1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

Proof. By ( 4.25), Stirling's formula, and the value of the factors at oo, one has

I

~ { j^3 ( z) _dx_dy 1

2

« A( 1l'~f _ 0 1l'~J 0 1l'~f , 1/2)

7r J Xo(l) (j, j)3/2 y2 A(sym27r f, 1/2)3

«e lttlelttl-^3 /2+e L(1l'J 0 1l'J 01l'f,1/2).

One has the factorization

L(1l'J 0 1l'J 0 1l'f, s) = L(sym^3 1l'f, s)L(1l'J, s)^2.
By [KiSh], L(sym^3 1l'f, s) is the L-function of a cusp form on GL4, hence the con-
vexity bound holds ,

Thus one finds that

{ cp3(z) dx~y «e: (1 + lttl)-i+e: L(1l'J, 1/2)2;

Jxo(l) Y

once again, any subconvex bound for L(1l'J, 1/2) in the spectral aspect is sufficient

to show that ~ f xo(l) ¢>3(z) d~gu --> 0. The first estimate of this kind was obtained

by Iwaniec [13] with the subconvex exponent 1/2 - 1/12, and it was improved

subsequently by Ivie to 1 /2 - 1/6 [Iv2]. D

Sarnak and Watson have addressed the problem of the fourth moment for mod-

ular curves. Using the triple product identities, subconvex estimates and a Voronoi

formula for GL 3 -forms [MiS], they show (assuming RPC at the moment) that the

fourth moment conjecture holds for dihedral forms, and that for non-dihedral forms

one has

for any c > 0.

5.4.4. QUE in the level aspect

Arithmetic QUE has a natural generalization to the level aspect. To fix ideas,

consider f E S~(q) a primitive holomorphic form of level q and fixed weight 2,

and the associated probability measure on X 0 (q):

μJ( ) Z··--(jlf(,j)y z)l2 ki dxdy y2'

Denote by 1l'q : X 0 (q) --> X 0 (1) the natural projection induced by the inclusion

ro(q) c ro(l).

Conjecture. For f E S~(q), the probability measure μf,l := 1l'q*μf (the direct image

of μf by 1l'q) weakly converges to μp = vol(_io(l)) d~gu as q--> +oo.

Remark 5.9. One can consider more general versions of QUE in the level aspect; for

instance, one can pull backμ f to some X 0 ( qq') and then push forward the resulting

measure to X 0 (q'). The conjecture is that the image of μf weakly converges to the

Poincare measure on X 0 (q').