Quantum variance for holomorphic Hecke cusp forms on the vertical geodesic

TL;DR

This paper calculates the quantum variance of holomorphic cusp forms along a vertical geodesic, linking it to shifted-convolution problems and revealing a surprising diagonal-off-diagonal match similar to L-function moments.

Contribution

It provides an explicit asymptotic evaluation of the quantum variance for holomorphic cusp forms on a vertical geodesic, connecting it to shifted-convolution sums and moments of L-functions.

Findings

Quantum variance computed explicitly for holomorphic cusp forms.

Off-diagonal terms match diagonal terms, indicating a moment-like phenomenon.

Results relate to shifted-convolution problems and L-function moments.

Abstract

We compute the quantum variance of holomorphic cusp forms on the vertical geodesic for smooth, compactly supported test functions. The variance is related to an averaged shifted-convolution problem that we evaluate asymptotically. We encounter an off-diagonal term that matches exactly with a certain diagonal term, a feature reminiscent of moments of $L$-functions.