Quantum variance for dihedral Maass forms

TL;DR

This paper derives an asymptotic formula for the quantum variance of dihedral Maass forms on modular surfaces, revealing connections to classical geodesic flow variance and arithmetic properties of quadratic fields.

Contribution

It provides the first explicit asymptotic formula for quantum variance of dihedral Maass forms, linking quantum chaos with arithmetic of quadratic fields.

Findings

Quantum variance asymptotics established for dihedral Maass forms.

Quadratic form relates to classical geodesic flow variance.

Factors sensitive to arithmetic of () included.

Abstract

We establish an asymptotic formula for the weighted quantum variance of dihedral Maass forms on $\Gamma_0(D) \backslash \mathbb H$ in the large eigenvalue limit, for certain fixed $D$. As predicted in the physics literature, the resulting quadratic form is related to the classical variance of the geodesic flow on $\Gamma_0(D) \backslash \mathbb H$, but also includes factors that are sensitive to underlying arithmetic of the number field $\mathbb Q(\sqrt{D})$.