1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

By elementary techniques (Cauchy/Schwarz, the Polya/Vinogradov completion

technique and the Weil bound for Kloosterman sums), it is possible to improve the

trivial bound as long as log M /log N is away from 1. In [DFIS], Duke/Friedlander/

Iwaniec succeeded in improving on the trivial bound in the remaining range (and

uniformly for 0 < la l ~ MN):

Theorem 4.7. For any c > 0, one has

B(M, N) «c (L laml2 L lbn l2)lf2(1al + M N)l4/29(M + N)l/58+c.

m n

In fact this technical estimate is a key point in the proof of Theorem 4.1. But

the most remarkable feature of this bound lies in its proof, which uses the ampli-

fication method in a completely unexpected way. Indeed, the bound follows by

amplification of the contribution of the trivial (a priori invisible) character, in the

averaged second moment

1 -

V(M,N)= 2::-(-) 2::12::x(l)cd^21 L x(n)bne(am)l2

m~M ~m x(m) l~L n~N n

(m,n)=l

where x ranges over the characters of moduli m!

4.3. Application to the Subconvexity Problem

In this section we explain how the amplification method can be used to solve Sub-

convexity Problems for GL 2 and GL 2 x GL 2 £-functions.

We start by establishing a subconvex bound for twisted £-functions in the q-

aspect, basically due to Duke/ Friedlander/ Iwaniec [DFil] (but see also [Ha2, Mi]):

Theorem 4.8. Let g E Sf ( q', x', it'), and let x .be a primitive Dirichlet character of

modulus q. Then

L(x.g ' s) « £,g,s q1/2-1;22+" '

where the implied constant depends polynomially on s and the parameters of g.

Unless otherwise specified, we assume that g is holomorphic.

4.3.1. The case of L(g.x, s): reduction to the Shifted Convolution Problem

For simplicity, we give the argument at s = 1 /2. By the approximate functional

equation (1.30), we have

L(g.x, 1/2) = L Ag(n;~(n) v('.12'.) + w(g.x) L >:";(n;~(n) v('.12:),

n n q n n q

were V is smooth with rapid decay. The crucial range being n ,.._, q, we will simply

assume the V is supported on the interval [1/2, 1]; we wish to bound non-trivially

the sum

I; (g x) ·= ~ >.g(n)x(n) v('.12'.) « q1/2 - o

v '. ~ n n 1/2 q k,E: '

for some positive 6. We follow the method of [DFil] (with a simplification of

Sarnak): rewriting x = xo the character we started with, we take for F the family