Rolf Berndt

L-functions for Jacobi forms à la Hecke

Heckes method to associate anL-function to a modular form by a Mellin transform is applied here to Jacobi forms. One comes up with a functional equation and some connection to Shimuras theory for modular forms of half integral weight.

Einführung in die symplektische Geometrie

Einige Aspekte der Theoretischen Mechanik - Symplektische Algebra - Symplektische Mannigfaltigkeiten - Hamiltonsche Vektorfelder und Poissonklammern - Die Impulsabbildung - Quantisierung - Anhang

On automorphicL-functions for the Jacobi group of degree one and a relation withL-functions for Jacobi forms

We try to attach anL-function to an automorphic representations of the Jacobi group by defining local factors via certain zeta-integrals. We come up with two kinds of factors which are compared to factors appearing in theL-functions associated to Jacobi forms (of index 1).

On local Whittaker models for the Jacobi group of degree one

The lowest dimensional Jacobi groupGJ sets a link between the theory of Siegel modular forms of degree two and the elliptic modular forms of integral and half integral weight. This note is meant to help finding a way to associateL-functions to automorphic representations of the groupGJ by using the approach via Whittaker models of these representations. Thus, here the question of existence and unicity of these models is discussed. This question may be reduced to a closer study of the Schrödinger and Weil representation of the Heisenberg resp. the metaplectic group and thus to the application of some results by Waldspurger.

The Jacobi Group

The Jacobi group is a semidirect product of a (semisimple) symplectic group with a (nilpotent) Heisenberg group. It comes along in several presentations which may be more or less appropriate for the different parts of the theory. So we will discuss here several realizations and change from one to the other from time to time. To keep track it is helpful to think of the Jacobi group as a certain subgroup of a bigger symplectic group.

Einige Aspekte der Theoretischen Mechanik

Symplektische Strukturen ergeben sich in naturlicher Weise in der theoretischen Mechanik, und zwar insbesondere bei der Quantisierung, d.h. dem Ubergang von der klassischen zur Quantenmechanik. Zur Motivation fur die symplektische Geometrie soll dies hier zu Beginn in groben Umrissen vorgestellt werden. Als Leitfaden kann dabei Ch. 1 aus Vaisman [V] genommen werden. Ausfuhrlichere Darstellungen der Prinzipien der klassischen Mechanik finden sich bei Arnold [A] Ch. 3 und Abraham-Marsden [AM] Ch.3 u.5, wobei allerdings zum Teil Dinge benutzt werden, die hier erst in den nachsten Kapiteln systematisch behandelt werden. Eine weitere sehr empfehlenswerte klassische Quelle ist Siegel-Moser [SM] Ch. 1. Fur den Quantisierungsprozes sei schon jetzt auf Kirillov [Ki] §15.4 verwiesen. Es ist ein Ziel dieses Textes, spater eingehender auf die hier in 0 angerissenen Dinge zuruckzukommen.

Hamiltonsche Vektorfelder und Poissonklammern

Schon in 0.4 und 0.5 wurde darauf hingewiesen, das sich die der theoretischen Mechanik zugrundeliegenden Hamiltonschen Gleichungen mit Hilfe des Formalismus der symplektischen Geometrie elegant formulieren (und dann auch bearbeiten) lassen. Diese Formulierung soll hier ausgefuhrt werden. Quellen dafur sind u.a. [Ki] p. 231–3, [GS] p. 88 ff sowie [AM] p. 187–208.

Local Representations: The p -adic Case

In this chapter we turn to the local non-archimedean case and study the representation theory of G J (F), where F is a finite extension of some \( {\mathbb{Q}_p} \).We will reach the goal of classifying all irreducible, admissible representations of G J (F) by using the fundamental relation \(\pi \simeq \tilde{\pi } \otimes \pi _{{SW}}^{m} \) and the classification of representations of the metaplectic group given by Waldspurger in [Wa1]. Roughly speaking, all non-supercuspidal irreducible representations of Mp are obtained by parabolic induction. This result can be taken over to the Jacobi group by making the isomorphism \(\pi \simeq \tilde \pi \otimes \pi _{SW}^m \) explicit in the context of standard models for induced representations (Theorem 5.4.2).

Local Representations: The Real Case

Here we rearrange and extend material from [Bel]—[Be4] and [BeBo]. By the general theory from the last section, we have as a fundamental object the Schrodinger-Weil representation \( \pi \begin{array}{*{20}{c}} m \\ s \end{array}W\) which is a genuine representation of the metaplectic cover \( {\tilde G^J}\left( \mathbb{R} \right)\) and may be identified with a projective representation of \( {G^J}\left( \mathbb{R} \right)\) If we tensorize \( \pi \begin{array}{*{20}{c}} m \\ s \end{array}W\) with another genuine representation \( \bar \pi\) of the metaplectic cover Mp \( \left( \mathbb{R} \right)\) (again to be identified with a projective representation of SL2 \( \left( \mathbb{R} \right)\) ) we get