1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. A REVIEW OF CLASSICAL AUTOMORPHIC FORMS 219

Mellin transform, the J Bessel function appearing as the inverse Mellin transform

of the ratio of the Gamma factors. O

Remark 2.1. S. D. Miller and W. Schmidt [MiS] have derived Voronoi's summa-

tion formula by a somewhat different method based on the boundary distribution

attached to a modular form. This new approach is smooth and provides a uniform

treatment for all types of GLrforms; moreover, it extends naturally to automorphic

forms of higher degree.

One can derive a similar identity for Eisenstein series: let E(z, s) be the stan-

dard Eisenstein series for the full modular group:

E(z,s) =

1Eroo\ro(l)

and let g be its derivative at 1 /2; g has the following Fourier expansion

a

g( z ) =

08

E(z, s) is=l/2 = y^112 logy+ L T(n)n-^112 4foYcos(2nnx)K 0 (21my).

n;;,l

Under the assumptions of Lemma 2.3.1, one has

c L T(n)eC;)w(n)

n;;,l

1

+00 y'x

= 2 (log - + 1)F(x)dx

0 c

r+= [ ( an) (4nJnx)

+ ~ T(n) Jo e --z (-2nY^0 ) c

• e(a;)(4Ko)(

4

7r~)] W(x) dx.

2.4. Trace formulae
In the sequel we will average heavily over families of automorphic forms; in order
to do this, we use several trace formulae that are consequences of the spectral
decomposition of the underlying spaces. The simplest of these formulae are the
Dirichlet orthogonality relations for the characters of a finite group G: in particular,
for G = Z/qZ one has

°'"""'1 k(m-n)

~ - e( ) = Om:=n(q)

k(q) q q

(2 .20)

and for G = (Z/qZ)* one has

1

c2.21) 2= <p(q) x(m)x(n) = o m=n(q).

x(q) (mn,q)=l

For modular forms, the most natural candidate seems to be Selberg's trace formula;
however, for analytic purposes, the following Petersson-Kuznetzov type formulae
are much more efficient. We give two versions, one in the holomorphic case and
one in the Maass case: