Automorphic Forms on SL2 (R)

Armand Borel

• Publisher: Cambridge University Press
• Year: 1997
• ISBN:
  9780511899881
• Topic: Mathématiques
• Subtopic: Théorie des nombres

This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upper-half plane X, with respect to a discrete subgroup G of G of finite covolume. The point of view is inspired by the theory of infinite dimensional unitary representations of G; this is introduced in the last sections, making this connection explicit. The topics treated include the construction of fundamental domains, the notion of automorphic form on G\G and its relationship with the classical automorphic forms on X, Poincare series, constant terms, cusp forms, finite dimensionality of the space of automorphic forms of a given type, compactness of certain convolution operators, Eisenstein series, unitary representations of G, and the spectral decomposition of L2 (G\G). The main prerequisites are some results in functional analysis (reviewed, with references) and some familiarity with the elementary theory of Lie groups and Lie algebras. Graduate students and researchers in analytic number theory will find much to interest them in this book.

Table of contents

• Cover
• Epigraph
• Half-title
• Title
• Copyright
• Dedication
• Contents
• Preface
• Introduction
• Prerequisites and notation
• Review of SL2(ℝ), differential operators, and convolution
• Action of G on X. Discrete subgroups of G. Reduction theory
• The unit disc model
• Growth conditions. Automorphic forms
• Poincaré series
• Constant term: the fundamental estimate
• Finite dimensionality of the space of automorphic forms of a given type
• Convolution operators on cuspidal functions
• Definition and convergence of Eisenstein series
• Analytic continuation of the Eisenstein series
• Eisenstein series and automorphic forms orthogonal to cusp forms
• Spectral decomposition of L 2(Γ\G) m with respect to C
• Generalities on representations of G
• Representations of G
• Spectral decomposition of L 2(Γ\G): the discrete spectrum
• Spectral decomposition of L 2(Γ\G): the continuous spectrum
• Concluding remarks
• References
• Notation index
• Subject index