The $L^2$ restriction of the Eisenstein series to a geodesic segment

TL;DR

This paper investigates the behavior of Eisenstein series when restricted to geodesic segments, providing bounds and supporting a conjecture related to quantum unique ergodicity (QUE) in this setting.

Contribution

It establishes bounds for the $L^2$ norm of Eisenstein series on geodesic segments and proves the restricted QUE conjecture for geodesics with rational endpoints.

Findings

Lower bound matches the predicted asymptotic.

Upper bound nearly matches the lower bound assuming RH.

Restricted QUE holds for geodesics with rational endpoints.

Abstract

We study the $L^2$ norm of the Eisenstein series $E(z,1/2+iT)$ restricted to a segment of a geodesic connecting infinity and an arbitrary real. We conjecture that on slightly thickened geodesics of this form, the Eisenstein series satifies restricted QUE. We prove a lower bound that matches this predicted asymptotic. We also prove an upper bound that nearly matches the lower bound assuming the Riemann Hypothesis (unconditionally, the sharp upper bound holds for almost all $T$). Finally, we show the restricted QUE conjecture for geodesics with rational endpoints.