1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. THE SUBCONVEXITY PROBLEM 245

l.J rl' ITr --b-u + bi. h h

1 rs not a k-t power (w ere we denote by k the order of x), one has

i=l u + i

/I: x((u + b1) ... (u + br))x((u + b~)... (u + b~)) / «r q^112.

u(q)

The number of (b 1 , ... , b~) not satisfying this criterion is bounded by «r B r, and

in this case we bound the sum trivially by q. Hence

L /2: x((u + b1)... )I« Brq + B 2rql/2.

b1,... ,br~B u(q)

b~, ... ,b~~B

The estimate of Theorem 4.3 follows by taking

A= Nq-l/2r, B = ql/2r.

D

The proof of Theorem 4.2 follows by integration by parts with r = 4.

4.1.2. Fouvry /lwaniec's extension of Burgess's argument

The method of Burgess, though qualitatively very powerful, has the disadvantage of

not being easily applicable to other types of arithmetic functions, and in particular,

to more general cases of the Subconvexity Problem. However, recently Fouvry

and Iwaniec have used a variant of Burgess's technique to solve the Subconvexity

Problem for certain types of Grossencharacter £-functions attached to imaginary

quadratic fields [Fol3]:

Theorem 4.4. Let D = 3( 4) be a prime number> 3 and let 'ljJ be a primitive character
modulo the different of Q( FD) that takes values
a
'l/J((a)) = x(~ea)( rar r

on principal ideals (a), for x a primitive Dirichlet character mod D, such that

x(-1) = (-lr. Thenfor ~es= 1/2,

L('l/J, s) «s,e: Dl/2-l/16+e:.

4.2. The Amplification method

The amplification method is a tricky variant of the method of moments (see the first
lecture); it was invented by Iwaniec to solve instances of the Subconvexity Problem
and this is to date the most general technique for obtaining subconvex estimates:
it may be used for the proof of all known cases of subconvexity (although with
weaker exponents than the ones obtained by more ad-hoc methods).

4.2.1. Principles of Amplification

Given 7ro, one wants a non-trivial bound for some given linear form in the Hecke
eigenvalues,

.C(7ro,N) = L an.A.,.. 0 (n),

n~N