1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. A REVIEW OF CLASSICAL AUTOMORPHIC FORMS 221

In fact, this formula is not quite sufficient for all purposes. In order to have a

function on the right hand side with rapid decay as c grows, one can perform an

extra averaging over r (see [DFIS] Sect. 14). Given A a fixed large real number,

we set

(2.26)

r sinh 27rr 7rr )-4A

q(r)= (r2+A2)8 (cosh2A.

Integrating q(r)h(t, r ) over r we form

(2.27) Ji(t) = l h(t, r)q(r )dr and I(x ) = l I(x , r)q(r )dr.

Correspondingly, one has

(2.28)

_ , ~ Sx(m, n;c),r(47ry'm71)

- CAUm = n + L_., .L

c:=O(q) c c

where CA is the integral of q(r) over R. We collect below the following estimates
for I and Ji (see [DFI8] sect.14 and 17).
For t real or purely imaginary, one has

(2.29)

For all j ~ 0, we have

(2.30)

Such formulae are obtained by considering two Poincare series Pm(z ) and
Pn(z), and by computing their scalar products (Pm, Pn) in two different ways. The
first one is direct and based on the definition of the Poincare series: the resulting
expression involves Kloosterman sums. The other way is by applying the spectral
decomposition (2 .5), Parseval's formula, and the fact that (Pm, f) is proportional
to p 1 ( m). In the case of Maass forms, there is some flexibility in the choice of
the Poincare series and one could obtain similar formulae with more general test
functions in place of h(t) or Ji(t), but this primitive version is sufficient for the ap-
plications given in these lectures. As was shown by Kuznetsov, it is also possible to
replace I by a fairly general test function; this in turn enables one to connect the
distribution of Kloosterman sums to automorphic forms. We refer to [DI, Du, Pr]
for more general versions of Kuznetsov's formula, in particular, for forms of half-
integral weight, and also to the book of Cogdell and Piatetski-Shapiro, where a rep-
resentation theoretic derivation of Petersson/ Kuznetsov's formula is given [CoPS].
Using Weil's bound for Kloosterman sums

(2.31) Is x (m ' ' n · c) I ~ ""'-;::: 2wccl(m ' , n c)^112 c^112 '

and the following bounds for the Bessel functions

xk- 1 x

Jk-1 (x ) « min(l, (k _ l)!) « ( k )a, for <r E [O, l],