Jacobi forms, finite quadratic modules and Weil representations over number fields
type: Item
instance of: scholarly article
title: Jacobi forms, finite quadratic modules and Weil representations over number fields
zbMATH Open document ID: 1317.11002
DOI: 10.1007/978-3-319-12916-7
author: Hatice Boylan
published in: Lecture Notes in Mathematics
publication date: 2014-12-04T00:00:00Z
review text: 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…
reviewed by: Shaul Zemel
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11F50
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11R80
zbMATH DE Number: 6376892
zbMATH Keywords: Weil representation
zbMATH Keywords: Jacobi forms
zbMATH Keywords: Hilbert modular forms
zbMATH Keywords: positive definite lattice
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-12916-7
OpenAlex ID: W586525648
description: scientific article; zbMATH DE number 6376892