248 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS
which shows that for unramified p, >. 1 (p) and >. 1 (p^2 ) cannot be simultaneously
small. Consequently, one chooses
{
-X. 10 (p), for n = p^2 ~ N , (p, q) = 1
(4.6) c1= >.1 0 (p), forn=p~N^112 , (p,q)=l
0 else,so that for p ~ L^112 , (p,q) = 1 one has cP2>.1 0 (p^2 ) + CpAJ 0 (p) = 1. This builts a
lacunar amplifier with a = 1/2.
Remark 4.2. The amplification method is completely "moral": despite appear-
ances, there is no contradiction between the fact that IA(n, L )I takes large values at
n =no (i.e. A(n 0 , L) »e L'Y.-e) and the fact that on average the IA(n, L)l^2 are small
(i.e. « e £"); indeed, an appropriate GRH for various Rankin-Selberg £-functions
(among others, for L(n ® n 0 , s)) shows that for any n =!=no, IA(n, L)I «e,d (QFL)"
for any c > 0. But of course, by positivity; one never need to use any unproved
hypotheses.
Remark 4.3. The reader should keep in mind that, the three basic strategies to
attack the Subconvexity Problem are essentially formal. In particular, whichever
method is chosen -higher moments, shortening or amplifying-the subconvexity
bound will never come for free! The large sieve inequalities, or other general tech-
niques may well bring us (rather easily) to the frontier between convexity and
subconvexity but not further; the tough part begins now, as one tries to cross the
border.
4.2.2. Improved bounds for ISf(q, x)I
Our first application of the amplification method is an improvement over Duke's
bound presented in the third lecture [MiV] :
Theorem 4.5. For any c > 0, and any character of modulus q, one has
isfxotic(q, x)I «,, q6/7+e.
Remark 4.4. Remark 3.6 and this theorem imply for q squarefree
ISfxotic(q)I «e q6/7+e_
Proof. We use the identification (2.6) between weight one holomorphic forms and
weight one Maass forms with parameter at infinity it = O; we consider { uj }J;;, 1 an
orthonormal basis of the space of weight 1 Maass forms containing { <l.~>
2
1? 2 , f E
Sfxotic(q, x)}. The estimate (2.33) and positivity show that for any sequence
(c1)1~L, one has
L H(O) I L c1>.1(l)l^2 = L 1i(O) I L c1VlP1(l)l2
fESf"otic(q,x) (f, f) l~L f ESf"otic(q,x) (f, f) l~L
(l,q)=l (l ,q)=l~ L1i(tj)I L c1Vlpj(l)l^2 +L
4
~1 H(t)I L c1VlPa(l, t)l2dt
j;;,l l~L a R l~L
(l,q)=l (l,q)=l