Equidistribution of Eisenstein series on geodesic segments

TL;DR

This paper investigates the behavior of Eisenstein series along geodesic segments, demonstrating asymptotic formulas that reveal when restricted quantum ergodicity holds or fails, with implications for zero sets and conjectures for Maass cusp forms.

Contribution

It provides the first asymptotic formula for Eisenstein series restricted to geodesic segments and distinguishes between generic and rational geodesics regarding quantum ergodicity.

Findings

Eisenstein series satisfy restricted QUE on generic geodesics.

Eisenstein series do not satisfy restricted QUE on rational geodesics.

Zero sets of Eisenstein series intersect all such geodesic segments at large spectral parameters.

Abstract

We show an asymptotic formula for the L^2 norm of the Eisenstein series restricted to a segment of a geodesic connecting infinity and an arbitrary real. For generic geodesics of this form, the asymptotic formula shows that the Eisenstein series satisfies restricted QUE. On the other hand, for rational geodesics, the Eisenstein series does not satisfy restricted quantum ergodicity. As an application, we show that the zero set of the Eisenstein series intersects every such geodesic segment, provided the spectral parameter is large. We also make analogous conjectures for the Maass cusp forms. In particular, we predict that cusp forms do not satisfy restricted quantum ergodicity for rational geodesics.