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Elements of the Representation Theory of the Jacobi Group

  • Rolf Berndt
  • Ralf Schmidt

Part of the Modern Birkhäuser Classics book series (MBC)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Rolf Berndt, Ralf Schmidt
    Pages 1-14
  3. Rolf Berndt, Ralf Schmidt
    Pages 15-30
  4. Rolf Berndt, Ralf Schmidt
    Pages 31-73
  5. Rolf Berndt, Ralf Schmidt
    Pages 75-103
  6. Rolf Berndt, Ralf Schmidt
    Pages 105-136
  7. Rolf Berndt, Ralf Schmidt
    Pages 137-172
  8. Rolf Berndt, Ralf Schmidt
    Pages 173-200
  9. Back Matter
    Pages 201-213

About this book

Introduction

The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group. It is an important example for a non-reductive group and sets the frame within which to treat theta functions as well as elliptic functions - in particular, the universal elliptic curve.

This text gathers for the first time material from the representation theory of this group in both local (archimedean and non-archimedean) cases and in the global number field case. Via a bridge to Waldspurger's theory for the metaplectic group, complete classification theorems for irreducible representations are obtained. Further topics include differential operators, Whittaker models, Hecke operators, spherical representations and theta functions. The global theory is aimed at the correspondence between automorphic representations and Jacobi forms. This volume is thus a complement to the seminal book on Jacobi forms by M. Eichler and D. Zagier.

Incorporating results of the authors' original research, this exposition is meant for researchers and graduate students interested in algebraic groups and number theory, in particular, modular and automorphic forms.

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The book is very well written and gives an up to date collection of the results known. It will be quite useful for everyone working in the field.

(Zentralblatt MATH)

 

This book is certainly recommended for researchers interested in modular and automorphic forms.

(Mathematical Reviews)

 

This book complements the book on Jacobi forms by M. Eichler and D. Zagier and includes many of the author's original results. It can be read, however, independently of other sources, and it will be very useful for researchers interested in algebraic groups and number theory.

(Mathematica)

Keywords

Grad Group Theory algebraic geometry algebraic group number theory representation theory

Authors and affiliations

  • Rolf Berndt
    • 1
  • Ralf Schmidt
    • 2
  1. 1., Department MathematikUniversität HamburgHamburgGermany
  2. 2., Department of MathematicsUniversity of OklahomaNormanUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-0283-3
  • Copyright Information Birkhäuser Verlag, reprint by Springer Basel AG 1998
  • Publisher Name Springer, Basel
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-0348-0282-6
  • Online ISBN 978-3-0348-0283-3
  • Buy this book on publisher's site
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