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