1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

the implied constant depending on rJ, j , A. When A is large, Vs,7foo (y) becomes very
small as y ---> +oo, so that the two sums of (1.30) are essentially of length Xv Q" ( t)
and -jQ"(t)/ X, respectively. In particular, with the most symmetric choice X = 1,
we obtain two sums oflength ~ vQ7f(t), and for ~es= 1 /2 we retrieve, by (1.23),
the convexity bound (1.22); this shows that a subconvexity bound is the result of a
cancellation in the sum of Hecke eigenvalues .A"(n) when n is close to -jQ"(t) in
the logarithmic scale.

Remark 1.13. Although we will not use it in these lectures, it is good to know

that there are cases where an asymmetric representation (i .e. X =f. 1) is better

adapted: in particular, when one has not enough control on the "analytic" root

number w(H, s) , one may take X > 1 to reduce the length of the second sum and

thus its influence; of course, the price one pays is then a longer first sum (see [Velk]

for an example).

1.3.3. The subconvexity problem and families of £-functions

Families of £-functions enter naturally into the Subconvexity Problem via the meth-

od of moments. Suppose one wants to solve some aspect of the Subconvexity Prob-

lem for a given L(Ho, 1 /2); the approximate functional equation reduces essentially

this question to a non-trivial bound for linear forms (in the Hecke eigenvalues

.A" 0 ( n)) of the type

where V is rapidly decreasing and X ~ VQ::. Suppose one can put the given

Ho into a "natural" family F = { 7f} endowed with some probability measure μF;

"natural" implying (among other things) that the analytic conductor Q" 0 is close

to the average analytic conductor, i.e. QF := E(Q",μF) ~ Q" 0 • The method of

moments consists of bounding the L^00 norm of the random variable 7f r-+ Ev(H, X)

by its L^2 k norm and then estimating the latter. That is, one wants a bound for the

2k-th moment:

Under GLH, one has for any c: > O

(1.34)

However, for sufficiently "nice" families and for sufficiently small k's, such a bound

can be obtained unconditionally. In these nice situations, we obtain by positivity,

(1.33), and (1.34), that

this bound breaks the trivial estimate, provided that one can take k strictly bigger

than -4log(μF(7ro))/log(Q" 0 ).