Mass Distribution for holomorphic cusp forms on the vertical geodesic

TL;DR

This paper computes the quantum variance of holomorphic cusp forms along the vertical geodesic and demonstrates that most such forms satisfy the QUE conjecture within a short weight interval.

Contribution

It provides the first computation of quantum variance for holomorphic cusp forms on the vertical geodesic and verifies QUE for almost all forms in a specified weight range.

Findings

Quantum variance computed for holomorphic cusp forms on the vertical geodesic.

Almost all holomorphic Hecke cusp forms in a short weight interval satisfy QUE.

Method advances understanding of quantum unique ergodicity in this setting.

Abstract

We compute the quantum variance of holomorphic cusp forms on the vertical geodesic for smooth compactly supported test functions. As an application we show that almost all holomorphic Hecke cusp forms, whose weights are in a short interval, satisfy QUE conjecture on the vertical geodesic.