Combining algebraic groups and number theory, this volume gathers material from the representation theory of this group for the first time, doing so for both local (Archimedean and non-Archimedean) cases as well as for the global number field case.

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.

Very well written monograph combining algebraic groups and number theory Recommended reading for researchers of modular and automorphic forms Up to date and structured collection of known results Includes supplementary material: sn.pub/extras

Autorentext

Rolf Berndt is a Professor of Mathematics at the University of Hamburg.

Ralf Schmidt is a Professor of Mathematics at the University of Oklahoma, Norman, OK, USA.

Klappentext

After Pyatetski-Shapiro[PS1] and Satake [Sa1] introduced, independent of one another, an early form of the Jacobi Theory in 1969 (while not naming it as such), this theory was given a de?nite push by the book The Theory of Jacobi Forms by Eichler and Zagier in 1985. Now, there are some overview articles describing the developments in the theory of the Jacobigroupandits autom- phic forms, for instance by Skoruppa[Sk2], Berndt [Be5] and Kohnen [Ko]. We refertotheseformorehistoricaldetailsandmanymorenamesofauthorsactive inthistheory,whichstretchesnowfromnumbertheoryandalgebraicgeometry to theoretical physics. But let us only brie?y indicate several sometimes very closely related topics touched by Jacobi theory as we see it: ?eldsofmeromorphicandrationalfunctionsontheuniversalellipticcurve resp. universal abelian variety structure and projective embeddings of certain algebraic varieties and homogeneous spaces correspondences between di?erent kinds of modular forms L-functions associated to di?erent kinds of modular forms and autom- phic representations induced representations invariant di?erential operators structure of Hecke algebras determination of generalized Kac-Moody algebras and as a ?nal goal related to the here ?rst mentioned mixed Shimura varieties and mixed motives. Now, letting completely aside the arithmetical and algebraic geometrical - proach to Jacobi forms developed and instrumentalized by Kramer [Kr], we ix x Introduction will treat here a certain representation theoretic point of view for the Jacobi theory parallel to the theory of Jacquet-Langlands [JL] for GL(2) as reported by Godement [Go2], Gelbart [Ge1] and, recently, Bump [Bu].

Inhalt
The Jacobi Group.- Basic Representation Theory of the Jacobi Group.- Local Representations: The Real Case.- The Space L2(?J\GJ (?)) and its Decomposition.- Local Representations: The p-adic Case.- Spherical Representations.- Global Considerations.