1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

for e, e' E a gss (F q ) , so that M^0 is the orthogonal complement of the divisor eo =

:Z:::e e/we. Moreover, Mand M^0 are acted upon by a Hecke algebra T(q) generated

by correspondences Tp, p f q of degree p + 1, which are symmetric with respect

to ( , ). Their definition can by given either directly via Brandt matrices ([Gr]) or

adelically by the identification of £ff^55 (F q) with either the set of left ideal classes of

a fixed maximal order (R say) inside the definite quaternion algebra over Q (B say)

ramified at q and oo, or with the double cosset space R_x \ Bx/ Bx (here ii (resp. R)

denotes the adelization of B (resp. R)). A special case of the Jacquet/ Langlands

correspondence states that M^0 ® C and S 2 (q) are isomorphic as T (q)_modules,

hence M^0 ® C (in fact M^0 ® R) admits an orthonormal basis { e f } fES~(q) indexed

by primitive forms such that

Tpef = Vfi>. 1 (p)e1

for p Jq. In particular, by the same analysis as above, in order to prove (5.10) it is

sufficient to prove that for any f E S~(q), one has

W(G, f) := l~I (~= r q(E 0 ), e1) = 01(1),

<JE G

which follows from

1

W('lj;, f) := IPic(Ox)I ( L "ii}(a)rq(Eg), e1) = o1(D- 1/2 4000 .5)

<JE P ic(Ox)

for every 'ljJ that is trivial on G. Again, the twisted Weyl sum is related to central

values of Rankin-Selberg £-functions through the formula (S.11), which in this

case was proved by Gross [Gr] (together with the fact that the action of Pic(Ox)

commutes with the reduction rq (see [BD2] p.120 )). It follows from Theorem 4.13

that in fact W('lj;, f) = Oi(D-1/2200).

D

Remark 5.6. The equidistribution theorem above can be widely generalized. It is a
special case of general equidistribution properties for small orbits of Heegner points
on Shimura curves associated to definite or indefinite quaternion algebras over Q.
In [Zl, Z2, Z3], Zhang provided very general formulas relating central values of
Rankin/Selberg £-functions to twisted Weyl sums corresponding to the appropriate
equidistribution problem. More generaly, these formulas and the corresponding
subconvexity bounds show the equidistribution of (small orbits of) Heegner points
associated with not necessarily non-maximal orders of large conductor, for exam-
ple, in the (orthogonal) case of Heegner points associated to orders inside a fixed
field of complex multiplication. However, this direction of inquiry can be treated
by other quite different (ergodic) techniques (see [Va, Cor, ClU]).

5.4. Subconvexity vs. Quantum Chaos

For a more thorough description of the applications given in this section, we refer
to the lectures [Sa2, Sa6].

5.4.1. The Quantum Unique Ergodicity Conjecture

Suppose we are giveniven X a compact Riemannian manifold and { <pj }j~o an or-
thonormal basis of L^2 (X) composed of Laplace eigenfunctions with eigenvalues