Equidistribution in Shrinking Sets and L^4-Norm Bounds for Automorphic Forms

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 $E(z,1/2+i{t}_g)$ 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 Lindelof hypothesis for Maass eigenforms but is unconditional for Eisenstein series. We also show that equidistribution for 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.

36 pages. To appear in Mathematische Annalen

References in corpus (2)

• Planck-scale mass equidistribution of toral Laplace eigenfunctions
• On the fourth moment of Hecke Maass forms and the Random Wave Conjecture

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• Quantum Unique Ergodicity for Eisenstein Series in the Level Aspect
• The cubic moment of Hecke--Maass cusp forms and moments of $L$-functions
• $L^p$-Norm Bounds for Automorphic Forms via Spectral Reciprocity
• Small scale equidistribution for a point scatterer on the torus
• The variance of closed geodesics in balls and annuli on the modular surface
• Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms
• On the $L^2$ restriction norm of large level