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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)
List 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.
About the author
Prof. Dr. Rolf Berndt ist am Mathematischen Seminar der Universität Hamburg tätig.
