Elements of the Representation Theory of the Jacobi Group / Edition 1

Elements of the Representation Theory of the Jacobi Group / Edition 1

by Rolf Berndt, Ralf Schmidt
     
 

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ISBN-10: 303480282X

ISBN-13: 9783034802826

Pub. Date: 01/05/2012

Publisher: Springer Basel

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

Overview

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.

Product Details

ISBN-13:
9783034802826
Publisher:
Springer Basel
Publication date:
01/05/2012
Series:
Modern Birkhauser Classics Series
Edition description:
1998
Pages:
213
Product dimensions:
6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

​Introduction.- 1 The Jacobi Group.- 2 Basic Representation Theory of the Jacobi Group.- 3 Local Representations: The Real Case.- 4 The Space and its Decomposition.- 5 Local Representations: The p-adic Case.- 6 Spherical Representations.- 7 Global Considerations.- Bibliography.- Index of Notations.- Index.

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