A Note on Holomorphic Quantum Unique Ergodicity

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.