Quantum Variance for Eisenstein Series

Abstract

In this paper, we prove an asymptotic formula for the quantum variance for Eisenstein series on PSL 2 (Z)\H. The resulting quadratic form is compared with the classical variance and the quantum variance for cusp forms. They coincide after inserting certain subtle arithmetic factors, including the central values of certain L-functions.for Re(s) > 1, and has a meromorphic continuation to s ∈ C, where Γwhere dµ(z) = dxdy y 2 . In this paper, we are interested in the fluctuation of µ t . In [21], Luo-Sarnak proved an analogue of quantum unique ergodicity (QUE) for Eisenstein series: for Date: November 8, 2018.

Quantum variance for dihedral Maass forms

“…(see [20,Lemma 11]). These estimates allow us to quickly derive a short Dirichlet polynomial approximation of 1/L(1, φ 2k ) 2 , by a contour integration argument (see [19,Lemma 3] for a similar result).…”

“…The latter case is exactly what we consider in this paper. The quantum variance for Eisenstein series was considered by the first author in [19]. The arithmetic correction factor in that case is L( 1 2 , ψ) 2 , and the covariance can be proved to be zero unconditionally.…”

Quantum variance for holomorphic Hecke cusp forms on the vertical geodesic

“…In the compact setting of quaternion algebras Nelson [Nel16], [Nel17], [Nel19] evaluated the quantum variance using the theta correspondence. Further papers indicating the active field of research are for example given by work of Huang [Hua21] for the variance of Eisenstein series, Huang-Lester [HL20] for the variance of dihedral Maass cusp forms and the work of Nordentoft, Petridis and Risager [NPR21] on small scale equidistribution at infinity. We explore for the first time the one-dimensional case of the vertical geodesic for holomorphic cusp forms and show Theorem 1.2.…”