Dissipation of correlations of holomorphic cusp forms

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.

Locations

• arXiv (Cornell University) - View - PDF