1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. THE SUBCONVEXITY PROBLEM 259

4.4.4. Integral of products of eigenfunctions

As we have seen, the key point in the above approach is the problem of getting the

correct exponential decay for the integral of the triple product:

( 4.22) (V(z), Uj (z)) = r (.e1y )k/^2 g(.e1z) (.e2y )kf^2 g(.e2z )Uj (z) dx~y

Jx 0 (N) Y

« 9 (N(l + ltjl))A exp(-~ltjl),

as ltj I ..__, +oo, for some constant A. In fact, in applications to the SCP, what one

really needs is an averaged bound of the form

L l(V(z), Uj (z))l^2 e7rlt; I + · · · « 9 (NT)A,

lt;l~T

where... denotes the corresponding contribution from the Eisenstein spectrum.
We will not describe in detail the methods involved in these bounds; this would
bring us too far off course. We merely list the main contributors and refer to their
works.
Sarnak was the first to obtain ( 4 .22) for general modular forms [Sa3]; in fact,
his method, which is based on the analysis of the triple product in the spherical
model, is smooth and applies in quite general situations: one can replace the prod-
uct V(z) = (.e 1 y)kf^2 g(.e 1 z)(.e 2 y)kf^2 g(.e 2 z) by an arbitrary product of automorphic
forms and Uj by any primitive Maass form of appropriate weight and nebentypus.
Moreover, this method works for non-arithmetic lattices, over more general number
fields, and in several higher rank situations [PS, CoPSS].
A little later, Bernstein and Reznikov [BR, BR2] gave another proof of ( 4.22)
(for g a Maass form at least, the case of holomorphic forms is treated in [KrS]).
Their method uses techniques of complexification and analytic continuation of real
analytic vectors of representations of SL 2 (R) (computed in their various models),
as well as an innovative technique based on the concept of invariant Sobolev norms.
Like Sarnak's, their method extends to non-arithmetic lattices and, moreover, pro-
vides an essentially optimal bound in the ltj I-aspect (A can be taken arbitrarily
small). More recently, and using related ideas, Krotz and Stanton [KrS] gave
a wide generalization of the Bernstein/ Reznikov method in particular, extending
these techniques to automorphic forms on SLn.

4.4.5. Questions of uniformity I.

The problem of the exponential decay in ( 4.22) having been solved, the next step
concern the uniformity of these bounds with respect to .e 1 , .e 2 or the parameters of g.
This technical question is crucial for the solution of new instances of the SCP, and
hence new instances of the ScP: for example, the ScP problem for the symmetric
square L(sym^2 f, s) in the level aspect, and, more generally, for Rankin-Selberg £-
functions L(f x g, s), where f and g both have levels varying arbitrarily.
Concerning this issue, Sarnak's method already provides reasonable (i.e. poly-
nomial) control in the remaining parameters^5 : for instance when g is primitive,

(^5) we will discuss here only the uniformity with respect to the level q' of g