1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 5. SOME APPLICATIONS OF SUBCONVEXITY 281

have been given for G L 3 [BFG], but so far they seem difficult to use effectively for

refined analytic purposes.

5.4.2. QUE on the sphere

It is also possible to formulate an analog of QUE for the 2-dimensional sphere 82 ,

although the geodesic flow is not ergodic. Recall that the spectrum of b. on 82 is

0 < 1 < 3 < · · · < k ( k + 1) / 2 ... , and that the corresponding eigenspace is the space

of harmonic polynomials of degree k - this space has dimension 2k + 1, which is

rather large. In this case, it is possible to prove a probabilistic version of QUE:

namely, QUE holds for almost all orthonormal eigenbasis of b.. However, as for

the holomorphic case above, the QUE conjecture cannot be true in its most naive

form. Recently, Bocherer/ Schulze-Pillot/ Sarnak formulated a deterministic version

of QUE for the sphere [BSSP]: 82 is viewed as the symmetric space attached to · a

definite quaternion algebras, D say,with class number one. As such, 82 is endowed

with an action of the Hecke algebra of D x that commutes with b.; thus the QUE

conjecture for the sphere is: for cpk a harmonic polynomial of degree k --> +oo,

which is also a Hecke-eigenform,

Jcpk(P) J^2 1

μ'Pk := f 32 Jcpk(P) J2dμ 32 μ 32 --> vol(8 2) μ 32 weakly.

Again, this conjecture can be reduced -via triple product identities due to Bocherer-
/ Schulze-Pillot [BS] - to the ScP in the k aspect for the £-functions (5.14), where
f and g are the holomorphic primitive forms corresponding, respectively, to cpk and
to some harmonic (Hecke eigen) polynomial via the Jacquet/ Langlands correspon-
dence between G L 2 and D x.

5.4.3. The Random Wave Conjecture for Hecke-eigenforms

Another famous conjecture in quantum chaos is the Random Wave Conjecture of
Berry and Hejhal. This conjecture predicts that

., viewed as a random variable on
X , converges to the normal Gaussian distribution N(O, 1/vol(X )^112 ) as >. --> +oo.
This is equivalent to the following moments conjecture:

Conjecture. For any integer m ~ 0,

lim r ( <p>.)mdx = Cm /2'

>--+oo} x Fol(X}

where Cm is them-th moment of the Gaussian distribution N(O, 1).

The cases m = 1, 2 are obvious. The first non-trivial case is the third moment
for which c 3 = 0: it has been solved by Watson for X = X 0 (1) and cp = f /(!, !)^112
for f a primitive Maass form, as a consequence of his triple product identity [Wat]
and a subconvex estimate for L(f, 1/2) in the spectral aspect due to Iwaniec.

Theorem 5.6. For f a primitive Maass form on X 0 (1 ), one has

~ r (f(z))3 dx dy = o(l)

7r j Xo(l ) (!, !)3/2 y 2

as tf--> +oo.