1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1

Mostly SL(2)

1.1. In this lecture we will describe, for the simplest group G = SL(2, Q), the

phenomena of spectral analysis that we will study later for more general groups. In

particular we recall, in this case, the different aspects of the Ramanujan conjecture,

and what is known about it or approximations of it.

We then explain the formulation of these results or problems in terms of lo-

cal representations theory, and how the basic formalism for (unramified) Hecke

operators extends to higher groups.

There is some overlap with the lectures of Sarnak, Cogdell and (for the end) of

Shahidi. We refer the reader to these lectures for more information, but we have

tried to give a self-contained introduction to our material in this simple case.

1.2. Assume H is the Poincare half-plane

H = {z EC: Im(z) > O} ,

and r is SL(2, Z). Then r acts on H by z 1--7 ~;:~. Consider the action of the

hyperbolic Laplacian

on A:= L^2 (r\H).

This is, of course, an unbounded operator, but it is known to be (essentially)

self-adjoint^1. Thus it has a spectral decomposition, which we write:

A r+oo 1

(1.1) A= C.il. EB EB Cfn EB Jo E(., 2 + it)dt.

ne::1 O
This is a Hilbert direct sum. The first term is composed of scalar multiples of
the constant function il.(z) = 1. The fn are the Maass cusp forms. The Eisenstein
series E(z, ~+it) (z EH) is obtained by analytic continuation to the line Re(s) = ~
of the non-holomorphic Eisenstein series
1
E(z, s) = 2 y^5 L lcz +di-Zs (z EH, Re(s) > 1).
(c,d)=l

(^1) For an excellent recent reference for the spectral theory see Iwaniec [29]. Another good reference
is Kubota [37].
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