Quantum variance for dihedral Maass forms

Abstract

We establish an asymptotic formula for the weighted quantum variance of dihedral Maass forms on Γ 0 ( D ) ∖ H \Gamma _0(D) \backslash \mathbb H in the large eigenvalue limit, for certain fixed D D . As predicted in the physics literature, the resulting quadratic form is related to the classical variance of the geodesic flow on Γ 0 ( D ) ∖ H \Gamma _0(D) \backslash \mathbb H , but also includes factors that are sensitive to underlying arithmetic of the number field Q ( D ) \mathbb Q(\sqrt {D}) .