depend only on s modulo L, we then have .; z/ D

X s2L# =L

s ./#L;s .; z/:


87

The functions s are holomorphic functions on H . Indeed, .; z/ is holomorphic and we have, for any s 2 L# =L, P s ./ D

y2L# =L

.; z C y/e fˇ.y; s/g

jL# =Lj #L;s .; z/

(see the proof of Proposition 3.33). Now by the second assumption and Lemma 3.39, we obtain X @ 0 D H .; z/ D s ./ #L;s .; z/: @ # s2L =L

Since the #L;s .; z/, for fixed as functions of z, are linearly independent (Proposition 3.33), we deduce that the s are constants, and hence that lies in L . t u The characterization of L as given in the preceding proposition enables us to prove now that L is invariant under Q . Proof of Theorem 3.1 We show that L is invariant under the jr=2;L action of Q . Let # in L and ˛ 2 Q , and set WD #jr=2;L ˛. We have to show that is an element of L . By Proposition 3.40, it suffices to show that is invariant under the action of H.L/, and that, for any in E , we have H D 0. Let h 2 H.L/. The first claim holds true, since we have (writing j for jr=2;L ) jh D #j˛ jh D #j˛h˛ 1 j˛ D #j˛ D : The third identity follows from the fact that Q leaves H.L/ invariant under conjugation, and the fact that H.L/ acts trivially on L (see Proposition 3.36). The second claim also holds true, since we have 2 H ./ D H #j˛ D .A; / .H #/j˛ D 0: Here the second identity follows from Proposition 3.31, and the last one follows from Lemma 3.39. This proves the first part of the theorem. For deducing the explicit formulas for the action of Q we need some further preparations. t u a b # Lemma 3.41 For every c d ;w ˛ D , there exists z˛ 2 L satisfying the congruence tr cdˇ.y/ tr dˇ.y; z˛ / mod Z for all y in Sc WD fy 2 L# W cy 2 Lg. Proof The map ' W Sc =L ! 12 Z=Z, y C L 7! tr cdˇ.y/ C Z is a group homomorphism. Indeed, for y, y 0 2 Sc , we have '.y C y 0 C L/ D tr cdˇ.y/ C cdˇ.y 0 / C cdˇ.y; y 0 / C Z;

88

3 Jacobi Forms

and cy 2 L implies then cˇ.y; y 0 / 2 d1 . We can continue ' to a group homomorphism 'Q W L# =L ! Q=Z [Ser73, Ch. VI,§ 1, Prop. 1]. Since the tr DL is non-degenerate (see Proposition 3.2) the map L# =L ! Hom.L# =L; Q=Z/, y C L 7! ˇ.y; _/ C Z is injective. Since a finite abelian group and its dual have the same order (see [Ser73, Ch. VI,§ 1, Prop. 2]), this map is an isomorphism. Hence there exists a z in L# such that '.y/ Q D ˇ.y; z/ C Z. Set z˛ D az. Then, for y in Sc , we have tr ˇ.y; z/ tr dˇ.y; z˛ / mod Z since ad 1 mod c. t u Proof of Theorem 3.1 (cont.) It remains to calculate the matrix coefficients of the Q -action on L . We prove first of all, that #L;0 j˛ D czL˛ .˛/

X

#L;z˛ j˛ 1 .0; y; 1/˛;

(3.16)

y2L# =L L

where cz˛ .˛/ is a constant, and z˛ is as in Lemma 3.41. Here and in the following we write j for jr=2;L . For the proof denote the sum on the right hand side by S . Note that each term depends indeed only on the coset of y in L# =L as follows easily from the invariance of #L;z˛ under H.L/. The claimed identity follows from the fact that both sides are invariant under the subgroup ˛ 1 0 L# 1 ˛ of H.L# /, and that the space of functions in L invariant under this subgroup is one dimensional (cf. Lemma 3.38). The invariance of the left hand side follows from the fact that #L;0 is invariant under 0 L# 1 (as follows from Proposition 3.36). The invariance of the right hand side follows from Proposition 2.15. For concluding the proof of the formula we still have to show that S is different from zero. Writing ˛ 1 .0; y; 1/˛ D .cy; dy; 1/, we obtain X X #L;z˛ j.cy; dy; 1/ D e fˇ.cy; dy/=2g e fˇ.dy; z˛ /g #L;z˛ Ccy SD y2L# =L

D

X

x2L# =L

#L;x

X

y2L# =L

e fcdˇ.y/ C dˇ.y; z˛ /g :

y2L# =L z˛ Ccyx mod L

From this we see that S 6D 0 since the #L;x are linearly independent, and since the inner sum is different from zero for x D z˛ . Indeed, in this case the inner sum runs over a complete set of representatives for Sc =L, and then, by the very definition of z˛ , the terms are all equal to 1 (since tr cdˇ.y/ C dˇ.y; z˛ / tr 2cdˇ.y/ 0 mod Z). Next, note that, for any x 2 L# , one has #L;x D #L;0 j.x; 0; 1/. Using this identity and (3.16), we obtain


89

#L;x j˛ D #L;0 j˛j ˛ 1 .x; 0; 1/˛ X D czL˛ .˛/ #L;z˛ j ˛ 1 .0; y; 1/˛ j ˛ 1 .x; 0; 1/˛ y2L# =L

X

D czL˛ .˛/

#L;z˛ j ax C cy; bx C dy; e fˇ.x; y/=2g :

y2L# =L

Applying again the formulas (3.15) for the H.L# /-action on L we obtain thus X

#L;x j˛ D czL˛ .˛/

e fˇ.bx C dy; z˛ /g

y2L# =L

e

˚

ˇ.ax C cy; bx C dy/ ˇ.x; y/ =2 #L;z˛ CaxCcy ;

(3.17)

which is the formula stated in the theorem. Hence, it remains to calculate the L constant cz˛ .˛/. L For obtaining a formula for cz˛ .˛/ we set x D dz˛ in (3.17), evaluate the resulting identity at z D 0 and D i t ˝ 1 with real t, and let t tend to infinity. For calculating the limit of the right hand side of (3.17) we note that #L;z˛ adz˛ Ccy D #L;c.bz˛ Cy/ , and that lim #L;c.bz˛ Cy/ .i t ˝ 1; 0/

t 7!1

X

D lim

t 7!1

e 2 t tr

sc.bz˛ Cy/ mod L

ˇ.s/

( D

1

if c.bz˛ C y/ 2 L

0

otherwise:

But c.bz˛ C y/ 2 L if and only if y 2 bz˛ C Sc . Note that .ax C cy; bx C dy/ D .z˛ C ct; dt/ for .x; y/ D .dz˛ ; bz˛ C t/. Thus, the limit of the right hand side of (3.17) (specialized to .; z/ D .it ˝ 1; 0/ and x D dz˛ ) becomes czL˛ .˛/

X

e fˇ.dt; z˛ /g e

˚ ˇ.z˛ C ct; dt/ ˇ.dz˛ ; bz˛ C t/ =2

t 2L# =L;ct 2L

D czL˛ .˛/e fbdˇ.z˛ /g jSc =Lj: Summarizing we have found czL˛ .˛/ D

e fbdˇ.z˛ /g lim #L;dz˛ j˛ .i t ˝ 1; 0/; t !1 jSc =Lj

which completes the proof of the theorem.

t u

90

3 Jacobi Forms

We conclude this section by some propositions which we shall need in the last section of this chapter when we shall discuss the relation of Jacobi forms and vectorvalued Hilbert modular forms. Proposition 3.42 The application

7 #L;x jr=2;L .˛; h/ WD #L;x jr=2;L ˛ jr=2;L h #L;x ; .˛; h/ !

defines a right JQ .L# /-module structure on L . Proof This can be verified by a straightforward calculation similar to the one in the proof of Proposition 3.27 on using the Proposition 3.36 and the Theorem 3.1. t u Definition 3.43 By h; i we denote the Hermitian scalar product on L which is anti-linear in the second argument, and which satisfies: ( h#L;x ; #L;y i D

1

if x D y

0

otherwise:

(3.18)

Proposition 3.44 The JQ .L# /-action on L is unitary with respect to the scalar product in (3.18). Proof For proving the invariance of the scalar product under the action of Q it suffice to prove the invariance under the generators Tb , I and S of Q . For the generators Tb and I the invariance is obvious. For P proving the invariance underPS , let # and # 0 be elements of L , say, # D x2L# =L c.x/#L;x and # D x 0 2L# =L c.x 0 /#L;x 0 . Using the formula for the S -action from Corollary 3.34, we have X X i nr=2 c.x/ e fˇ.y; x/g #L;y ; #jr=2;L S D p jL# =Lj x2L# =L # y2L =L and similarly for # 0 . Using these formulas we can write ˝ ˛ #jr=2;L S ; # 0 jr=2;L S D

1 jL# =Lj

X x;x 0 2L# =L

c.x/c.x 0 /

X

˚ e ˇ.y; x 0 x/ :

y2L# =L

By Proposition 1.11, the inner sum equals zero unless x 0 D x, when it P # equals jL =Lj. The right hand side becomes thus x;2L# =L c.x/c.x/, which equals indeed h#; # 0 i. The invariance under H.L# / can be easily deduced using the formulas for the action on L from Proposition 3.36. t u

3.6 Definition and Basic Properties of Jacobi Forms

91

3.6 Definition and Basic Properties of Jacobi Forms In the present section we give finally the definition of Jacobi forms over totally real number fields, and we shall discuss their Fourier developments and theta expansions. Definition 3.45 Let L D .L; ˇ/ be a totally positive definite even O-lattice and let k 2 12 Z. Moreover, let be a subgroup of finite index in Q , and let be a linear character of whose kernel is of finite index in Q . A Jacobi form over K of weight k, index L and character on is a holomorphic function W H LC ! C satisfying (i) jk;L ˛ .; z/ D .˛/.; z/ (˛ 2 ) (ii) jk;L h .; z/ D .; z/ (h 2 H.L/) . If K D Q, we assume furthermore that the function is holomorphic at all cusps (see [EZ85]). K .; /. The C-vector space of all Jacobi forms over K is denoted by Jk;L (For the notion of O-lattices we refer to Sect. 3.1, and for the space LC we refer to Sect. 3.2. Moreover, for the actions of Q and H.L/ on the space Hol.H LC / we refer the reader to Propositions 3.28 and 3.25, respectively.) K K If D Q , we simply write Jk;L . / for Jk;L .; /, and call this space the space of Jacobi forms over K of weight k, index L and character . In the following we K shall mainly concentrate on the spaces Jk;L . /. If the number field in question is clear from the context, we refer to the Jacobi forms over K simply as Jacobi forms, K and we write Jk;L .; / instead of Jk;L .; /. Remark Applying the transformation law (i) to ˛ D .1; 1/, we obtain, for 2k in Jk;L . /, that .˛/ D 2kjk;L ˛ D .1/ . Hence Jk;L . / is trivial unless we have .1; 1// D .1/ . If k is integral and .1; 1// D .1/2k .D C1/, then factors through a linear character of . In this case we can rewrite the transformation law (i) as jk;L A D .A/ (A 2 ), and we shall also write Jk;L . / for Jk;L . /. If k is not integral and .1; 1// D .1/2k .D 1/, then does not factor through a linear character of (see Proposition 2.4). Proposition 3.46 Every Jacobi form in Jk;L . / possesses a Fourier development of the form .; z/ D

X

c.t; s/ q t e fˇ.s; z/g :

(3.19)

s2L# t 2hCd1

Here h is an element of K such that .Tb / D e fhbg for all b 2 O. Proof Set .; z/ D e fh g .; z/. From the transformation laws in Definition 3.45, we have that .; z/ is periodic in and z with respect to O and L,

92

3 Jacobi Forms

respectively. Since .; z/ is holomorphic, it can be written as infinite sum of the functions e ft C ˇ.z; s/g, where t and s run through d1 and L# , respectively. u t Theorem 3.2 (Köcher Principle for Jacobi Forms) Assume K ¤ Q. In the Fourier expansion (3.19) one has c.t; s/ D 0 unless t ˇ.s/ 0 or t D ˇ.s/. The proof of this theorem will be given in the next section, since it requires some extra tools which we have to develop first. Remark For K D Q, the statement c.t; s/ D 0 unless t ˇ.s/ 0 or t D ˇ.s/ is a part of the definition. If is a Jacobi form of weight k and index L on a subgroup of finite index in Q , then possesses also a Fourier development of the form (3.19), where, however, the range of t will in general be different. The Köcher principle holds then also in this case. The proof for these statements is the same as in the case of Jacobi forms on the full Hilbert modular group, and is left to the reader. Furthermore, it Q where G D SL.2; K/, the function jk;L is again is easy to see that, for in G, a Jacobi form of the same weight k and index L but on 1 \ (which has also finite index in Q ). In particular, jk;L has a Fourier expansion for which the Köcher principle holds.

C

Definition 3.47 Let be in Jk;L .; P /. If, for each in SL.2; K/, the function jk;L has a Fourier development c.t; s/q s e fˇ.s; z/g such that c.t; s/ D 0 unless t ˇ.s/ 0; then is called a Jacobi cusp form.P If has a Fourier development c.t; s/q s e fˇ.s; z/g satisfying c.t; s/ D 0 unless t D ˇ.s/; then is called a singular Jacobi form. Remark The vanishing condition for jk;L in the definition depends only on the Q D, Q where is the projection double coset of the first component of in nG= onto and G D SL.2; K/ as before, and D is the subgroup of triangular matrices in SL.2; K/. To count these double cosets we identify G=D with the projective line P1 .K/ via the application ac db 7! Œa W c , so that the double coset space becomes the orbit space nP1 .K/ with respect to the natural action of G on the projective line. Note that there are only finitely many orbits since has finite index in . Indeed, nP1 .K/ is in one to one correspondence with the ideal classes of K via the application which maps Œa W b to the ideal class of Oa C Ob (see e.g. [vdG88, Chap. I, Prop. 1.1]). In particular, if K has class number one and is a Jacobi form on the full modular group , then is a cusp form if and only if its “singular part”

3.6 Definition and Basic Properties of Jacobi Forms

sing WD

X

93

c.ˇ.s/; s/ q s e fˇ.s; z/g

s2L# ˇ.s/2d1

vanishes identically. Example 3.48 Let L be a totally positive definite even O-lattice of rank r. For all x 2 L# =L, the Jacobi theta functions #L;x associated to L as defined in (3.14) are singular Jacobi forms on the subgroup L of Q (see Corollary 3.35) of weight r=2. The invariance under the Q -action follows from Corollary 3.35. The fact that they are singular and of weight r=2 is immediate from their very definition. Theorem 3.3 Let L be a totally positive definite even O-lattice and in Jk;L . /. Then can be written in the form .; z/ D

X

hx ./#L;x .; z/;

(3.20)

x2L# =L

where hx ./ D

X

c ˇ.x/ d; x q d

d 2ˇ.x/hCd1

[with c.t; s/ as in (3.19)]. The function hx depends only on x modulo L. In the following we call the expansion (3.20) the theta expansion of . Remark Note that the theorem implies that the Fourier coefficient c.t; s/ of a Jacobi form depends only on ˇ.s/ t and the coset s C L. Proof of Theorem 3.3 Writing d D ˇ.s/ t and setting C.d; s/ WD c ˇ.s/ d; s , we can write the Fourier development (3.19) in the form .; z/ D

X

C.d; s/ q ˇ.s/d e fˇ.s; z/g

s2L# d 2ˇ.s/hCd1

D

X

X

x2L# =L

s2L# sx mod L

q ˇ.s/ e fˇ.s; z/g

X

C.d; s/ q d :

(3.21)

d 2ˇ.s/hCd1

Using the second transformation law in Definition 3.45 for elements .x; 0; 1/ (x 2 L), we obtain e fˇ.x/ C ˇ.x; z/g .; z C x/ D .; z/:

94

3 Jacobi Forms

Inserting the Fourier development of into the left hand side, we obtain X

e fˇ.x/ C ˇ.x; z/g

D

C.d; s/ q ˇ.s/d e fˇ.s; z C x/g

s2L# d 2ˇ.s/hCd1

X

C.d; s/ q ˇ.sCx/d e fˇ.s C x; z/g :

s2L# d 2ˇ.s/hCd1

Replacing s by s x and comparing the Fourier coefficients we obtain C.d; s/ D C.d; s x/

.x 2 L/:

In other words, C.d; s/ depends only on s mod L. Thus the inner sum in (3.21) depends only on s mod L and equals hence hx . But then (3.21) reads .; z/ D

X

hx ./

x2L# =L

X

q ˇ.s/ e fˇ.s; z/g :

s2L# sx mod L

t u

This proves the theorem.

3.7 Jacobi Forms as Vector-Valued Hilbert Modular Forms In the present section our main aim will be to set up an isomorphism between spaces of Jacobi forms and spaces of vector-valued Hilbert modular forms. In particular, this will imply the Köcher principle for Jacobi forms and that the spaces of Jacobi forms are finite dimensional. For explicit formulas for the dimensions of the spaces of Jacobi forms, the reader is referred to [SS14]. In the sequel we shall make use of various facts and notions concerning representations of groups which were recalled in Sect. 2.1. Recall from Theorem 3.1 that the space L spanned by the functions #L;x is invariant under Q with respect to the jr=2;L -action. Thus, for any ˛ in Q , there are numbers !x;y .˛/ such that #L;x jr=2;L ˛ D

X

!.˛/x;y #L;y

.x 2 L# =L/:

(3.22)

y2L# =L

Note that the coefficients !x;y .˛/ are unique since the #L;y are linearly independent. Theorem 3.4 Let L D .L; ˇ/ be a totally positive definite even O-lattice of rank r with level l.

3.7 Vector-Valued Hilbert Modular Forms

95

(i) The map ! W Q ! GL.CŒL# =L /;

X

!.˛/.ex / WD

!.˛/y;x ey

y2L# =L

[where !y;x .˛/ denote the coefficients in (3.22)] defines a representation of Q . (ii) The representation ! is unitary with respect to the scalar product (2.12). It factors through a representation of the finite group Q =L , where L is the normal subgroup of Q defined in Corollary 3.35. (iii) One has !.Tb /ex D e fbˇ.x/g ex

.b 2 O/; X e fˇ.y; x/g ey ;

1 !.S /ex D .DL / p jL# =Lj y2L# =L !.I /ex D .1/r ex :

Proof First of all, we show that for all ˛; ˛ 0 2 Q , one has !.˛˛ 0 / D !.˛/!.˛ 0 /: To prove this identity, it is in fact enough to show !.˛˛ 0 /y;x D

X

!.˛/y;y 0 !.˛ 0 /y 0 ;x :

y 0 2L# =L

But since jr=2;L defines an action on L (see Theorem 3.1), we easily recognize the above identity. This proves (i). The fact that ! is unitary follows immediately from Proposition 3.44. This proves the first statement of (ii). The second part of (ii) is immediate by the very definition of L . Since we have that .DL / equals i nr=2 (see Milgram’s formula [MH73, p. 127]), part (iii) is immediate by Corollary 3.34. t u Definition 3.49 Let L D .L; ˇ/ be a totally positive definite even O-lattice. Let W Q ! GL.V / be a finite dimensional representation of Q whose kernel has finite index in Q . Let k 2 12 Z. A holomorphic function F W H ! V satisfying F jk ˛ D .˛/F

.˛ 2 Q /

is called a vector-valued Hilbert modular form.Here .˛/F denotes that function on H which at in H takes on the value .˛/ F ./ . If K D Q we require F ./ in addition to be bounded on each subset of H of the form =./ r > 0. The C-vector space of all such functions is denoted by Mk ./.

96

3 Jacobi Forms

Let U WD fb 2 O W .Tb / D 1g and UQ be the dual of U with respect to trace. Then, for any F 2 Mk ./, we have F . C b/ D F jk Tb D F , and hence we have a Fourier expansion X cF .t/ q t (3.23) F ./ D t 2UQ

for suitable cF .t/ 2 V . Note that, for K D Q, we have cF .t/ D 0 unless t 0, as follows from the boundedness condition. Lemma 3.50 (Köcher Principle for Vector-Valued Hilbert Modular Forms) Suppose K ¤ Q and F 2 Mk ./. The coefficients cF .t/ in (3.23) are equal to zero unless t 0 or t D 0. Proof If ej (1 j d ) is a basis for the space V , we can write F ./ D

d X

Fj ./ ej :

j D1

Here the Fj arePholomorphic functions for all j . If ˛ lies in the kernel of , then F D F jk ˛ D j Fj jk ˛ ej , i.e. for all j , we have Fj D Fj jk ˛. In other words, Fj is a Hilbert modular form of weight k on the kernel of . By [Fre90, Prop. 4.9] the Fj satisfy the Köcher principle (loc. cit. even weight automorphic forms are considered, but it is easy to modify the proof in loc. cit so that it also covers our case). Therefore, we have X Fj ./ D cFj .t/ q t ; t 2UQ t 0 or t D 0

and hence

0

X

F ./ D

@

X

t 2UQ t 0 or t D 0

Since cF .t/ D

P j

1 cFj .t/ej A q t :

j

cFj .t/ej , the lemma follows.

t u

Before we prove the main result of this section, we need a lemma. Lemma 3.51 Let V be a finite dimensional G-module, and let be the representation afforded by this G-module. Let vi (i D 1; : : : ; n) denote a basis for V and define a map W G ! GL.V / by .˛/vi D

n X j D1

Then is a representation of G.

.˛/ji vj :


97

Proof The lemma follows by a straightforward calculation.

t u

Theorem 3.5 Let L D .L; ˇ/ be a totally positive definite even O-lattice of rank r, and let ! be the representation (3.22). The application D

X

X

hx #L;x 7! “ 7! F ./ WD

x2L# =L

hx ./ex00

x2L# =L

defines an isomorphism W Jk;L . / ! Mk 2r . ! /. Here ! denotes the representation associated to ! with respect to the basis ex (x 2 L# =L) as in Lemma 3.51. Proof Let 2 Jk;L . /. We need to show, first of all, that ./ 2 Mk r2 . ! /. If ˛ is in Q , then we have X

.˛/ D jk;L ˛ D D

hx jkr=2 ˛ #L;x jr=2;L ˛

x2L# =L

X

#L;y

y2L# =L

X

hx jkr=2 ˛ !.˛/x;y :

x2L# =L

Since, for fixed , the functions z 7! #L;x .; z/ are linearly independent (see Proposition 3.33), we have hy D .˛ 1 /

X

hx jkr=2 ˛ !.˛/x;y

x2L# =L

for y 2 L# =L. Applying ˛ 1 to both sides and then writing ˛ for ˛ 1 in the resulting identities, we obtain X hy j˛ D .˛/ hx jkr=2 !.˛ 1 /x;y : x2L# =L

Using these identities we find F jkr=2 ˛ D

X

hx jkr=2 ˛ ex D .˛/

x2L# =L

X y2L# =L

hy

X

!.˛ 1 /y;x ex :

x2L# =L

Since ! is unitary (see Theorem 3.4) and the ex form an orthonormal basis, we have !.˛ 1 /y;x D !.˛/x;y for all x and y in L# =L. Hence, we have F jk 1 ˛ D .˛/

X

2

y2L# =L

which was to be proven.

hy ! .˛/ ey D ! .˛/F;

98

3 Jacobi Forms

The injectivity of follows from the fact that ex (x 2 L# =L) form a basis for the space CŒL# =L . Next we prove the surjectivity of . Suppose F 2 Mk r2 . ! /. We need to find some 2 Jk;L . / such that F DP./. For each 2 H , we have F ./ 2 CŒL# =L . So, we can write F ./ D x2L# =L cx ./ex . Since F is holomorphic, the functions cx are holomorphic functions on H for all x 2 L# =L. We set P WD x2L# =L cx #L;x . We obviously have F D ./. It remains to show that is an element of the space Jk;L . /. The invariance under H.L/ is obvious from Proposition 3.36, since H.L/ acts trivially on L . The invariance under Q follows on reversing the arguments of the first part of the proof. We leave the details to the reader. t u For a subgroup of finite index in Q , we use Mk .; / for the space of Hilbert modular forms of weight k 2 12 Z and character on . If is trivial, we shortly write Mk ./ for Mk .; 1/. Corollary 3.52 We continuewith the notations of Theorem 3.5. For any x 2 L# =L, the function hx lies in Mk r2 Ker. ! / . P r Proof From Theorem 3.5, we have F ./ D x2L# =L hx ./ ex 2 Mk 2 . ! /. Hence, for any ˛ 2 Q , the following holds true

X hx jkr=2 ex :

! .˛/F D F jkr=2 ˛ D x2L# =L

But this obviously implies that for any ˛ 2 Ker. ! /, we have hx jk r2 ˛ D hx which proves the corollary. u t Corollary 3.53 The space of Jacobi forms is finite dimensional. Proof By Theorem 3.5, we have Jk;L . / ' Mk r2 . ! /. By Corollary 3.52 the application F D

X

hx ex 7! .hx /x2L# =L

x2L# =L

L defines an embedding Mk r2 . ! / ! x2L# =L Mk r2 Ker. ! / . The corollary is now immediate from the subsequent Lemma 3.54. t u Lemma 3.54 For a subgroup of Q of finite index in Q the dimension of the space of Hilbert modular forms Mk ./ is finite. Proof By [Fre90, Thm. 6.1] the space of Hilbert modular forms of even weight is finite. If k is not even, then let # be a Hilbert modular form on some congruence subgroup, say 1 , of Q of weight 1=2, and consider the embedding


99

Mk ./ ! MkC3=2 . \ 1 /;

f 7! f # 3

if k 2 1=2 C 2Z;

Mk ./ ! MkC1=2 . \ 1 /;

f 7! f #

if k 2 3=2 C 2Z;

Mk ./ ! MkC1 . \ 1 /;

f 7! f # 2

if k is odd;

which in each case implies again that Mk ./ is finite dimensional. As function # one can take (see Example 3.48) #L;0 .; 0/ for any even L of rank one and, which defines a Hilbert modular form on L (see Corollary 3.35 for L ). t u P Proof of Theorem 3.2 We write D x2L# =L hx #L;x , where, for each x, we have P hx D d 2ˇ.x/hCd1 C.d; x/q d , where h is an element of K such that .Tb / D e fhbg for b 2 O, and where C.d; x/ D c.ˇ.x/ d; x/ (see Theorem 3.3). From Corollary 3.52, the functions hx are vector-valued Hilbert modular forms. Hence, by Lemma 3.50 we have that C.d; x/ D 0 unless d 0 or d D 0, i.e. that c.t; x/ D 0 unless t ˇ.x/ 0 or t D ˇ.x/. This proves the claimed statement. t u

Appendix: Jacobi Forms of Odd Index In this appendix we discuss briefly the notion of Jacobi forms whose index is a not necessarily even O-lattice. Moreover, we shall prove a proposition which links Jacobi forms over number fields with Hilbert modular forms, and which justifies the informal description of Jacobi forms which is given in the introduction. Let L D .L; ˇ/ denote a (totally positive definite) O-lattice. Assume that L is odd, i.e. that ˇ takes on values in d1 as for all lattices considered in this treatise, but that there exist elements x in L such that ˇ.x/ D 12 ˇ.x; x/ is not in d1 . Note that for such x there exist a in O such that e faˇ.x/g D 1. Let k be a half integer and let be a linear character of Q . If L D .L; ˇ/ is odd, then there is no nonzero function which satisfies jk;L ˛ D .˛/

.˛ 2 Q /

(3.24)

jk;L h D

.h 2 H.L//:

(3.25)

Q Indeed, in this case H.L/ is not normalized by . Namely, for h 2 L, we have ˛ 1 h˛ 2 H.L/ 0; fA .x; y/ , where fA .x;y/ D e fabˇ.x/ C cdˇ.y/g and .x; y/ is the first component of h and A D ac db is the first component of ˛. Hence, if satisfies (3.24) and (3.25), then applying ˛ 1 , h and ˛ successively to yields 1 j.˛ h˛/ D (where we used j for jk;L ). On the other hand, if we write ˛ 1 h˛ D 0 h 0; fA .x; y/ , then h0 is in H.L/ and hence j.˛ 1 h˛/ D fA .x; y/, so that, since is different from zero, we have fA .x; y/ D 1. But since L is odd we can find A and x and y such that fA .x; y/ D 1, a contradiction.

100

3 Jacobi Forms

Instead we can ask for functions satisfying (3.24) and jh D .x; y/.h 2 H.L// for a linear character of H.L/ ' L L. If such a character and nonzero exist then, by a similar reasoning as before, we conclude that .x; y/ D .x; y/A e fabˇ.x/ C cdˇ.y/g

.x; y 2 L; A 2 /:

(3.26)

It is not hard to show that the character is uniquely determined by these identities, namely, one finds .x; y/ D e fˇ.x/ C ˇ.y/g (see [BS14b]). This function defines a character of H.L/, but it does not necessarily satisfy (3.26). It is not obvious when .x; y/ D e fˇ.x/ C ˇ.y/g satisfies (3.26); it does it for instance, if a2 C ab C b 2 1 mod 2 for all relatively prime elements a and b in O (which depends on the splitting and ramification of 2 in K). We call a lattice L weakly-odd if .x; y/ D e fˇ.x/ C ˇ.y/g satisfies the identity (3.26). An even lattice is, of course, weaklyodd. Definition 3.55 For a totally positive definite, not necessarily even O-lattice we let Jk;L . / denote the space of holomorphic functions on H LC which satisfy (3.24) and jk;L h D e fˇ.x/ C ˇ.y/g

for all h D .x; y; e fˇ.x; y/=2g/ 2 H.L/: (3.27)

Remark Note that, for even O-lattices, this definition coincides with the one given in Sect. 3.6. The discussion above shows that Jk;L . / D 0 unless L is weakly odd. Examples of Jacobi forms with weakly-odd index can be found in the next chapter. We conclude this appendix by a proposition in which a Jacobi form .; z/ specialized in the z-variable to division points of L C L yields Hilbert modular forms. More precisely, we have: Proposition 3.56 Let be in Jk;L . /, where L denotes a not necessarily even Olattice. We set .; x; y/ WD .; x C y/ e fˇ.x/g. Then we have: (i) The function .; x; y/ is quasi-periodic in the variables x and y in R with respect to the O-module L. More precisely, for any , in L, we have .; x C ; y C / D e fˇ.; y/ C ˇ. C /g .; x; y/. (ii) For fixed x and y in K, the map 7! .; x; y/ defines a Hilbert modular form of weight k and character ıx;y on the inverse image of .a2 / in Q , where a denotes the ideal of all a in O such that ax and ay are in L, and where ıx;y is trivial if L is even, and trivial or quadratic otherwise. Remark Note that, for L D .c; !/, a is the least common multiple of the denominators of xc1 and yc1 . Proof of Proposition 3.56 We use L D .L; ˇ/ for the O-lattice .c; !/. It is easy to see using Proposition 3.25 that .; x; y/ D jk;L .x; y; e fˇ.x; y/=2g/ .; 0/:


101

We prove (i). Using the multiplication law in the Heisenberg group [see (3.4)] and the invariance of under H.L/, we find (writing j for jk;L ) .; x C ; y C / D j.x C ; y C ; e fˇ.x C ; y C /=2g/ .; 0/ D e fˇ.; y/g j.; ; ˇ.; /=2/.x; y; e fˇ.x; y/=2g/ .; 0/ D e fˇ.; y/ C ˇ. C /g j.x; y; e fˇ.x; y/=2g/ .; 0/ D e fˇ.; y/ C ˇ. C /g .; x; y/: Next we prove (ii). Let ˛ be in Q , and let A denote the first component of ˛. Using the multiplication in the Jacobi group and that j˛ D .˛/, we have .; x; y/jk ˛ D j.x; y; e fˇ.x; y/=2g/˛ .; 0/ D j˛ .x; y/A; e fˇ.x; y/=2g .; 0/ ˚ D e Œˇ .x; y/A ˇ.x; y/ =2 .˛/ ; .x; y/A : Now, suppose that x and y are in K. Assume, first of all, that L is even. Then by part (i) we see that .; x; y/ is periodic with respect to aL L. Assume that A is in .a2 /. We then have .x; y/A .x; y/ mod aL L. Moreover, ˇ .x; y/A =2 ˇ.x; y/=2 D abˇ.x/ C bcˇ.x; y/ C cdˇ.y/. But each of the three termsPon the right is in d1 . Indeed, to prove, e.g., that bˇ.x/ is in d1 , we write P b D bj aj2 with numbers bj in O and aj in a (Lemma 1.14), so that bˇ.x/ D bj ˇ.aj x/, which is in d1 since aj x is in L and L is even. It follows that .; x; y/j˛ D

.˛/ .; x; y/ as claimed. If L is odd, then 2 is a Jacobi form of even index L.2/ D .L; 2ˇ/. By what we have already proved we conclude that .; x; y/2 is then in M2k Q .a2 /; 2 , where Q .a2 / denotes the inverse image of .a2 / in Q . It follows that .; x; y/ is 2 0 in Mk Q .a /; , where 0 is a character of .a2 / whose square equals 2 . t u

Chapter 4

Singular Jacobi Forms

As in the previous chapter, K will denote a totally real number field. Similarly, O, d will denote the ring of integers and different of K, respectively. Moreover, we shall use D SL.2; O/ and Q for the metaplectic cover of . In the present chapter we shall study singular Jacobi forms over number fields. The main result of this chapter will be the explicit description of all singular Jacobi forms whose indices are totally positive definite rank one O-lattices (see Theorems 4.2 and 4.3). In Sect. 4.1, we shall observe that singular Jacobi forms are in one to correspondence with the one-dimensional Q -submodules of the spaces of Jacobi theta functions. In Sect. 4.2, we shall present that the spaces of Jacobi theta functions are isomorphic to the Weil representations associated to certain discriminant modules. Using the results of Sects. 2.4 and 2.5, we shall finally be able to describe explicitly all singular Jacobi forms whose indices are totally positive definite rank one O-lattices. This will be carried out in Sect. 4.4.

4.1 Characterization of Singular Jacobi Forms In this section, we shall characterize the singular Jacobi forms as the one dimensional Q -submodules of Weil representations. Let L D .L; ˇ/ be a totally positive definite even O-lattice of rank r and in Jk;L . /. (We refer Definition 3.45 for the space Jk;L . /.) Recall from Definition 3.47 that is a singular Jacobi form if and only if c.t; s/ D 0 unless t D ˇ.s/. Here the c.t; s/ are the Fourier coefficients of as given in Theorem 3.2. For a linear character of Q , we define Q ;

L

WD f# 2 L W #jr=2;L ˛ D .˛/# for all ˛ 2 Q g:


103

104

4 Singular Jacobi Forms Q ;

Clearly, the space L

is a Q -submodule of L .

Proposition 4.1 Let L D .L; ˇ/ be a totally positive definite even O-lattice of rank r and 2 Jk;L . /. The following statements are equivalent: (i) is a singular Jacobi form (ii) k D r=2 (iii) 2 L Q ;

(iv) 2 L

Proof Recall from Theorem 3.3 that we have the following expansion .; z/ D

X

hx ./#L;x .; z/

(4.1)

C.d; x/q d ;

(4.2)

x2L# =L

with hx ./ D

X d 2ˇ.x/hCd1

where C.d; x/ D c.ˇ.x/ d; x/ and the c.t; x/ are the Fourier coefficients of , and where h 2 K is such that .Tb / D e fhbg (b 2 O). .i / H) .ii/; .iii/. Suppose is a singular Jacobi form. Fix x 2 L# =L. Since is a singular Jacobi form, from (4.2) we have that hx ./ D C.0; x/, i.e. hx is a constant. Hence is in L and has, in particular, weight r=2. .ii/ H) .i /. Suppose k D r=2. Fix x 2 L# =L. From Corollary 3.52, we have that hx is a Hilbert modular form of weight k r=2 D 0. From [Fre90, Prop. 4.7], we have that hx is a constant. From Example 3.48, we have that #L;x is a singular Jacobi form (for some subgroup of Q ). Hence , being a linear combination of singular Jacobi forms (see (4.1)), is also a singular Jacobi form. .iii/ H) .ii/. Suppose 2 L . Hence is a linear combination of forms of weight r=2, hence has weight r=2. .iii/ H) .iv/. Suppose 2 L . Since is in Jk;L . / we have jk;L ˛ D .˛/ Q ;

(˛ 2 Q ). Hence, 2 L . .iv/ H) .iii/. This is obvious. t u

4.2 Theta Functions and Weil Representations The main purpose of this section will be to set up natural isomorphisms between Weil representations associated to certain discriminant modules and the Q -modules L of Jacobi theta functions. Moreover, as preparation for the complete decomposition of the L in the next section, we shall translate via these

4.2 Theta Functions and Weil Representations

105

isomorphisms the essential ingredients of the representation theory of the Weil representations to the Q -modules L . Let L D .L; ˇ/ be a totally positive definite even O-lattice of rank r. From Theorem 3.1, we know that L is a right Q -module. So, the space L equipped with the following Q -action .˛; #/ 7! #jr=2;L ˛ 1

.˛ 2 Q /

becomes a left Q -module. This space will be denoted in the following by L} . For the definition of the jr=2;L -action, the reader is referred to Proposition 3.28. Proposition 4.2 Let L D .L; ˇ/ be a totally positive definite even O-lattice. The linear continuation of the map L W W DL1 ! L} ;

exCL 7! #L;x

defines a Q -linear isomorphism. Proof Clearly, L is a well-defined linear map. From Proposition 3.33 we know that for fixed , the functions z 7! #L;x .; z/ (x 2 L# =L) are linearly independent. Hence, L is injective. Since L} is spanned by the functions #L;x (x in L# =L), the map L is also surjective. It remains to show that L is Q -linear. Since the group Q is generated by Tb (b 2 O), S and I (see Proposition 3.13), it is enough to prove for those types of elements ˛, the following identity: L ˛exCL D #L;x jr=2;L ˛ 1 : Applying Theorem 3.1 to the element .Tb /1 , we see that the claimed identity holds true for .Tb /1 , since we have #L;x jr=2;L .Tb /1 D e fbˇ.x/g #L;x D L Tb exCL : Proceeding as in the proof of Corollary 3.34 (ii), we can easily obtain X 1 #L;x jr=2;L .S /1 D .i /nr=2 p e fˇ.y; x/g #L;y ; # jL =Lj y2L# =L where r stands for the rank of L. To prove the claimed identity for .S /1 , it remains to show that .DL1 / D .i /nr=2 . But this follows from Milgram’s formula [MH73, p. 127]. The claimed identity obviously holds true for the element I . t u As preparation for the next section we append here two lemmas.

106

4 Singular Jacobi Forms

Lemma 4.3 Let L D .L; ˇ/ be a totally positive definite even O-lattice, let U be an isotropic submodule of the discriminant module DL and let L=U D 1 .U /; ˇ (see Definition 3.4). Then the following diagram of Q -homomorphisms is commutative:

Here U is the embedding defined in Sect. 2.3, L and L=U are the isomorphisms from Proposition 4.2, and jU is the inclusion map. Moreover, ' denotes the isomorphism induced from the isomorphism ' from Proposition 3.5. Proof We set L1 WD 1 .U /. The map ' is defined by exCL1 7! e.x/CU , where W L# ! L# =L is the canonical projection (see Proposition 3.5). To show that the diagram commutes, we need to prove the following identity of maps: jU ı L=U D L ı U ı ': On the left we have X

jU ı L=U .exCL1 / D jU .#L=U;x / D #L=U;x D

#L;y :

y2L#1 =L

yx mod L1

For the last identity we used X q ˇ.r/ e fˇ.r; z/g #L=U;x D r2L#1 rx mod L1

D

1 jL1 j

X

X

q ˇ.r/ e fˇ.r; z/g D

y2L#1 =L r2L#1 yx mod L1 ry mod L

1 jL1 j

X

#L;y :

y2L#1 =L yx mod L1

On the right we have L ı U ı '.exCL1 / D L ı U .e.x/CU / D

X

L .eY /

Y 2U # =U Y .x/ mod U

D

1 jL1 j

X

#L;y :

y2U # yx mod L1

For the last identity we did the substitution Y 7! .y/, where Y D y C U . But since U # D L#1 =L, the diagram commutes.

4.3 Decomposition of the Q -Modules L

107

The map U is Q -linear (see Proposition 2.30) and also the maps L and L=U ı ' are Q -linear (see Proposition 4.2). t u 1

4.3 Decomposition of the Q -Modules L In the present section, we shall decompose the spaces of Jacobi theta functions L into irreducible Q -submodules, where L is a totally positive definite even O-lattice of rank one. Our main observation is that the discriminant modules DL (see Definition 3.3) of such lattices are cyclic finite quadratic O-modules (which follows from Propositions 3.10 and 3.7). The same propositions also imply that if the level of the lattice L (the level of DL ) is l, then the modified level and the annihilator of L equals l=4 and l=2, respectively. Definition 4.4 We define the new part Lnew of L as the orthogonal complement P of U 6D0 L=U with respect to the scalar product (3.18), where U runs through the nonzero isotropic submodules of DL . Let l and a denote the level and annihilator of L, respectively. Recall that E.DL / consists of all " C a 2 .O=a/ such that " 1 mod h and " C1 mod ah1 for some exact divisor h of a. The group E.DL / acts on L via linear continuation of the map .g; #L;x / 7! g#L;x WD #L;gx . Since it acts obviously unitarily and leaves the subspaces L=U invariant (since L=U has as the underlying O-module 1 .U /, where is the canonical projection from L# ! L# =L, and is O-linear), it leaves also Lnew invariant. For a square-free divisor f of m, we define new;f

L

D f# 2 Lnew W g# D

f .g/#

for all g 2 E.M /g:

Here f denotes the linear character of E.M / such that f ." C a/ D .1/t , where t is the number of primes in .h; f/ and h is as above (see Proposition 1.23). Theorem 4.1 Let L D .L; ˇ/ be a totally positive definite even O-lattice of rank one with annihilator a, level l and modified level m. (i) For every square-free divisor f of m, the space Lnew;f is Q -invariant and irreducible. (ii) One has the following decompositions L D

M

new ab 1 L# CL

(4.3)

b2 jm

Lnew D

M

fjm f square-free

new;f

L

:

(4.4)

108


For the proof, which will be a consequence of the decomposition of W .DL / given in Sect. 2.4, we need a lemma. Lemma 4.5 Let L be a totally positive definite even O-lattice of rank one whose modified level is m. Let L be the Q -module isomorphism in Proposition 4.2. For a square-free divisor f of m, one has L W .DL1 /new D .L} /new ;

L W .DL1 /new;f D .L} /new;f :

P } iD Proof Firstly, let v 2 W .DL1 /new . We need to show that hL .v/; U ¤0 L=U 1 0, where U runs through isotropic submodules of DL . Using the fact that the scalar product (2.12) on W .DL1 / satisfies for all v; v 0 2 W .DL1 /, hL .v/; L .v 0 /i D hv; v 0 i (since L is an isomorphism), it is enough then to P } } show hv; U ¤0 L1 L=U i D 0. But since L=U D L=U ı ' 1 .W .DL1 =U //, and L1 ı L=U ı ' 1 D U (see Lemma 4.3), the claimed identity holds true, since v lies in the new part of W .DL1 /. P } i D 0. But then Secondly, let v 2 .L} /new . Then we have hv; U ¤0 L=U 1 applying L (it leaves the scalar product invariant) to this identity and using the two identities in the previous paragraph (which follows from Lemma 4.3), we see that L1 .v/ must lie in the space W .DL1 /new , which proves the first identity in the statement of the lemma. Since L is obviously an E.M /-module isomorphism, using the first identity in the statement of the lemma, the second identity holds true. t u Proof of Theorem 4.1 Proof of part (ii). First we prove the identity (4.3). We decompose W .DL1 / into Q -submodules using Theorem 2.5 (i). Applying the isomorphism L (which is given in Proposition 4.2) to the decomposition of W .DL1 / and using Lemma 4.5, we obtain a decomposition of L} into Q -submodules as the one in (4.3). But the underlying spaces of L and L} are equal. Hence, the claimed identity holds true. Next we prove (4.4). We decompose the space W .DL1 /new using Theorem 2.5 (ii) into irreducible Q -submodules. Again by applying the isomorphism L to the decomposition of W .DL1 /new and using Lemma 4.5, we obtain a decomposition of L} as of the kind in (4.4). But the underlying spaces of L and L} are the same, hence the claimed identity holds true. Proof of part (i). Part (i) is an immediate consequence of Lemma 4.5. The fact new;f that the spaces L are irreducible follows from the proof of (4.4). t u

4.4 The Singular Jacobi Forms of Rank One Index

109

4.4 The Singular Jacobi Forms of Rank One Index In this section we shall describe explicitly all singular Jacobi forms whose indices are totally positive definite rank one O-lattices. Recall that, for even L, the discriminant modules DL (see Definition 3.3) of such lattices are cyclic finite quadratic O-modules (see Propositions 3.10 and 3.7). Recall also from the same propositions that if the level of the lattice L (the level of DL ) is l, then the modified level and the annihilator of L equals l=4 and l=2, respectively. From Proposition 3.13, we know that the group Q is generated by the elements Tb (b 2 O), S and I . Hence, the abelianized group Q ab D Q =C of Q is generated by the elements Tb C (b 2 O), S C and IC . Here C denotes the commutator subgroup of Q . But since .S T /3 D .S /2 , and the group Q ab is abelian, we have that .S /3 C.T /3 C D .S /2 C . This implies that S C D .T /3 C , i.e. Q is in fact has a smaller set of generators, namely the elements Tb C (b 2 O) and IC . Therefore, any character of Q is uniquely determined by the value of at Tb (b 2 O) and at I , since any homomorphism of Q factors through a homomorphism of Q ab . Recall that an odd character ideal is an integral O-ideal which is a (possibly empty) product of pairwise different prime ideals of degree one over 3. Definition 4.6 Let .c; !/ be as defined in Definition 3.6. Suppose 2 splits completely in K, and let g be an odd character ideal such c2!d D g. We denote ˚ that 2 by ".c;!/ that character of Q such that ".c;!/ .Tb / D e b! =8 and ".c;!/ .I / D 1, where C 4c is a generator for cg1 =4c. Remark The fact that ".c;!/ defines indeed a character of Q is a consequence of Theorem 4.2. Note that the character ".c;!/ does not factor through a character of , whereas its square does ( since it maps .1; ˙1/ to 1). A complete classification of linear characters of including explicit formulas is given in [BS13], which can be used for obtaining explicit formulas for ˙".c;!/ .˛/ for an arbitrary ˛ in Q . If g is not one, then the order of ".c;!/ is 24, and if g D 1, then it has order 8, as follows from 3!d 2 3!dc2 g2 D 3g1 O and the remark preceding Definition 4.6. We state the three main results of this section. Theorem 4.2 Let c be a fractional O-ideal and ! a totally positive element in K such that c2 !d D g for an odd character ideal g. Suppose 2 splits completely in K. Set X 1 2 #.c;!/ .; z/ WD

4g .s 0 /q 8 !s e f!sz=2g : (4.5) s2cg1

Here s 0 2 O is so that s s 0 mod 4c, where C 4c is a generator for cg1 =4c and 4g is the totally odd Dirichlet character modulo 4g (see Definition 2.44).

110


Then #.c;!/ is a Jacobi form on the full modular group of weight 1=2, index .c; !/ with character ".c;!/ . Note that #.c;!/ depends also on the generator . However, a different generator changes #.c;!/ only by a sign. Therefore, we suppress the dependency on the choice of in the notation. Theorem 4.3 Let L D .L; ˇ/ be a totally positive definite (not necessarily even) O-lattice of rank one. The space J1=2;L . / is trivial unless 2 splits completely in K, there is a homomorphism from L into a lattice .c; !/ of the kind which occurs in Theorem 4.2, and D ".c;!/ . If 2 splits completely in K, if the map ' W L ! .c; !/ is a homomorphism into a lattice .c; !/ as in Theorem 4.2, and if D ".c;!/ , then J1=2;L . / D C #.c;!/ ; '.z/ . (Here '.z/ denotes the value at z of the C-linear extension of ' to LC .) Proposition 4.7 The number of indices modulo isomorphism which admit a nonzero singular Jacobi form equals j F.K/j j ClC .K/Œ2 j, where F.K/ is the subset of the principal genus containing ideals of the form gd1 with g an odd character ideal, and where ClC .K/Œ2 is the kernel of the squaring map of the narrow class group. Proof Let J denote the group of fractional O-ideals, and PC denote the subgroup of principal O-ideals which have totally positive generators. It is easy to see that the following sequence ı ' 1 ! Ker.'/ ! f.c; !/ W ! 0g f a1 ; a2 W a 2 K g ! J2 PC ! 1; where ' W .c; !/ f a1 ; a2 W a 2 K g 7! c2 !, is exact. (Here the set of pairs .c; !/ of ideals and numbers ! 0 is considered as a group via componentwise multiplication.) Using Theorems 4.2 and 4.3, the number of indices modulo isomorphism which admit a nonzero singular Jacobi form equals j F.K/j j Ker.'/j. We calculate the number of elements in Ker.'/. The following sequence is also exact

1 ! Ker./ ! Ker.'/ ! Ker

W Cl.K/ ! ClC .K/ ! 1;

where is the map which maps .c; !/ f a1 ; a2 W a 2 K g to the ideal class of c. Hence, j Ker.'/j D j Ker./j j Ker. /j. By direct calculation, we find that the number of elements in Ker./ equals Œ.O /C W .O /2 , where .O /C denotes the group of totally positive units in K. Therefore, the number that we are looking for is j F.K/j Œ.O /C W .O /2 j Ker. /j. However, by [vdG88, I. 4], we have that the number Œ.O /C W .O /2 j Ker. /j equals j ClC .K/Œ2 j. This proves the proposition. t u The rest of this section is devoted to the proofs of the previously stated theorems.


111

Proof of Theorem 4.2 The sum in (4.5) can be rewritten in the following way: X

#.c;!/ .; z/ D

4g .s 0 /

s2 12 cg1 =2c

X

D

X

1

2

q 2 !y e f!yzg

y2 12 cg1 ys mod 2c

4g .s 0 / #.2c;!/;x :

x2 12 cg1 =2c

But the last identity shows that #.c;!/ is the image of the vector which spans the onedimensional Q -subspace of W .D.2c;!/ / (see Theorem 2.6, (iii)) under the Q -module isomorphism .2c;!/ (which is given in Proposition 4.2). Hence, by Proposition 4.1, we have that #.c;!/ is a singular Jacobi form. Since #.c;!/ is a singular Jacobi form, it transforms under the Q -action with a suitable character. Next we show that this character is in fact ".c;!/ . According to the observation about the abelianized group of Q which is explained in the beginning of the present section, it suffices to prove for any b 2 O, the following identity ˚ #.c;!/ j1=2;.c;!/ Tb D e b! 2 =8 #.c;!/

.b 2 O/;

(4.6)

since we obviously have #.c;!/ j1=2;.c;!/ I D #.c;!/ (see the action given in Proposition 3.28). On the left of (4.6), we have #.c;!/ . C b; z/ D

X

1 2 ˚

4g .s 0 /q 8 !s e b!s 2 =8 e f!sz=2g :

s2cg1 1 0 If˚we can show that ˚ for2 every b 2 O and every s 2 cg with .s ; 4g/ D 01, we have 2 e b!s =8 D e b! =8 , then (4.6) obviously holds true. Write s D s C 4c for c 2 c. Then we have

!s 2 =8 ! 2 =8 D !.s 0 C 4c/2 =8 ! 2 =8 D !s 02 2 =8 C !s 0 c C 2!c 2 ! 2 =8: Note that we have !s 0 c !cg1 c D d1 , and also we have 2!c 2 2!c2 D 2gd1 , since c2 !d D g (by the assumption of the theorem). Hence it remains to show that !d 2 .s 02 1/=8 is integral. Since 2 cg1 , it is enough to show that !d.cg1 /2 .s 02 1/=8 is integral. But since c2 !d D g, it suffices to show that 8g divides s 02 1. Let q be a prime divisor of 2. Since q has degree one, we have O=q3 ' Z=8Z, and hence the group .O=q3 / has exponent 2. Therefore, by the assumption .s 0 ; 4g/ D 1, we have s 02 1 mod q. If g D 1, there is nothing left to prove. Suppose g ¤ 1. Let p be a prime divisor of g. Since p has degree one, we have O=p ' Z=3Z, and hence the group .O=p/ has order 2. Again by the same assumption, we have s 02 1 mod p. Therefore, the claimed identity holds true, and thus (4.6) holds true.

112


To prove that #.c;!/ is of index .c; !/, we show that #.c;!/ transforms under the H.c/-action via ˚ #.c;!/ j1=2;.c;!/ h D e !.x C y/2 =2 #.c;!/ ;

(4.7)

where h D x; y; e f!xy=2g with x and y in c. Recall that H.c/ is generated by ˚1 the elements x; y; e 2 !xy (x; y 2 c). By applying the action of the Heisenberg group to #.c;!/ , we have ˚ #.c;!/ j1=2;.c;!/ h D e !x 2 =2 C !xz #.c;!/ .; z C x C y/: By evaluating #.c;!/ at .; z C x C y/, we obtain X 1 2 1 #.c;!/ .; z C x C y/ D

4g .s 0 /q 8 !s C 2 !sx e f!sy=2g e f!sz=2g : s2cg1

First we show that every b 2 O and every s 2 cg1 with .s 0 ; 4g/ D 1, we have ˚ for 2 e f!sy=2g D e !y =2 . We have !sy=2 !y 2 =2 D !.s 0 C 4c/y=2 !y 2 =2 D !s 0 y=2 C 2!cy !y 2 =2: 2 Since 2!cy 2!c2 D 2gd1 (see the of the theorem, i.e. c !d D assumption 0 1 2 g), it is enough to show that the ideal !ds cg c !dc =2 is integral. From the assumption of the theorem (i.e. from c2 !d D g) again, it suffices to show that s 0 g is divisible by 2. Let q be a prime divisor of 2. By the assumption .s 0 ; 4g/ D 1, we have that q does not divide s 0 . Obviously, q does not divide g either. But since q has degree one over 2, we have O=q ' Z=2Z, and hence qjs 0 g. Thus, the claimed identity holds true. Therefore, we have

˚ ˚ #.c;!/ .; z C x C y/ D e !y 2 =2 e !x 2 =2 X 1 2

4g .s 0 /q 8 !.sC2x/ e f!sz=2g s2cg1

˚ ˚ D e !y 2 =2 e !x 2 =2 e f!xzg X 1 2

4g .s 00 /q 8 !s e f!sz=2g ; s2cg1

where s 2x s 00 mod 4c. But then we have ˚ ˚ e !x 2 =2 C !xz #.c;!/ .; z C x C y/ D e !y 2 =2 X 1 2

4g .s 00 /q 8 !s e f!sz=2g : s2cg1

(4.8)


113

First note that if x and y in 2c, then we have s 00 s 0 mod 4g. Indeed, multiplying the congruence s 00 s 0 mod 4c with c1 g and noting that c1 g is integral and relatively prime to 4g (recall cg1 D O C 4c), we see that the claimed statement holds true. Then the identity (4.8) becomes #.c;!/ j1=2;.c;!/ h D #.c;!/ ; which proves (4.7). Next suppose that x and y are not in 2c. If we show that g .s 00 / D g .s 0 / and

4 .s 00 / ˚D 4 .s0 2/, then using also the easily deduced identity 4 .s 0 2/ D

4 .s 0 /e !x 2 =2 , the identity (4.8) becomes ˚ #.c;!/ j1=2;.c;!/ h D e .x C y/2 =2 #.c;!/ ; which proves (4.7), and hence the theorem. It remains to prove s 00 s 0 mod g and s 00 s 0 2 mod 4. Multiplying the congruence s 00 s 0 2x mod 4c similarly as above with c1 g, we obtain s 00 s 0 2gxc1 . c1 g/1 mod 4g. Here note that xc1 and . c1 g/1 are odd. Therefore, the claimed statement holds true. t u For the proof of Theorem 4.3, we need a lemma and a proposition. Lemma 4.8 Let L D .L; ˇ/ be a totally positive definite even O-lattice of rank one with level l. The space Lnew contains one-dimensional Q -submodules if and only if 2 splits completely in K and l D 8g, where g is an odd character ideal. If 2 splits completely in K and l D 8g, then Lnew contains exactly one one-dimensional Q -submodule, namely new;2g . L

Proof Suppose 2 splits completely in K and l D 8g. By applying Lemma 2.46 to the O-CM DL1 , we observe that the space W .DL1 /new;2g is the unique one-dimensional Q -submodule of Lnew . Hence, by Lemma 4.5, the space Lnew;2g is the unique onedimensional Q -submodule of Lnew . Suppose on the other hand that the space new contains one-dimensional Q L

submodules. From Lemma 4.5 we know that the space W .DL1 /new also contains one-dimensional Q -submodules. Hence, by Lemma 2.46, l must be a character ideal, i.e. l D gh3 , where g is an odd character ideal and h is a (possibly empty) product of pairwise different prime ideals above 2 of degree one and ramification index one. From Propositions 3.10 and 3.7, we know that l is divisible by 4, i.e. we have h D 2O. But this implies that 2 splits completely in K and l D 8g. t u Proposition 4.9 Let L D .L; ˇ/ be a totally positive definite even O-lattice of rank one with level l and modified level m. The following statements hold true. (i) The space L contains one-dimensional Q -submodules if and only if 2 splits completely in K, and l D 8gb2 , where g is an odd character ideal, and b is an integral O-ideal such that b2 jm.

114


(ii) The space L contains at most one one-dimensional Q -submodule. As a consequence, the space of singular Jacobi forms with index L is at most one dimensional. Proof Proof of part (i). Suppose that 2 splits completely in K and l D 8gb2 . In the new following we denote by a, the annihilator of L. By Lemma 4.8, the space ab 1 L# CL contains one-dimensional Q -submodules, since the level of ab1 L# Cı L equals 8g. Indeed, the level of ab1 L# C L equals the level of the O-FQM DL1 .ab1 L# =L/ (see Proposition 3.5) which has level lb2 D 8g by Corollary 1.19. Hence, by (4.3), the space L contains one-dimensional Q -submodules. Now suppose that the space L contains a one-dimensional Q -submodule, say W . By combining the identities (4.3) and (4.4), we obtain a decomposition of L into irreducible Q -submodules. From Proposition 2.16, we have W ' new;f ab1 L# CL for some square-free divisor f of m, and an integral O-ideal b such that b2 divides m. But by Lemma 4.8, we have that 2 must split completely in K, and that the level of ab1 L# C L which equals lb2 (see the above paragraph) must be equal to 8g, which proves (i). Proof of part (ii). By Theorem 2.6 we have that the space W .DL1 / contains at most one one-dimensional Q -submodule. Hence, by Proposition 4.2, the space L contains at most one one-dimensional Q -submodule. Proposition 4.1 implies that the space of singular Jacobi forms are in one-toone correspondence with the one-dimensional Q -submodules of L . Therefore, the space of singular Jacobi forms of index L is at most one dimensional. t u Proof of Theorem 4.3 By Proposition 3.10 we have that L is isomorphic to an Olattice of the form .c0 ; ! 0 /, where c02 ! 0 d is integral and ! 0 0. Suppose J1=2;L . / is non-zero. We set L.2/ WD .2c0 ; ! 0 /. Hence, by Proposition 4.1 we have that the space L.2/ contains one-dimensional Q -submodules (since J1=2;L . / can be identified with a subspace of J1=2;L.2/ . /). Proposition 4.9 (i) implies then that 2 splits completely in K and that the level of L.2/ (which is 8c02 ! 0 d) must be equal to 8gb2 for some integral O-ideal b whose square divides the modified level of the O-lattice L.2/. Hence, we have the identity .c0 b1 /2 ! 0 d D g. However, this implies that c0 b1 ; !0 is of the kind which occurs in Theorem 4.2. Since .c0; ! 0 / obviously embeds into c0 b1 ; ! 0 , the O-lattice L also embeds into c0 b1 ; ! 0 . Now we prove that D ".c0 b1 ;! 0 / . By Theorem 4.2 we have #.c0 b1 ;! 0 / in J1=2;.c0 b1 ;! 0 / .".c0 b1 ;! 0 / /. Proposition 4.9 (ii) implies that #.c0 b1 ;! 0 / spans the space J1=2;.c0 b1 ;! 0 / .".c0 b1 ;! 0 / /. But this space can be viewed as a subspace of J1=2;L . /, since L can be embedded into c0 b1 ; ! 0 , i.e. we have the claimed identity. Assume D ".c;!/ , 2 splits completely in K, and ' denotes a homomorphism from L to .c; !/, where .c; !/ is of the kind which occurs in Theorem 4.2. From Theorem 4.2 we know that #.c;!/ is a singular Jacobi form of index .c; !/, and from Proposition 4.9 (ii) we know that the space of Jacobi forms of a given index is at most one-dimensional, hence J1=2;.c;!/ . / D C #.c;!/ . However,

4.5 Constructing Jacobi forms

115

from J1=2;.c;!/ . / J1=2;'.L/ . /, we have J1=2;'.L/ . / D C #.c;!/ (see also Proposition 4.9 (ii)). Since ' is injective (from Sect. 3.1 we know that every homomorphism between totally positive definite O-lattices is injective), the map .; z/ 7! .; '.z// defines an isomorphism from J1=2;L . / to J1=2;'.L/ . / which finally proves the theorem. t u

4.5 Constructing Jacobi Forms of Non-Singular Weight In this last section we take up the definition of general Jacobi forms of Sect. 3.6 and show how to construct explicit examples of non-singular weight forms. We shall be in particular interested in Jacobi forms of integral weight, even rank one index and trivial character since it is expected that these forms will lift to usual K Hilbert modular forms. We shall simply write Jk;L for the space of Jacobi forms on SL.2; O/ of weight k, index L and trivial character. We shall show how to generate such Jacobi forms from forms of singular weight using three simple functorial principles. We shall end this section with several concrete examples. However, we know of at least two other methods for constructing examples, whose detailed investigation does not quite fit into the range of this monograph, and, in particular, not into the context of Jacobi forms of singular weight. For the sake of completeness we will also sketch these methods here in the hope that they find interest for doing explicit calculations to obtain further interesting examples of Jacobi forms. We start with a sketch of the two “non-singular” methods. The first one is the well-known method of averaging some simple function over the Jacobi group J.L/ associated to a lattice L D .L; ˇ/ as in Definition 3.18. As an example we take the constant function 1 and set X Ek;L .; z/ WD 1jk;L g: g2J.L/1 nJ.L/

Here k is an integer and L an even O-lattice, and J.L/1 is the subgroup of all g in J.L/ such that 1jk;L g D 1. Using [Fre90, I. Lemma 5.7] is not hard to show that this infinite series converges for k 3 (see also [Boy14]). Moreover, the series is identically zero unless N.a/k D remains all units a in O (since the sum aC1 for k unchanged if we replace g by a1 g, which multiplies it by N.a/ ). One can closely follow the method in [EZ85, § 2, p. 18–20] to calculate the Fourier expansion of Ek;L . The result is Theorem 4.4 ([Boy14]) Let L D .L; ˇ/ be an even O-lattice, and let k be an integer. Assume N.a/k D C1 for all a in O . Then, for some constant C , one has Ek;L D #L;0 C C

X t 2d1 ; s2L# t ˇ.s/ 0

N.t ˇ.s//k3=2 DL .t; sI k 1/ q t e fˇ.s; z/g ;

116


where, for any sufficiently large complex , we use ˇ˚ ˇ X ˇ x 2 L=aL W ˇ.x C s/ ˇ.s/ t mod ad1 ˇ ; DL .t; sI / D N.a/ a the sum running over all non-zero integral ideals of K. The constant C , which depends on k and L can be made explicit but is not important here. In the case that L is a rank one lattice the Dirichlet series can be further simplified and related to Hecke L-series. For details we refer the reader to [Boy14]. In general, one could also average functions like q t s , where t, s are fixed elements in d1 , L# , respectively, and where we use s for the function e fˇ.s; z/g on LC . This would lead into a theory of Poincaré series, i.e. Jacobi forms which correspond to the functionals 7! c .t; s/ ( .t; s/th coefficient of ) K if we identify Jk;L . / with its dual via a suitable Petersson type scalar product (see [GKZ87, II.2]) for the case K D Q and scalar index). However, we shall not pursue this here. Arithmetically interesting are those series Ek;LIt;s which we obtain by averaging q t s with t D ˇ.s/, i.e. the averages Ek;LIˇ.s/;s of q ˇ.s/ s where s runs through the elements in L# such that ˇ.s/ 2 d1 . One would call these series Jacobi Eisenstein series since they represent the functionals 7! c .ˇ.s/; s/ which vanish identically on the subspace of cusp forms, i.e. they would be perpendicular to the space of cusp forms with respect to a Petersson type scalar product. They are arithmetically interesting since the condition t D ˇ.s/ entails that the Fourier coefficients are special values of Dirichlet series of the type encountered in the discussion of Ek;L D Ek;LI0;0 . This can be seen by mimicking the computation of the Fourier coefficients of the Eisenstein series in [EZ85, §2] (or see [Boy14]). We shall not pursue this any further here since the calculation of the Fourier coefficients seems to be tedious and will eventually become easier with the help of a suitable Hecke theory, which has still to be developed. It is not difficult to count the number of Eisenstein series Ek;LIˇ.s/;s . Indeed it is easy to see that the average of q ˇ.s/ s over J.L/ is the same as the average of q ˇ.s/ s jk;L g, where g D .1; x; 0; 1/ or g D . a a1 ; 0; 0; 1/ (x 2 L, a 2 O ). In other words, one has Ek;LIˇ.s/;s D Ek;LIˇ.sCx/;sCx and Ek;LIˇ.s/;s D N.a/k Ek;LIˇ.as/;as (see Propositions 3.25, 3.28). Thus the dimension of the space of Eisenstein series of the form Ek;LIˇ.s/;s (s 2 L# , ˇ.s/ 2 d1 ) is bounded to above by the number of orbits ˚ O n s C L 2 L# =L W ˇ.s/ 2 d1 ; N.Os /k D 1 ;

(4.9)

where Os is the stabilizer in O of s C L in L# =L. The condition N.Os /k D 1 is, of course, void if k is even or if K does not possess units with norm equal to 1. The dimension of the space spanned by the Eisenstein series is even equal to this number of orbits. Namely, following the proof of [EZ85, Thm. 2.4], one finds that


117

Ek;LIˇ.s/;s D const.

X

N.a/k #L;as C :

a2O =Os

Here “ ” indicates that part of the Fourier development of Ek;LIˇ.s/;s consisting only of terms q t s with t ˇ.s/ 0, and const. is a nonzero number. The “singular parts” of the Ek;LIˇ.s/;s (s running through a complete set of representatives R for the orbits in (4.9)) are linearly independent as follows from the linear independence of the #L;x (Proposition 3.33). Consequently, the Ek;LIˇ.s/;s (s 2 R) are linearly independent too. If K has class number one and L has rank and modified level one, then there is only one Eisenstein series, namely, Ek;L;0;0 D Ek;L , i.e. the one discussed in the preceding theorem. K The transformation law in Definition 3.45 of a Jacobi form in Jk;L applied to a diagonal matrix in SL.2; O/ implies that its Fourier coefficients c.t; s/ satisfy c.a2 t; as/ D N.a/k c.t; s/ for all units a in O. Furthermore, we know from Theorem 3.3 that c.t; s/ depends only on ˇ.s/ t and s C L. Therefore, if the class number of K is one, we find that is a cusp form if and only if c.ˇ.s/; s/ D 0 for all s in a complete system of representatives for the orbits in (4.9). (Here we also use the remark after Definition 3.47). Summarizing, we have proved Proposition 4.10 Let L be an even O-lattice, k 3 be an integer, and suppose that the class number of K is one. Then the co-dimension of the subspace of cusp forms K in Jk;L equals the cardinality of the set of orbits in (4.9). If the class number of K is strictly greater than one we have to take into account more than one cusp, which can be done by considering the average of functions q t s jk;L g, where again t is in d1 , s is in L# such that t D ˇ.s/, and where now g is an arbitrary element in SL.2; K/. We can proceed as before to count the coK dimension of the subspace of cusp forms in Jk;L . This is not difficult but the resulting formulas will become more tedious. The second method uses the fact that the lth Taylor coefficient l ./ D

@l1 C

Clm @zl11

@zlmm

.; z/jzD0

l D .l1 ; : : : ; lm /

of a Jacobi form of index L (with respect to any given system of coordinate functions zj on LC ) in the Taylor expansion of a Jacobi form .; z/ around z D 0 is not an arbitrary holomorphic function of in H . Indeed, 0 ./ equals the nullwert .; 0/ of , and the transformation formulas for Jacobi forms imply that this is a Hilbert modular form on the full Hilbert modular group of the same weight as . For general l one obtains functions which have a transformation law under SL.2; O/ similar to the quasi-modular forms in the theory of elliptic modular forms. These are distinguished functions equal to or generalizing usual Hilbert modular forms, and, for certain K, the spaces spanned by these functions can be explicitly described. Since .; z/ as a function of z is multivariate one would not a priori

118


expect that finitely many coefficients l suffice to recover . However, if one picks a finite set S of r D rank.L/ many multi-indices which satisfies If l is in S then all h 2 Zm 0 with h l and h l mod 2 are contained in S

˚ then, under certain additional conditions, the map 7! l l2S is injective. (The relations “” and “” have to be read componentwise.) Even more, there is an explicit finite closed formula to express any of a given L in terms of the finitely many Taylor coefficients l (l in S ). The “certain additional conditions” are that W .DL / is irreducible and that the Wronski type determinant det .#L;xi /l .j / 1i;j r does not vanish identically (here fx1 ; : : : ; xr g is a complete system of representatives for L# =L and S D fl .1/ ; : : : ; l .r/ g, and, as before, .#L;xi /l denotes the lth Taylor coefficient of #L;xi around 0 with respect to the coordinates zj ). Note that W .DL /, for a rank one lattice L, is irreducible if and only if its modified level is one (see Theorem 2.5). A rank one lattice .c; !/ has modified level c2 !d=2 (Prop. 3.7), and thus there exists rank one lattices of modified level one if and only if the different lies in the principal genus of K. The method of the preceding paragraph can be extended (by passing to certain K natural subspaces of Jk;L . /), so to dispose of the assumption that W .DL / is irreducible. For details the reader is referred to [BHS14], where the indicated method is described in detail and explicit working examples are given. For the (classical) case K D Q and scalar index this method was developed in [Sko85, §7]. The very starting point for this method are the considerations in [EZ85, §3]. We finally explain the three general principles mentioned in the introduction. The first one is multiplication of Jacobi forms. However, since the indices of the Jacobi forms in question might be different, we have to be a bit more We define precise. the product of two Jacobi forms 1 and 2 of index Lj D Lj ; ˇj (j D 1; 2), respectively, as .1 ˝ 2 /.; .z1 ; z2 // D 1 .; z1 /2 .; z2 /

.zj 2 .Lj /C /:

(4.10)

The following proposition is immediate from the Definition 3.45 of Jacobi forms. Proposition 4.11 The multiplication (4.10) defines a map ˝ W JkK1 ;L1 . 1 / JkK2 ;L2 . 2 / ! JkK1 Ck2 ;L1 ˚L2 . 1 2 /: Here L1 ˚ L2 denotes the orthogonal sum of L1 and L2 (i.e. the lattice given by .L1 L2 ; .x; y/ 7! ˇ1 .x/ C ˇ2 .y//). The described multiplication is not always interesting since it increases the rank of the index lattice. One can combine it with the next principle for reducing again the resulting rank. The second principle is a pullback in the lattice-variable. More precisely, assume that ˛ W L ! L0 is an isometric map from the lattice L D .L; ˇ/


119

0 0 0 0 into a lattice L D .L ; ˇ /, i.e. a homomorphism of groups ˛ W L !0 L such that 0 ˇ ˛.x// D ˇ.x/ for all x in L. If is a Jacobi form of index L , we define a function on H LC by setting

˛ .; z/ WD .; ˛.z//:

(4.11)

(In this formula ˛ is linearly continued to a map from LC to L0C ). Again the following proposition is an immediate consequence of Definition 3.45. Proposition 4.12 For any isometric map ˛ W L ! L0 , the application (4.11) defines a map K K ˛ W Jk;L 0 . / ! Jk;L . /:

(4.12)

We can combine the two described principles as follows. For a positive integer a set L.a/ D .L; aˇ/ and ˚a L D L ˚ ˚ L the a-fold orthogonal sum of L with itself. The diagonal map W L.a/ ! ˚a L which maps x in L to .x; : : : ; x/ is K isometric. The composition of with the product ˝ defines a map from Jk;L . / to K Jk;L.2/ . /. This can, of course, be iterated, so to obtain the map K K K Jk;L . / Jk;L . / ! Jk;L.a/ . /

which, in explicit terms, is nothing else but the simple application which maps Jacobi forms 1 , . . . , a of the same index L to 1 .; z/ a .; z/. The last principle varies the underlying number field by a pullback in the upper half plane variable of a Jacobi form. For this let K1 K be a subfield of K with ring of integers O1 . Note that K1 is then totally real too (since every embedding of K1 into C can be continued to an embedding of K into the complex numbers). We then have C1 WD C ˝Q K1 C D C ˝Q K, and accordingly LC1 WD L ˝O1 C1 LC . If K . /, we can restrict to a function 1 on H1 LC1 , where is a Jacobi form in Jk;L H1 denotes the set of elements in C1 with positive imaginary part (see Sect. 3.2). If we examine the definition of the Jacobi group actions 3.24 and 3.25 we find that 1 transforms like a Jacobi form over K1 of index ResO1 L WD L; trK=K1 ıˇ ; respectively. However, in the transformation law 3.24 under A D ac db in SL.2; O1 / the factor ˙ N.c C d /k with in H1 shows up, where N is the norm of the Calgebra C and not the norm N1 of C1 . It is quickly checked that, for in H1 , one has N./ D N1 ./d with d D ŒK W K1 (since the infinite places of K1 in the totally real extension K do not ramify). We have thus the group homomorphism W Q1 ! Q ;

.A; w/ 7! .A; wd /

120


of the metaplectic cover Q1 of SL.2; O1 / into the metaplectic cover of D SL.2; O/. Finally, for a character of Q , we use ResO1 . / for its composition with . The following proposition is now clear. Proposition 4.13 Let K1 be a subfield of K, and let d D ŒK W K1 . The application which maps a Jacobi form over K of index L to its restriction to H1 LC1 defines a map K 1 ResK1 W Jk;L

! JdKk;Res O

1

L

ResO1 . / :

Note that the map ResK1 increases the weight. We can apply the restriction map for example to the singular forms of Theorem 4.2 for obtaining non-singular forms over subfields of K. p Example 4.14 Let K D Q. D/ be the real quadratic field of discriminant D. Assume that D 1 mod 8 (i.e. that 2 splits completely in K), and that the fundamental units have norm 1. The first examples of discriminant D fulfilling these assumptions are D D 17; 41; 65; 73; 89; 97; : : : . Let " denote the fundamental unitpwith " > 1. Here, for an a in K we use a0 for its conjugate. The number ! WD "= D is totally positive (since "0 D 1=" < 0) and !d D 1. Therefore the pair .O; !/ defines a totally positive rank one lattice which satisfies the assumption of Theorem 4.2. Hence we have available the singular Jacobi form . Note that the unique totally odd Dirichlet character modulo 4 is 4#.O;!/ . We therefore have given by N.a/ #.O;!/ .; z/ D

X s2O

4 N.s/

p 1 !s 2 Q. D/ q 8 e f!sz=2g 2 J 1 ;.O;!/ ".O;!/ : 2

For obtaining forms of non-singular weight with trivial character one can take ˝k powers #.O;!/ with k divisible by 8 (recall from the remark following Definition 4.6 that ".O;!/ has order 8). This gives forms of even weight k=2, however, with index L WD .O k ; !.x1 y1 C C xk yk //, which is odd and of rank k. For obtaining forms with even rank one index let a1 , . . . , ak be elements of O such that 2m WD a12 C C ak2 is divisible by 2. Then x 7! .a1 ; : : : ; ak /x defines an isometric map of the lattice .O; 2m!/ into L. Note that the lattice .O; 2m!/ is even of modified level .m/. Applying Proposition 4.12 we see that p Q. D/

#O;! .; a1 z/ #O;! .; ak z/ 2 Jk=2;.O;2m!/ : For improving the preceding method one can think of dividing the products considered in this example by suitable Hilbert modular forms to decrease the weight. However, this would need a careful analysis of the zeros of such products which seems to be not obvious. Another possibility is to consider also singular Jacobi forms on the full modular group whose index has rank greater than one and which ˝k are not just products of singular forms of rank one like the functions #.O;!/ of the


121

preceding example. The next example constructs singular Jacobi forms of higher rank index. Example 4.15 Assume that the narrow ideal class of the different d of K is trivial, i.e. that d1 D .!/ for some ! 0. We have then the even rank one lattice L D .O; 2!/; its dual is 12 O. The Q -module L spanned by the theta functions #L;x=2 .x 2 O=2O) has dimension d D 2ŒKWQ . To construct a Jacobi form Von the full Hilbert modular group D SL.2; O/ we consider the Q -module d L . It is one-dimensional, spanned by K WD

X

sign./ #L; .x1 /=2 ˝ ˝ #L; .xd /=2 ;

where xj (j D 1; ; d ) runs through a complete set of representatives for O=2O, where runs through the permutations of O=2O, and where sign denotes the nontrivial linear character (signature) of this group of permutations. (We allow us here the slight abuse of language to write .xj / for the representative xj 0 of .xj C2O/.) From Proposition 4.11 (which is, of course, also valid for Jacobi forms on subgroups of Q ) we see that K is a (singular) Jacobi form of weight d=2 on the full modular group. Its character maps 10 b1 to e f!bN=4g, where N D x12 C C xd2 ; it is trivial if N 0 mod 4 (cf. Sect. 4.4), which is for example the case if 2 is a prime in K and d 4 as in the subsequent example (since then, by an obvious argument, .a2 1/N 0 mod 4 for any unit a modulo 2, and a2 1 is a unit modulo 4 if a 6 1 mod 2 and O=2O is a field). Summing up we have proved K K 2 Jd=2;˚ d L . /

.d D 2ŒKWQ /:

We can rewrite the defining equation of K in the form K .; z1 ; : : : ; zd / X D

sign

x1

x2

xd r1 r2

rd

2

2

q !.r1 C

Crd /=4 e f!.r1 z1 C C rd zd /g :

r1 ;:::;rd 2O

Here the matrix argument of sign stands for the application of O=2O to itself which maps x1 C 2O to r1 C 2O, x2 C 2O to r2 C 2O etc., and where we set sign.P / D 0 if the matrix P does not define a permutation. One can now apply Propositions 4.12, 4.13 to pull back and restrict K and obtain forms of non-singular weight and index of rank one. Example 4.16 We continue the preceding example by the first totally real number p by field different from Q, which is K D Q. 5/ (if one orders number fields p p 1C 5 degree and discriminant). Here we can choose ! D "= 5, where " D 2 . As representatives for O=2O we can choose 0; 1; "; "0 . For any a; b; c; d in O D ZŒ" we obtain, using the pullback method, the Jacobi form

122


Q.p5/ .; az; bz; cz; dz/ X 2 2 2 2 0 sign 0r 1s "t "u q !.r Cs Ct Cu /=4 e f!.ra C sb C tc C ud/zg D r;s;t;u2ZŒ" p Q. 5/

2 J2;.O;2! m/

.m D a2 C b 2 C c 2 C d 2 /:

Depending on the choice of a; b; c; d this form might vanish identically. We leave it to the reader to verify that, for e.g. a; b; c; d D 0; 1; "; "0 , we obtain a non-zero form, whose singular part equals #L;1 C #L;" C #L;"0 (L D .O; 8!/), and whose index has modified level 4. It is clear that these examples, though at the first glance somewhat more “ad hoc” than conceptual, are not at all exceptional. They deserve eventually their proper theory. In any case, one can use these methods to produce a plethora of Jacobi forms over number fields from the singular forms considered in this monograph or from their generalizations to higher rank.

Appendix

Tables This chapter contains tables which list the first number fields (ordered by increasing discriminant) of degrees 2, 3 and 4 which admit nonzero singular Jacobi forms. More precisely, we searched all of the Bordeaux number field tables of the PARI group [Bor95] for totally real number fields fulfilling the conditions for admitting nonzero singular Jacobi forms. For number fields K of degree 2 and 3 over Q we list the first 30 number fields where we find nonzero singular Jacobi forms (Tables A.1 and A.2). The percentage of number fields of degree 2 and 3 admitting nonzero singular Jacobi forms among all fields of these degrees in the Bordeaux tables is 17:87 % and 4:75 %, respectively. For number fields K of degree 4 over Q we list all number fields of the Bordeaux tables admitting nonzero singular Jacobi forms (Table A.3). The corresponding percentage is in this case 0; 25 %. We searched the Bordeaux tables also for number fields of degrees n D 5; 6; 7 which admit nonzero singular forms. However, in the available range we could not find any such fields. The columns of the tables display from left to right the discriminant DK of K, the number s of nonzero singular Jacobi forms modulo isomorphism, the minimal polynomial f of K, whether the different dK of K is a square in the narrow ideal class group or not, the number of g satisfying the assumption of Theorem 4.2, and the number of prime ideals p of degree one above 3, respectively.

© Springer International Publishing Switzerland 2015 H. Boylan, Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields, Lecture Notes in Mathematics 2130, DOI 10.1007/978-3-319-12916-7

123

124 Table A.1 Number fields K with ŒK W Q D 2

Appendix DK 17 41 57 65 73 89 97 113 129 137 145 185 193 201 217 233 241 257 265 273 281 305 313 337 353 377 401 409 417 433

s 1 1 2 2 4 1 4 1 2 1 4 2 4 2 4 1 4 1 4 4 1 2 4 4 1 2 1 4 2 4

f x2 x 4 x 2 x 10 x 2 x 14 x 2 x 16 x 2 x 18 x 2 x 22 x 2 x 24 x 2 x 28 x 2 x 32 x 2 x 34 x 2 x 36 x 2 x 46 x 2 x 48 x 2 x 50 x 2 x 54 x 2 x 58 x 2 x 60 x 2 x 64 x 2 x 66 x 2 x 68 x 2 x 70 x 2 x 76 x 2 x 78 x 2 x 84 x 2 x 88 x 2 x 94 x 2 x 100 x 2 x 102 x 2 x 104 x 2 x 108

@K X X X X X X X X X X X

X X X X X X X X X X X X X

g 1 1 1 1 4 1 4 1 1 1 2 1 4 1 2 1 4 1 2 1 1 1 4 4 1 1 1 4 1 4

p 0 0 1 0 2 0 2 0 1 0 2 0 2 1 2 0 2 0 2 1 0 0 2 2 0 0 0 2 1 2

Appendix Table A.2 Number fields K with ŒK W Q D 3

125 DK 961 1;849 3;969 4;481 7;057 7;441 8;281 8;289 8;713 9;153 10;641 11;137 11;665 11;881 13;689 14;129 14;609 15;641 15;961 16;129 16;369 16;649 16;689 17;689 18;201 19;441 20;073 20;385 21;281 23;321

s 1 1 2 2 1 1 1 2 1 2 4 1 1 1 2 2 2 2 1 1 1 2 4 1 4 1 4 2 2 2

f x 3 x 2 10x C 8 x 3 x 2 14x 8 x 3 21x 28 x 3 17x 8 x 3 x 2 22x C 32 x 3 x 2 22x 16 x 3 x 2 30x C 64 x 3 21x 12 x 3 25x 32 x 3 21x 4 x 3 x 2 22x C 16 x 3 x 2 22x C 8 x 3 x 2 26x C 40 x 3 x 2 36x C 4 x 3 39x 26 x 3 x 2 26x 16 x 3 x 2 26x C 32 x 3 29x 36 x 3 x 2 30x 32 x 3 x 2 42x 80 x 3 x 2 26x 8 x 3 x 2 34x 48 x 3 x 2 26x C 24 x 3 x 2 44x C 64 x 3 x 2 30x C 48 x 3 37x 68 x 3 x 2 30x 24 x 3 33x 48 x 3 x 2 42x C 104 x 3 x 2 30x 16

@K X X X X X X X X X X X X X X

X X X X X X X X X X X X

g 1 1 2 1 1 1 1 2 1 2 4 1 1 1 2 1 1 2 1 1 1 2 4 1 4 1 4 2 2 1

p 0 0 1 1 0 0 0 1 0 1 2 0 0 0 1 1 1 1 0 0 0 1 2 0 2 0 2 1 1 1

126 Table A.3 Number fields K with ŒK W Q D 4

Appendix DK 122;825 164;441 171;377 274;625 282;353 310;985 314;721 317;033 340;857 356;337 379;457 389;017 393;129 393;329 471;537 485;809 500;033 506;617 532;521 624;529 626;441 663;833 668;457 674;057 704;969 751;409 754;769 756;313 768;713 830;297 860;353 906;593 996;761

s 2 1 1 2 1 2 2 1 2 2 2 1 2 1 2 1 1 1 4 1 1 1 2 1 1 1 1 1 1 4 2 1 1

f x 4 x 3 23x 2 C x C 86 x 4 2x 3 13x 2 C 14x C 32 x 4 2x 3 19x 2 C 20x C 32 x 4 x 3 24x 2 C 4x C 16 x 4 x 3 35x 2 C 41x C 202 x 4 2x 3 13x 2 C 14x C 8 x 4 25x 2 C 16 x 4 2x 3 17x 2 C 18x C 64 x 4 2x 3 13x 2 C 14x C 16 x 4 x 3 37x 2 C 25x C 268 x 4 2x 3 23x 2 C 24x C 76 x 4 x 3 27x 2 C 41x C 2 x 4 x 3 37x 2 C 97x C 4 x 4 x 3 39x 2 C 9x C 302 x 4 x 3 25x 2 C 25x C 64 x 4 29x 2 C 36 x 4 2x 3 21x 2 18x C 8 x 4 2x 3 21x 2 C 22x C 104 x 4 x 3 27x 2 7x C 82 x 4 2x 3 27x 2 C 28x C 128 x 4 21x 2 8x C 20 x 4 x 3 51x 2 C 49x C 514 x 4 2x 3 33x 2 C 34x C 136 x 4 23x 2 2x C 88 x 4 x 3 33x 2 39x C 8 x 4 23x 2 6x C 80 x 4 x 3 26x 2 C 8x C 64 x 4 x 3 53x 2 C 33x C 596 x 4 21x 2 4x C 32 x 4 x 3 57x 2 C x C 664 x 4 2x 3 55x 2 C 56x C 172 x 4 2x 3 31x 2 C 32x C 188 x 4 2x 3 29x 2 C 30x C 208

@K X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

g 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Glossary

We list in roughly alphabetical order the basic notations which are used throughout this monograph. .a; b/ C1 .V / dK , d dR dimK V e fcg , Q GL.V / H I Hol.V / Im./ Ker./

l 1 NK=Q .a/

1

The greatest common divisor of the integral OK -ideals a and b. The space of functions which are differentiable for all degrees of differentiation defined on a C-vector space V . The different of the number field K. The principal ideal generated by the element d of the ring R. The dimension of the K-vector space V over the field K. If K D C, we shortly write dim V. The value exp 2 i tr.c/ , where c is an element of C ˝Q K 1 (K a number field). The group SL.2; O/ and its metaplectic cover (see Sect. 3.3), respectively. The group of all automorphisms of a C-vector space V . The upper half plane. The element .1; 1/ in the metaplectic cover of SL.2; O/. The space of holomorphic functions of a C-vector space V . The image of the map . The kernel of the map . The Möbius -function, i.e. the multiplicative function on the semigroup of integral OK -ideals which for prime ideal power pn assumes the values 1, 1 and 0 accordingly as n D 0 or n D 1 or n 2. The group of lth roots of unity. The group of all roots of unity. The norm of an element a in the number field K.

For the definition of the trace of an element in C ˝Q K, we refer to Sect. 3.2.


127

128

N.a/ OK , O qt R Rn S SL.n; R/ S1 Stab.x/ 0 .a/ Tb tr .a/ pK=Q z

2

Glossary

The norm of the ideal a. The ring of integers of the number field K. The function on H 2 defined by e ft g. The invertible elements of the ring R under multiplication. The R-module ofcolumn vectors of length n over the ring R. The matrix 10 1 0 . The subgroup of elements of GL.n; R/ which have determinant 1, where R is a ring. The group of all complex numbers whose absolute value equals one. The stabilizer of x under a given group action. The numberof ideals dividing the integral OK -ideal a. The matrix 10 b1 , where b is an element of the ring R. The trace of an element a in the number field K. The root of z 2 C whose argument lies in the interval . . 2 2

Here H is a the upper half plane in C ˝Q K (K a number field) as defined in Sect. 3.2.

References

[BHS14] [Bor95] [Boy14] [BS13] [BS14a] [BS14b] [Ebe02]

[EZ85] [FH91] [Fre90] [FT93] [GKZ87] [Hec81]

[Kub67] [MH73] [Neu99]

H. Boylan, S. Hayashida, N.-P. Skoruppa, On the computation p of Jacobi forms over totally real number fields with an explicit example over Q. 5/ (2014, preprint) Bordeaux Computational Number Theory Group, The number field tables (1995). http://pari.math.u-bordeaux.fr/pub/pari/packages/nftables.tgz H. Boylan, Jacobi Eisenstein series over number fields. (2014, preprint) H. Boylan, N.-P. Skoruppa, Linear characters of SL2 over Dedekind domains. J. Algebra 373, 120–129 (2013) H. Boylan, N.-P. Skoruppa, Explicit formulas for Weil representations of SL(2). (2014, preprint) H. Boylan, N.-P. Skoruppa, Jacobi forms over number fields. (2014, in preparation) W. Ebeling, Lattices and Codes. Advanced Lectures in Mathematics, revised edn. (Friedr. Vieweg & Sohn, Braunschweig, 2002). [A course partially based on lectures by F. Hirzebruch] M. Eichler, D. Zagier, The Theory of Jacobi Forms. Progress in Mathematics, vol 55 (Birkhäuser Boston Inc., Boston, 1985) W. Fulton, J. Harris, Representation Theory. Graduate Texts in Mathematics, vol 129 (Springer, New York, 1991) [A first course, Readings in Mathematics] E. Freitag, Hilbert Modular Forms (Springer, Berlin, 1990) A. Fröhlich, M.J. Taylor, Algebraic Number Theory. Cambridge Studies in Advanced Mathematics, vol 27 (Cambridge University Press, Cambridge, 1993) B. Gross, W. Kohnen, D. Zagier, Heegner points and derivatives of L-series. II. Math. Ann. 278(1–4), 497–562 (1987) E. Hecke, Lectures on the Theory of Algebraic Numbers. Graduate Texts in Mathematics, vol 77 (Springer, New York, 1981) [Translated from the German by George U. Brauer, Jay R. Goldman and R. Kotzen] T. Kubota, Topological covering of SL.2/ over a local field. J. Math. Soc. Jpn. 19, 114–121 (1967) J. Milnor, D. Husemoller, Symmetric Bilinear Forms (Springer, New York, 1973) [Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73] J. Neukirch, Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 322 (Springer, Berlin, 1999) [Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder]


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130 [Ser73] [Ser77]

[Sko85]

[Sko10]

[SS14] [SW14] [O’M00] [Vas72] [vdG88]

References J.-P. Serre, A Course in Arithmetic (Springer, New York, 1973) [Translated from the French, Graduate Texts in Mathematics, No. 7] J.-P. Serre, Linear Representations of Finite Groups (Springer, New York, 1977) [Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, vol. 42] N.-P. Skoruppa, Über den Zusammenhang zwischen Jacobiformen und Modulformen halbganzen Gewichts. Bonner Mathematische Schriften [Bonn Mathematical Publications], 159. Universität Bonn Mathematisches Institut, Bonn, 1985. Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn (1984) N.-P. Skoruppa, Finite quadratic modules, weil representations and vector valued modular forms. Notes of courses given at Universität Siegen, Harish-Chandra Research Institute and Durham University (2010) N.-P. Skoruppa, F. Strömberg, Dimension formulas for vector-valued Hilbert modular forms (2014, in preparation) N.-P. Skoruppa, L. Walling, Hecke operators for Jacobi forms over number fields (2014, in preparation) O. Timothy O’Meara, Introduction to Quadratic Forms. Classics in Mathematics (Springer, Berlin, 2000) [Reprint of the 1973 edition] L.N. Vaseršte˘ın, The group SL2 over Dedekind rings of arithmetic type. Mat. Sb. (N.S.) 89(131), 313–322, 351 (1972) G. van der Geer, Hilbert Modular Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] (Springer, Berlin, 1988)

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Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields

Quadratic Forms Over Semilocal Rings

Representations of GL(2) and SL(2) over finite fields

Quadratic reciprocity (after Weil)

Matrices and Quadratic Forms

Arithmetic of quadratic forms

Automorphic forms and representations

Representations of rings over skew fields

Automorphic Forms and Representations

Number Theory Arising From Finite Fields Knopfmacher

Introduction to quadratic forms

Arithmetic of quadratic forms

Higher-dimensional geometry over finite fields

Finite-dimensional division algebras over fields

Finite-Dimensional Division Algebras over Fields

Higher-Dimensional Geometry Over Finite Fields

Equations over Finite Fields An Elementary Approach

Foundations of analysis over surreal number fields

Foundations of analysis over surreal number fields

The Theory of Jacobi Forms

Finite Fields

Number fields

2-universal O-lattices over real quadratic fields

Modules over operads and functors

Number Theory Arising From Finite Fields: Analytic And Probabilistic Theory

Invariants of quadratic differential forms

Invariants of Quadratic Differential Forms

Tame Algebras and Integral Quadratic Forms

Quadratic forms, linear algebraic groups, and cohomology

Primary Testing and Abelian Varieties Over Finite Fields

Primality Testing and Abelian Varieties over Finite Fields

Jacobi Forms, Finite Quadratic Modules and Weil Representations over Number Fields

Quadratic Forms Over Semilocal Rings

Representations of GL(2) and SL(2) over finite fields

Quadratic reciprocity (after Weil)

Matrices and Quadratic Forms

Arithmetic of quadratic forms

Automorphic forms and representations

Representations of rings over skew fields

Automorphic Forms and Representations

Number Theory Arising From Finite Fields Knopfmacher

Introduction to quadratic forms

Arithmetic of quadratic forms

Higher-dimensional geometry over finite fields

Finite-dimensional division algebras over fields

Finite-Dimensional Division Algebras over Fields

Higher-Dimensional Geometry Over Finite Fields

Equations over Finite Fields An Elementary Approach

Foundations of analysis over surreal number fields

Foundations of analysis over surreal number fields

The Theory of Jacobi Forms

Finite Fields

Number fields

2-universal O-lattices over real quadratic fields

Modules over operads and functors

Number Theory Arising From Finite Fields: Analytic And Probabilistic Theory

Invariants of quadratic differential forms

Invariants of Quadratic Differential Forms

Tame Algebras and Integral Quadratic Forms

Quadratic forms, linear algebraic groups, and cohomology

Primary Testing and Abelian Varieties Over Finite Fields

Primality Testing and Abelian Varieties over Finite Fields

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