Quantum variance for dihedral Maass forms

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 $\mathbb{Q}(\sqrt{D})$.

42 pages