Abstract
We study two closely related problems stemming from the random wave conjecture for Maaß forms. The first problem is bounding the L^4-norm of a Maaß form in the large eigenvalue limit; we complete the work of Spinu to show that the L^4-norm of an Eisenstein series E(z,1/2+it_g) restricted to compact sets is bounded by sqrt{log t_g}. The second problem is quantum unique ergodicity in shrinking sets; we show that by averaging over the centre of hyperbolic balls in Gamma backslash mathbb {H}, quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the Planck scale. This result is conditional on the generalised Lindelöf hypothesis for Hecke–Maaß eigenforms but is unconditional for Eisenstein series. We also show that equidistribution for Hecke–Maaß eigenforms need not hold at or below the Planck scale. Finally, we prove similar equidistribution results in shrinking sets for Heegner points and closed geodesics associated to ideal classes of quadratic fields.
Highlights
1.1 Randomness of Maaß newforms1.1.1 Random wave conjectureLet B0( ) denote the set of Hecke–Maaß eigenforms of weight zero and level 1 on the modular surface \H, where = SL2(Z) and H denotes the upper half-plane; we normalise g ∈ B0( ) to be such thatCommunicated by Kannan Soundararajan.P
A well-known conjecture of Berry [1] and Hejhal and Rackner [20] states that a Hecke–Maaß eigenform g ∈ B0( ) of large Laplacian eigenvalue λg = 1/4 + tg2 ought to behave like a random wave
The former is a special case of the Gaussian moments conjecture, while the latter is a refinement of quantum unique ergodicity
Summary
C f f (z), λ≤λ f ≤λ+η(λ) where η(λ) → ∞ as λ → ∞ and η(λ) = o(λ), each f is a normalised Hecke–Maaß eigenform, and the coefficients c f are independent Gaussian random variables of mean 0 and variance 1. These are a randomised model of eigenfunctions of the Laplacian in the large eigenvalue limit λ → ∞, and it is easier to prove (almost surely) results for random waves than for true eigenfunctions. We study two aspects of this conjecture: bounds for the L4-norm of an automorphic form, and quantum unique ergodicity in shrinking balls. The former is a special case of the Gaussian moments conjecture, while the latter is a refinement of quantum unique ergodicity
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