1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

one obtains the averaged bound for T ~ 1:

(4.21) L l(uj, V)l^2 e7rlt; I +

4

~ L 1T l(E 0 (. , ~+it), V )l^2 e7rltldt

lt;l~T a O

«k,e (NT)"llYk/2gll~N2T2k+4.

Applying Stirling's formula, Cauchy-Schwarz, (4.18) and (4.21) in (4.16), one con-
cludes the proof of Theorem 4 .11. 0

4.4.3. The SCP for Maass forms and for Eisenstein series

A similar analysis can be carried out when g is a Maass cusp form or an Eisenstein
series. When g is a primitive Maass forms of weight zero with Laplace eigenvalue
1/4 + t^2 , one defines (see [Sa4]):

Theorem 4.12. Assume H 2 (B) holds for all Maass cusp forms of weight zero with

trivial nebentypus. For any c > 0, D (g, s; e 1 , £ 2 , h) extends holomorphically to C5 : =

3tes ~ 1/2 + B + c and in this region it satisfies the upper bound

where s = e5 + it and the implied constant depends only on c and g only.

For g an Eisenstein series, Theorems 4.11 and 4.12 are still valid, with the main

difference being that D(g, s; £ 1 , £ 2 , h) acquires a pole at s = 1 of order at most 3:

this pole comes from the constant terms of the Eisenstein series and its exact value

can be computed by a (standard) regularization process, which, for example, is

carried out in the paper of Tatakhjan and Vinogradov [TV].

Remark 4.11. However, let us mention that for g non-holomorphic, there is still a

difficulty in relating the estimate of Thm. 4 .12 to the original shifted convolution

problem (i.e. in replacing

by a more general test function W(£ 1 m,£ 2 n )). This difficulty can be solved if h

is small with respect to Y by expanding the hypergeometric function in its Taylor

series. The general case (i.e. lhl as large as Y) is currently being investigated with

some success by G. Harcos.