1549380232-Automorphic_Forms_and_Applications__Sarnak_

238 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF L-FUNCTIONS

has for such 11 's, a good approximation of L(11, 1y by a very short Dirichlet poly-
nomial (for z an arbitrary fixed complex number). One can use these approxi-
mations (together with the trivial bound for the 11 in the exceptional subset) and
quasi-orthogonality relation for :F (in small ranges of m , n) to derive an asymptotic
formula for the averaged moment:

M(:F,z ) := L L(11, l)z.

nEF

(The contribution of the exceptional 11's for which the zero-free region is not large

enough is bounded trivially.) Eventually, these computations show that the random

variable 11 E :F --> L(11, 1) has a limiting distribution function when IFI --> +oo,

and in some cases, one can also study its extreme values. This kind of applica-

tion was initiated by Barban in the case of (real) Dirichlet L-functions (improving

and simplifying earlier results of Bateman, Chawla and Erdos). Later, the ques-

tions of the distribution function and of the extreme values of L(x, 1) were stud-

ied further by Montgomery/Vaughan and more recently in the exhaustive work of

Granville/Soundararajan [Mo Va, GrS].

For instance (thanks to H 2 (7 / 64)) the zero density result above applies to sym-

metric square L-functions for families of GLrmodular forms. Results on the distri-

bution of the symmetric square L(sym^2 11, 1) have been obtained by Luo [Lu3] and

then by Royer [Ro, Ro2] by using zero density estimates, Kuznetzov/ Petersson's

formula and a combinatorial analysis. An example is the following result[Ro, Ro2,

RoWu]:

Theorem 3.12. For q prime, consider the set of primitive holomorphic forms with triv-

ial nebentypus, S~ ( q, xo ), as a probability space endowed with the uniform measure.

When q --> +oo, the two random variables on

f--> L(sym^2 f ,1) andf--> L(sym^2 f ,1)-^1 E R~ 0

admit a limiting distribution function. Moreover; for any q prime, sufficiently large,

there exists f and f' in S 2 (q) such that

L(sym^2 f ,1) » (loglogq)^3 , L(sym^2 f ',1) « (loglogq)-^1 ,

where the implied constants are absolute.

Remark 3.8. This is essentially sharp: under GRH, one has, for any f E S~(q, xo),

(log logq)-^1 « L(sym^2 f ,1) « (log logq)^3.

It is also possible to handle similarly an algebraic family of L-functions: con-
sider
E t : y^2 = x^3 + A(t)x + B(t), A , BE Z[t],
a non geometrically trivial family of elliptic curves over Q. By the work of Wiles,
Taylor/Wiles and Breuil/Conrad/Diamond/Taylor[Wi, TW, BCDT] each non-singu-
lar fiber is modular, hence (by Gelbart/Jacquet) the L-function of the symmetric
square of each fiber has an analytic continuation to C with a possible pole at s = 1
occurring only for the finitely many t's such that E t is CM. One has the following
([KM3]):