Andre Reznikov (Bar-Ilan University)

Title

ANALYTIC ASPECTS OF AUTOMORPHIC FORMS AND REPRESENTATION

Abstract

The study of periods of automorphic forms plays an important role in the analytic theory of L-functions and more generally in analytic number theory.

Our aim is to explain how methods from the representation theory of the group SL(2,R) allow one to obtain meaningful bounds for the Fourier coe±cients of cusp forms and for various automorphic L-functions. In particular, we will explain the representation-theoretic counterpart of the highly successful method developed by Peter Sarnak in order to treat Dirichlet series naturally appearing in the theory of automorphic forms.

We start with an elementary review of the representation theory of the group SL(2,R) making emphasis on a geometric description of irreducible representations and on the fundamental uniqueness principle. This will allow us to develop techniques which will simplify the analysis once we turn to automorphic forms.

Recommended literature

◆ Still the best source for a beginner is the classic book

Gelfand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I.
Representation theory and automorphic functions. Generalized Functions, 6.
Academic Press, 1969 or 1990 editions.

◆ Other excellent books treating automorphic forms through representation theory

Borel, A. Automorphic forms on SL(2,R).
Cambridge Tracts in Mathematics, 130. Cambridge, 1997.

An introduction to the Langlands program.
A collection edited by J. Bernstein and S. Gelbart.
Birkhuser Boston, 2003.

Gelbart, S. Automorphic forms on adele groups.
Annals of Mathematics Studies, No. 83. Princeton University Press, Princeton, 1975.

And a very extensive book
Bump, D. Automorphic forms and representations.
Cambridge Studies in Advanced Mathematics, 55, Cambridge, 1997.

There are also available on-line notes of the course given by W. T. Gan at Hangzhou (Summer 2004).

These notes cover the representation-theoretic background and its relation to the classical language of modular forms. Although we will take a more analytic approach, these notes should be very useful. Notes are available at: www.cms.zju.edu.cn (and at http: www.math.ucsd.edu/~wgan/).