1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

weak-* converges on X 0 (q)(C ) to the normalized Poincare measure

dμ p = 1 dx dy_

vol(Xo(q)) y^2

Remark 5. 7. In the case of Maass forms, asking f to be a Hecke eigenform shouldn't

be too severe a restriction, when compared with the general QUE. Indeed, one ex-

pects that the dimension of the Laplace eigenspace with eigenvalue ,\ is bounded

by Oc:(X') for any E:. The latter bound seems to be difficult to reach, and there is

so far essentially no better bound than the trivial bound coming from Weyl's law

(compare with what is known for dim S 1 ( q, x )). However, if this bound were true,

then it would not be difficult to show, in the case of X 0 (1) for instance, that QUE

is implied by Arithmetic QUE, together with a power saving estimate for the dis-

crepancy. On the other hand, the dimension of the space of holomorphic forms of

weight k is large ( » k), and QUE certainly cannot hold for an arbitrary holomor-

phic form: f (z) = ~m(z), them-th power of Ramanujan's function, is a weight

12 m holomorphic form not satisfying QUE as m ~ + oo. Hence, it makes sense

in this case to restrict to Hecke eigenforms. It is very possible that the condition

of being a Hecke eigenform can be relaxed to the condition of being an eigenform

for two Hecke operators Tp, Tp', for distinct fixed primes p , p', not dividing q (see

[ClU]).

Remark 5.8. In the case of holomorphic forms, Arithmetic QUE has the following

nice consequence, due to Z. Rudnick [Ru]: if f is holomorphic of weight k, then f

has ~ qk f / 12 zeros on X 0 ( q); this leads naturally to the question of the distribution

of such zeros. It turns out that the convergence of dμ 1 (z) to dμp implies that

the zeros of f are equidistributed with respect to dμp. A corollary is that the

multiplicity of any zero off is o(kJ) as k 1 ~ +oo, which (again) is stronger than a

trivial application of Riemann/ Roch. Note that in the case of Maass forms, it is not

clear how to deduce from Arithmetic QUE a similar equidistribution for the nodal

lines (i.e. the lines on the surface defined by the equation </Jj(z) = 0).

In the non-compact case, quantum limits of the Eisenstein series can be studied

as well (although their associated measure does not have finite mass). For the full

modular curve Xo(l ), Luo/ Sarnak [LS] proved the analog of QUE for E 00 (z, 1 /2 +

it) as t ~ +oo:

Theorem 5.5. Set dμt(z) := IE 00 (z, 1/ 2 + it)i^2 dx~vy. For V a continuous function

compactly supported away from the cusp oo, one has, as t ~ +oo:

( V(z)dμt(z) =

48

( V(z ) dx~y logt + ov(logt ).

} Xo(l) 7r J Xo(l) Y

Proof. (Sketch) By density, it is sufficient to obtain the above identity for V either
an incomplete Eisenstein series or a Maass/ Hecke-eigenform g. Note that the for-
mer case is not trivial, since it requires both a subconvexity bound for ((1/ 2 +it)
(for instance ( 4 .1)), and the Hadamard/ de la Vallee-Poussin/Wey! bound (here the
savings of the log log t fa ctor is necessary),

('. logt

(^7) .,, (l+it)« (^1) og (^1) ogt '