Edited by J. Coates
Edited by M. J. Taylor
Publisher: Cambridge University Press
Print Publication Year: 1991
Online Publication Date:December 2009
Online ISBN:9780511526053
Paperback ISBN:9780521386197
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511526053.003
Subjects: Number theory
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PREFACE
This article follows the format of five lectures that we gave on automorphic L-functions. The lectures were intended to be a brief introduction for number theorists to some of the main ideas in the subject. Three of the lectures concerned the general properties of automorphic L-functions, with particular reference to questions of spectral decomposition. We have grouped these together as Part I. While many of the expected properties of automorphic L-functions remain conjectural, a significant number have now been established. The remaining two lectures were focused on the techniques which have been used to establish such properties. These lectures form Part II of the article.
The first lecture (§I.1) is on the standard L-functions for GLn. Much of this material is familiar and can be used to motivate what follows. In §I.2 we discuss general automorphic L-functions, and various questions that center around the fundamental principle of functoriality. The third lecture (§I.3) is devoted to the spectral decomposition of L2(G(F) \ G(A)). Here we describe a conjectural classification of the spectrum in terms of tempered representations. This amounts to a quantitative explanation for the failure of the general analogue of Ramanujan's conjecture.
There are three principal techniques that we discuss in Part II. The lecture §II.1 is concerned with the trace formula approach and the method of zeta-integrals; it gives only a skeletal treatment of the subject.
pp. i-iv
pp. v-vi
pp. vii-vii
pp. viii-viii
Lectures on automorphic L-functions: Read PDF
pp. 1-60
Gauss sums and local constants for GL(N): Read PDF
pp. 61-74
L-functions and Galois modules: Read PDF
pp. 75-140
Motivic p-adic L-functions: Read PDF
pp. 141-172
The Beilinson conjectures: Read PDF
pp. 173-210
Iwasawa theory for motives: Read PDF
pp. 211-234
Kolyvagin's work for modular elliptic curves: Read PDF
pp. 235-256
Index theory, potential theory, and the Riemann hypothesis: Read PDF
pp. 257-270
Katz p-adic L-functions, congruence modules and deformation of Galois representations: Read PDF
pp. 271-294
Kolyvagin's work on Shafarevich-Tate groups: Read PDF
pp. 295-316
Arithmetic of diagonal quartic surfaces I: Read PDF
pp. 317-338
On certain Artin L-Series: Read PDF
pp. 339-352
The one-variable main conjecture for elliptic curves with complex multiplication: Read PDF
pp. 353-372
Remarks on special values of L-functions: Read PDF
pp. 373-392