Automorphic forms on SL₂(R)

by Borel, Armand

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Title Statements

Full title :Automorphic forms on SL₂(R) / Armand Borel.

Summary

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.

Edition Statements

Publication Date:1997.

Publication Statements

Publisher(s):Cambridge University Press
Place of Publication:Cambridge
Language:eng
Publication Date:1997.

Notes

Bibliographic Level Mode of Issuance: Monograph.

The Physical Object

Description:1 online resource (x, 192 pages)

Edition Identifiers

ISBN:0511896069, 9780511896064, 9780521580496, 0521580498, 9780521072120, 0521072123
oclc:on1180856089
LC Classification:QA331
Dewey Decimal:515/.9

Part I. Basic Material On SL2(R), Discrete Subgroups and the Upper-Half Plane: -- 1. Prerequisites and notation -- 2. Review of SL2(R), differential operators, convolution -- 3. Action of G on X, discrete subgroups of G, fundamental domains -- 4. The unit disc model -- Part II. Automorphic Forms and Cusp Forms: -- 5. Growth conditions, automorphic forms -- 6. Poincare series -- 7. Constant term:the fundamental estimate -- 8. Finite dimensionality of the space of automorphic forms of a given type -- 9. Convolution operators on cuspidal functions -- Part III. Eisenstein Series: -- 10. Definition and convergence of Eisenstein series -- 11. Analytic continuation of the Eisenstein series -- 12. Eisenstein series and automorphic forms orthogonal to cusp forms -- Part IV. Spectral Decomposition and Representations: -- 13. Spectral decomposition of L2(G\G)m with respect to C -- 14. Generalities on representations of G -- 15. Representations of SL2(R) -- 16. Spectral decomposition of L2(G\SL2(R)): the discrete spectrum -- 17. Spectral decomposition of L2(G\SL2(R)): the continuous spectrum -- 18. Concluding remarks.