1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 4. THE SUBCONVEXITY PROBLEM 247

The second possibility of choosing a smaller subfamily of F containing n 0 , may

be available in some cases: for example, for Riemann's (, a subconvex bound for

((1/2 + it 0 ) follows from the upper bound

granted that one can choose a fixed < 1. However, this option is not available in

some other interesting situations, such as the level aspect.

In the remaining cases, there is a third solution, the amplification method, which

is a partial form of the first option: one keeps k = k 0 , but one introduces an extra

(short) linear form

A(n, L) := L c1A.rr(n),

l~L

where the coefficients are a priori arbitrary. Because the (an) are smooth, one can

hope, at least for (general) (c 1 ) with sufficiently small support, to get a quasiorthog-

onality type bound for the modified average

(4.5) l~I LI L bn>.'"'(n) l^21 L CnA'"'(n)l^2 «c (QFN)c( L lanl^2 )ko L lcd^2.

7r n~Nko l~L n~N l~L

Now suppose we can choose (c 1 ) satisfying

for some a > 0, one obtain by positivity that

£(no, N) «c (QFNL)c N^112 L-^0 ( L lanl^2 )^112 ,

n~N

which improves the trivial bound whenever log L » log N.

Such a sequence ( c 1 ) is called an amplifier for no since A( n, L) evaluated at

n = n 0 is large and thus amplifies the contribution of our preferred n 0 inside ( 4.5);

the amplifier acts very similarly to the detector (a~) encountered in Section 3.1.1.

The basic problem is then to construct an amplifier: when n 0 = xo is a Dirichlet

character, one can choose

c1 =x 0 (l), l ~ L

and we clearly have an amplifier with a = 1. When no = Jo E SP(q, x, it) is a

modular form, the most natural choice would be to take c 1 = "XJ 0 (l); but a difficulty

crops up: since L will be a small power of Qf 0 , proving that

L l>-to(l)l^2 »c (Q7ro)-c Ll-c

l~L

would require a strong form of subconvexity for L(f 0 f , s) (unless Jo is dihedral).
A way to get around this difficulty was found by Iwaniec and is based on the ele-
mentary identity