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Siegel Modular Forms

A Classical and Representation-Theoretic Approach

  • Book
  • © 2019

Overview

Authors:
  1. Ameya Pitale

    1. Mathematics, University of Oklahoma, Norman, USA

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  • 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

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

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-ix
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  2. Introduction to Siegel Modular Forms

    • Ameya Pitale
    Pages 1-7

  3. Examples

    • Ameya Pitale
    Pages 9-14

  4. Hecke Theory and L-Functions

    • Ameya Pitale
    Pages 15-22

  5. Nonvanishing of Fourier Coefficients and Applications

    • Ameya Pitale
    Pages 23-29

  6. Applications of Properties of L-Functions

    • Ameya Pitale
    Pages 31-37

  7. Cuspidal Automorphic Representations Corresponding to Siegel Modular Forms

    • Ameya Pitale
    Pages 39-48

  8. Local Representation Theory of \(\mathrm{GSp}_4({\mathbb Q}_p)\)

    • Ameya Pitale
    Pages 49-62

  9. Bessel Models and Applications

    • Ameya Pitale
    Pages 63-74

  10. Analytic and Arithmetic Properties of \(\mathrm{GSp}_4 \times \mathrm{GL}_2\) L-Functions

    • Ameya Pitale
    Pages 75-82

  11. Integral Representation of the Standard L-Function

    • Ameya Pitale
    Pages 83-91

  12. Back Matter

    Pages 93-138
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Keywords

About this book

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 areaas 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.



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)

Authors and Affiliations

  • Mathematics, University of Oklahoma, Norman, USA

    Ameya Pitale

Bibliographic Information

  • Book Title: Siegel Modular Forms

  • Book Subtitle: A Classical and Representation-Theoretic Approach

  • Authors: Ameya Pitale

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-15675-6

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2019

  • Softcover ISBN: 978-3-030-15674-9Published: 08 May 2019

  • eBook ISBN: 978-3-030-15675-6Published: 07 May 2019

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: IX, 138

  • Number of Illustrations: 112 b/w illustrations

  • Topics: Number Theory, Group Theory and Generalizations

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