1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

Theorem 2.1. Fork? 2, let Bk(q, x ) denotes an orthogonal basis of Sk(q, x). Then
for any m, n? 1 one has

f(k - 1) ~ l/2--

( 47r)k-l ~ (mn) PJ(m)pJ(n) = Om=n + b.(m,n)

fEBk(q,x)

with

b.(m, n) := 27fi-k L Bx(m,n;c) Jk-1(47fvmn)

c c

c=:O(q)

c>O

and Bx(m, n ; c) being the (twisted) Kloosterman sum

Bx(m,n;c)= L x(x)e(mx+nx).

x(c),(x,c)=l c

Remark 2.2. There is no such formula for holomorphic forms of weight 1.

We note the following property of Bessel functions ( see [Wats], p. 206)

(2.22)

where

W(x ) = ei~~I<-~; (2 r= e-Y(y(l + iy ))"-~dy.

r '"' + 2 v ;;:;; J 0 2x

When '"' is a positive integer, we derive (using the Taylor expansion for J"(x) if

0 < x :::;; 1, or the above integral expression for W ( x ) if x? 1) the following

bounds for the derivatives of W:

(2.23) x1. W ( ") J (x ) « (1 + x x)3/2

for any j? 0, the implied constant depending on j and '"'·

In the case of Maass forms, we borrow the following version of the Kuznetsov

formula from [DFI8]. Given Bk(q,x) = {uj}J~ 1 an orthonormal basis of Ck(q,x)

composed of Maass cusp forms with eigenvalues >..j = 1/4 + t; and Fourier coeffi-

cients Pj(n); or any real number rand any integer k, we set

47f^3 1

h(t) = h(t, r) = .-------

II'(l - ~ - ir)l^2 cosh 7r(r - t) cosh 7r(r + t)

(2.24)

Theorem 2.2. For any positive integers m, n and any real r, one has

(2.25)

= Om=n + L Bx(m, n ; c) J(47fvmn),

c=:O(q) c c

where I ( x ) is the Kloosterman integral

I(x ) = I(x, r) = -2x 1-ii (-i()k-l K2ir((x)d(.