Equidistribution of Eisenstein Series in the Level Aspect
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Communications in
Mathematical Physics
Equidistribution of Eisenstein Series in the Level Aspect Shin-ya Koyama The Institute of Mathematical Sciences, Ewha Womans University, Daehyun-dong 11-1, Sedaemoon-ku, 120-750 Seoul, South Korea. E-mail: [email protected] Received: 29 July 2008 / Accepted: 4 December 2008 Published online: 28 February 2009 – © Springer-Verlag 2009
Abstract: We prove an equidistribution property of the Eisenstein series for congruence subgroups as the level goes to infinity. This is an analogy of the phenomenon called quantum ergodicity.
1. Introduction Equidistribution of Eisenstein series was discovered by Luo and Sarnak [LS] in 1995. In the remarkable paper they showed for the Eisenstein series E(z, s) for the modular surface X = PSL(2, Z)\H 2 with H 2 the upper half plane that lim A
t→∞
B
|E(z, 21 + it)|2 d V (z) |E(z,
1 2
+ it)|2 d V (z)
=
vol(A) vol(B)
(1.1)
1 d xd y with A, B compact Jordan measurable subsets of X and d V = vol(X ) y 2 the normalized d xd y π volume measure with vol(X ) = X y 2 = 3 . The phenomenon (1.1) is an analogue of so-called quantum ergodicity, which is described as follows. Let u j be eigenfunctions of the Laplacian and all Hecke oper ators with the normalization X u i (z)u j (z)d V (z) = δi, j . Quantum ergodicity means the property that the probability measures dµ j = |u j (z)|2 d V (z) converge weak-∗ to d V (z). For any X = \P S L 2 (R) with any congruence lattice, Lindenstrass [L] proved quantum ergodicity for compact cases. He also proved for non-compact cases, that is when is a congruence subgroup, that any limit point of the sequence µ j is a multiple of d V .
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S. Koyama
The result (1.1) is regarded as a version of quantum ergodicity for continuous spectra. They actually proved 2 E(z, 1 + it) d V (z) ∼ 48 vol(A) log t 2 π vol(X ) A as t → ∞. In the proof they essentially used a subconvexity estimate for the automorphic L-function for a Maass cusp form u: 1
L( 21 + it, u) = O(t 3 +ε )
(t → ∞),
due to Meurman [M]. Such equidistribution was generalized to three-dimensional hyperbolic spaces by the author [K] by use of the subconvexity estimate of Petridis and Sarnak [PS]. According to various aspects of estimates of L-functions, it is possible to consider analogues of such equidistribution. For example, Rudnick and Sarnak [RS] posed a conjecture of quantum ergodicity in the weight aspect ([KMV] Conjecture 1.4), and Sarnak [S] obtained a subconvexity bound for the Rankin-Selberg convolution L-function along this direction. An analogue of this conjecture in the level aspect is formulated for holomorphic cusp forms by Kowalski, Michel and Vanderkam in [KMV] as follows. Here we put πq : X q → X 1 to be the canonical projection with X q = 0 (q)\H 2 . Conjecture 1.1 ([KMV] Conjecture 1.5). For k ≥ 2 even and fixed, let { f j } j≥1 be any sequence of primitive holomorphic forms of weight k with increasing levels q j . As j → ∞ the sequence of probability measure πq j ,∗ (µ f j ), j ≥ 1 converge weakly to the Poin
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