Dissipation of correlations of holomorphic cusp forms

TL;DR

This paper extends Quantum Unique Ergodicity to holomorphic cusp forms on the modular surface, showing that correlations dissipate as the weight increases, by combining spectral theory with shifted convolution sums and subconvexity methods.

Contribution

It introduces a novel approach combining spectral theory with Holowinsky-Soundararajan methods to analyze correlations of holomorphic cusp forms at high weights.

Findings

Correlations of masses dissipate with increasing weight.

New bounds for Fourier coefficients of weight k cusp forms.

Application of Ichino's formula for triple product integrals.

Abstract

We obtain a generalisation of the Quantum Unique Ergodicity for holomorphic cusp forms on $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ in the weight aspect. We show that correlations of masses coming from off-diagonal terms dissipate as the weight tends to infinity. This corresponds to classifying the possible quantum limits along any sequence of Hecke eigenforms of increasing weight. Our new ingredient is to incorporate the spectral theory of weight $k$ automorphic functions to the method of Holowinsky-Soundararajan. For Holowinsky's shifted convolution sums approach, we need to develop new bounds for the Fourier coefficients of weight $k$ cusp forms. For Soundararajan's subconvexity approach, we use Ichino's formula for evaluating triple product integrals.