Jacobi forms, finite quadratic modules and Weil representations over number fields
The classical theory of Jacobi forms, and its connections to elliptic modular forms, have been a constant subject of research for many decades. The relation to the Weil representations of \(\mathrm{SL}_{2}(\mathbb{Z})\) (and its metaplectic double cover) arising from finite quadratic modules and to vector-valued modular forms have also been established, enriching the theory in another intriguing direction. Now, elliptic modular forms have a natural generalization, which is defined over larger (totally real) number fields -- Hilbert modular forms. A natural question is thus generalize the other notions in a way which will be defined over larger number fields. Such generalizations, however, encountered many technical difficulties, which inhibited the development of the theory for many years. One such problem arises from the fact that not every finite quadratic module over the ring of integers in a larger number field arises from a lattice over that ring. In addition, the definition of index of a Jacobi form did not seem to extend well to this more general context, making it difficult to understand what kind of functions should be called Jacobi forms over number fields. The book under review manages to overcome most of these technical problems, and develop the basics of this desired theory. Moreover, it carries on this task in a clearly written, self-contained, and elementary way, not assuming former knowledge of the classical theory over \(\mathbb{Q}\). For example, an idea of the author of the book and \textit{N.-P. Skoruppa} [J. Algebra 373, 120--129 (2013; Zbl 1281.20050)], suggests to replace the index of a Jacobi form by a positive definite lattice. This notion extends in a natural manner to lattices over rings of integers in larger fields, making the definition of Jacobi forms over totally real number fields feasible. Moreover, many results from the classical case over \(\mathbb{Q}\), carry over, with almost no additional difficulty, to this more general setting. In more detail, the book is divided into 4 chapters. The first chapter considers the basic object required for the Weil representation -- finite quadratic modules over rings of integer in a number field \(K\). These are finite, torsion modules over such rings, which are endowed with a quadratic form taking values in \(K\) modulo the inverse different. A basic, rank 1 case of these modules are the cyclic ones, a detailed investigation of which occupies most of this chapter. The next chapter then investigates the Weil representations associated to finite quadratic modules over \(K\). It starts with a section collecting the results of the basic theory of complex representations of finite groups and their characters, for the benefit of the reader who may not be familiar with this theory. It then turns to the Weil representations of \(\mathrm{SL}_{2}\) of the ring of integers (or its metaplectic cover). This representation may be defined by the action of the generators \(T^{b}=\binom{1\;\;b}{0\;\;1}\) and \(S=\binom{0\;\;-1}{1\;\;\;\;0}\), whose formulae is simple and classical (and follows from Weil's general theory). The remainder of the chapter is concerned with the reducibilty properties of these representations. The third chapter defines the main objects of interest of this book, which are Jacobi forms over a totally real number field \(K\). It begins with introducing lattices over such rings and their discriminant forms (which are finite quadratic modules), and the metaplectic double cover of \(\mathrm{SL}_{2}(K\otimes\mathbb{R})\). To such a lattice (if it is even) one then assigns a Heisenberg group and a Jacobi group (both with interpretations as differential operators), as well as a Jacobi theta function. Following the latter example, one defines the general Jacobi form having this lattice as an index, and relates them to vector-valued Hilbert modular forms. The chapter closes with an appendix describing the case of odd lattices. The last chapter is dedicated to the investigation of \textit{singular} Jacobi forms, in which only special Fourier coefficient may not vanish. They can also be characterized by their weight or by being spanned by Jacobi theta functions. It continues with investigating the decomposability of the space of theta functions, and with a detailed analysis of the space case of rank 1. The book concludes with presenting ways to construct other Jacobi forms (e.g., Eisenstein and Poincaré series) as well as operations on them (products and pullbacks), and supplies some tables of fields admitting singular Jacobi forms. As mentioned above, the book is well written, and it presents the problems, as well as their solutions, in a manner which is clear and understandable also to newcomers into the subject. The reader should be aware though of the fact that some conventions are stated in the beginning (under ``notations''), while others appear at the end (under ``Glossary''), with the reference in the text not always being accurate. This, however, should not affect the overall impression, that this book is valuable contribution to the mathematical society, and serves as a welcoming invitation to anyone who finds interest in engaging him/herself in researching this beautiful new theory.
scientific article; zbMATH DE number 6376892