Summary: Lectures on automorphicL-functions
JAMES ARTHUR AND STEPHEN GELBART
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 I1 of the article.
The first lecture (51.1) is on the standard L-functions for GLn. Much of this
material is familiar and can be used to motivate what follows. In $1.2 we
discuss general automorphic L-functions, and various questions that center
around the fundamental principle of functoriality. The third lecture ($1.3) 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 represen-
tations. This amounts to a quantitative explanation for the failure of the
general analogue of Ramanujan's conjecture.