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Siegel Modular Forms

A Classical and Representation-Theoretic Approach

  • Explores the connections between the classical and representation theoretic approaches in the study of Siegel modular forms

  • Illustrates how basic concepts can be applied to solve contemporary problems in the study of modular forms

  • Based on lecture notes delivered during a workshop at the Indian Institutes of Science Education and Research, Pune, India from 2017

Book
  • 7.8k Downloads

Part of the Lecture Notes in Mathematics book series (LNM, volume 2240)

Table of contents

About this book

Introduction

This monograph introduces two approaches to studying Siegel modular forms: the classical approach as holomorphic functions on the Siegel upper half space, and the approach via representation theory on the symplectic group. By illustrating the interconnections shared by the two, this book fills an important gap in the existing literature on modular forms. It begins by establishing the basics of the classical theory of Siegel modular forms, and then details more advanced topics. After this, much of the basic local representation theory is presented. Exercises are featured heavily throughout the volume, the solutions of which are helpfully provided in an appendix. Other topics considered include Hecke theory, Fourier coefficients, cuspidal automorphic representations, Bessel models, and integral representation.

Graduate students and young researchers will find this volume particularly useful. It will also appeal to researchers in the area as a reference volume. Some knowledge of GL(2) theory is recommended, but there are a number of appendices included if the reader is not already familiar.

Keywords

Siegel modular forms Symplectic group Automorphic representations L-functions Fourier coefficients Deligne's conjecture Representation theory Number theory Shimura-Taniyama-Weil conjecture Fermat's Last Theorem Hecke algebra

Authors and affiliations

  1. 1.MathematicsUniversity of OklahomaNormanUSA

Bibliographic information

  • Book Title Siegel Modular Forms
  • Book Subtitle A Classical and Representation-Theoretic Approach
  • Authors Ameya Pitale
  • Series Title Lecture Notes in Mathematics
  • Series Abbreviated Title Lect.Notes Mathematics
  • DOI https://doi.org/10.1007/978-3-030-15675-6
  • Copyright Information Springer Nature Switzerland AG 2019
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-030-15674-9
  • eBook ISBN 978-3-030-15675-6
  • Series ISSN 0075-8434
  • Series E-ISSN 1617-9692
  • Edition Number 1
  • Number of Pages IX, 138
  • Number of Illustrations 112 b/w illustrations, 0 illustrations in colour
  • Topics Number Theory
    Group Theory and Generalizations
  • Buy this book on publisher's site
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Reviews

“This book does a very good job of introducing, clearly and concisely, key definitions, concepts and foundational results, necessarily leaving much of the detail to exercises and references, while also giving a flavour of current research … . It seems very much suited to its original target audience, but others, even those like me who couldn't be bothered doing the exercises, will find it an enlightening read, and will benefit from the well-judged choice of topics.” (Neil P. Dummigan, Mathematical Reviews, December, 2019)