A Note on Holomorphic Quantum Unique Ergodicity

TL;DR

This paper presents a new proof of the Quantum Unique Ergodicity conjecture for holomorphic modular forms, using minimal assumptions and extending to broader classes of cusp forms, with implications for Lehmer's conjecture.

Contribution

It provides a novel proof of the conjecture that requires only partial Ramanujan conjecture results and applies to more cusp forms than previously considered.

Findings

New proof of Quantum Unique Ergodicity for holomorphic forms

Extension of results to broader cusp form classes

Corollaries related to Lehmer's conjecture on Fourier coefficients

Abstract

In this paper we give a new proof of the Quantum Unique Ergodicity conjecture for holomorphic integral weight modular forms on the upper half plane. The proof requires only partial results towards the Ramanujan conjecture and the shifted convolution problem. Furthermore the proof is applicable to a wider class of cusp forms other than Hecke eigenforms. We also prove some interesting corollaries, particularly towards the Lehmer's conjecture on the non vanishing of the Fourier coefficients.