1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 5. SOME APPLICATIONS OF SUBCONVEXITY 279

which follows from the Hadamard/ de la Vallee-Poussin zero-free region. For V = g

a (primitive) Maass/ Hecke-eigenform, one has to show that

(5.13)

1 1

g(z)dμt (z ) = IE^2 dxdy

00 (z, 1/ 2 + it)I g(z )- 2 = o 9 (1ogt);

Xo(l) Xo(l) Y

by the unfolding method, one has

r g(z)dμt(z )

Jxo(l)

. jr(i+2it)l2 r(1-2it,-4it)r(i+2it 9 - 4it)
=27r-2itj((1+42it)j2 4 jr(~)l 2 4 L(g, l /2 )L(g,l/2- it).

By Stirling's formula and the lower bound ((1+2it) » (logt)-^1 (which again is a

consequence of the zero-free region), the latter is bounded by

«c:,g tc:-^1 /^2 L(g, 1/ 2 -it).

Hence, any subconvexity exponent for L(g, 1/ 2 - it) in the t-aspect, is sufficient to

prove (5.13). This case of the ScP was solved by Meurman [Me] with the subconvex

exponent 1/3:

L(g, 1 /2 - it) «c:, 9 jtj^1 /3+c:.

D

The determination of quantum limits of cuspidal Hecke-eigenforms is deeper.
In [BL], Bourgain and Lindenstrauss showed that the Hausdorff dimension of the
support of these quantum limits is at least 1 + 2/9 by mixing combinatoric tech-
niques3 together with number theoretic methods (interestingly the identity
j-\ 1 (p^2 ) - ,\ 1 (p )^21 = 1 for p Aqf that was been used in Section 4 .2.1 for the construc-
tion of amplifiers is also useful here). However, Arithmetic QUE can be related
directly to subconvexity for a collection of L-functions: for simplicity, consider the
case of the full modular curve X 0 (1). By the spectral decomposition (2 .5) and
Weyl's equidistribution criterion, it is sufficient to show that

( g(z)dμ 1 (z ), ( E 00 (z, 1/2 + it)dμ1(z ) _, 0, as AJ (resp. k1) _, +oo,

lxo(l) Jxo(l)

for any primitive Maass form g and any t E R. By the unfolding method, one has

1

( I

. ) ( )_L 00 (7rJ®1rJ, l /2+it)L(7rJ®1rJ,l/2+it)
E^00 z, 1 2 +it dμJ z - (f f).
Xo(l) '
By (2.18), (2.19), the expression for the local factor at oo is (depending on whether
f is a Maass form or holomorphic):

or

Loo(1fJ ® 1ff, s) = rR(s)^2 rrrR(s ± 2it1 ),

±

Loo(1fJ ® 1ff, s ) = rR(s)rR(s + l)rR(s + k - l)rR(s + k).

(^3) Recently; using ergodic theoretic methods, E. Lindenstrauss [Lin] has essentially proven that Arith-
metic QUE holds for compact arithmetic surfaces and for non-compact ones, that any quantum limit
is proportional to the hyperbolic measure. In his proof, the fact that the quantum limits have positive
entropy is crucial.