1549380232-Automorphic_Forms_and_Applications__Sarnak_

212 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS

for every 1 = ( : ~ ) E r, and that vanish at every cusp). This space is finite

dimensional and is equipped with the Petersson inner product:

1

• kdxdy
  (F, G)k = F(z )G(z)y - 2.
  r\H Y
  Such a form has a Fourier expansion at oo,

(2.2) F(z) = L pp(n)n~ e(nz).

n):l

2.1.2. Maass forms

A function f : H _.., C is said to be r -automorphic of weight k and nebentypus x iff

it satisfies

(2 .3)

for all 1 E r. Following Maass, one introduces the two differential operators of

order one

k .o o k .o o

Rk := 2 + y(i ox+ oy), Lk := 2 + y(i ox - oy),

which transform smooth automorphic functions of weight k into smooth automor-
phic functions of weight k + 2 and k - 2 respectively. Thus the Laplace operator of
weight k, given by

k k o^2 o^2 0
6.k = -Rk-2Lk - -(1--)Id= y^2 (-+ -) -iky- ,
2 2 ox^2 oy^2 ox
acts on smooth automorphic functions of weight k. The Laplacian is a self-adjoint
operator with respect to Petersson's inner product

1

dxdy

(f,g) = f(z )g(z)- 2 ,

r \ H Y

which is bounded from above by -~ ( 1 - ~), i.e.

k k

(2.4) (6.kf, f) :s; -2 (1 - 2) (!, f).

In particular, 6.k admits a self-adjoint extension to the L^2 -space of square-integrable
automorphic functions, Ck(q, x) (say); moreover, this space has a complete spectral
resolution, which we describe below.

2.1.2.1. Eisenstein series. An important class of automorphic functions is the set of
Eisenstein series (although these are not square-integrable): Eisenstein series are
indexed by the singular cusps {a} and are given by the (absolutely convergent for
~es > 1) series
Ea(z, s) = L x(!)J(T;;-1,(z)-k(~m(u;;-^11 z))8
1Era \r

where u a is the scaling matrix of the cusp a. Recall that the scaling matrix u a is the
unique matrix (up to right translations) such that

uaoo=a, u;;-

1

raua=roo={±(

1

~ ) , bEZ},