1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. THE SUBCONVEXITY PROBLEM 261

in the special case f, 1 R 2 = 1 and N = q'. By averaging over the primitive forms of

level q', one obtains

(4.26) L l(Yklg(z)l2, (f J)1/2)l2 «c:,k (QJQg)c:qt9/4(1 + lt11)2k+2,

lt1 1 ::;;r ,

still weaker than ( 4 .24) by a factor of T^2 q'^114. Eventually ( 4 .26) can be improved

further with subconvexity estimates, in particular the known subconvexity bounds

for L(nJ, s), but one can do much better. Indeed, by the approximate functional

equations, the central values L(sym^2 n 9 0 n 1 ,1/2) and L(n 1 , 1 /2) are represented

by linear forms in the >.1(n) of respective length (1 + lt11)^3 q'^2 and (1+1t 1 1)q'^112 ;

hence, after Cauchy/Schwarz, one bounds this product of linear forms on average

by the large sieve inequalities (3.8), resulting in

(4.27) L l(Yklg(z)l2, (f,J)l/ 2 )12e7rlt11 «c:,k (q'T)c:T2k+1/2q13/2,

lt1l::;;T

where the above sum ranges over primitive forms of level q' exactly. The above

bound is stronger than ( 4 .24) (in the q' aspect) by a factor of q'^1 /^2 and could be

improved further if necessary. Of course, for applications to the SCP one needs

to perform the average over old forms and in the case when f, 1 f, 2 is not equal to

one. One can reasonably expect that the contribution from the old forms is not

larger than the contribution of the new forms, and that dependence in f, 1 f, 2 is at

most polynomial; in particular, the following bound is certainly within reach of the

current techniques:

(4.28) L l(uj, V)l^2 e7rltJI +Eisenstein contr. «k (R 1 R 2 )Aq'^3 i^2 T^2 k+l/^2 ,

ltJl::;;T

for some absolute constant A. This is more than sufficient to solve completely many

new cases of the ScP (see below). Proving ( 4.28) along the above lines requires a

generalization of the triple product identities. More precisely, one expects a formula

similar to (4.25), relating

where f is a primitive Maass form oflevel q"IR1 f2q', and where f3lf1 R2q' /q", to the

L-value

A(sym^2 n 9 , 1)^2 A(sym^2 n 1 , 1)'

with possibly some extraneous local factors at the primes dividing q' f, 1 f, 2.

4.5. Subconvexity for Rankin-Selberg £-functions

We conclude this lecture with the Subconvexity Problem for Rankin-Selberg con-
volutions L(f 0 g, s), more precisely, when g is fixed and one of the parameters
of f varies. In this section, we describe the q aspect; in fact the other aspects
are very similar to this one and, in particular, all are related to some form of the
SCP. However, the level aspect presents an additional interesting feature, which
we would like to point out. The investigations of the level aspect began with the