Sign changes along geodesics of modular forms

TL;DR

This paper investigates the sign change behavior of cusp forms and Eisenstein series along geodesics on the modular surface, providing new bounds and extending results under certain spectral assumptions.

Contribution

It establishes unconditionally sharp lower bounds for Eisenstein series sign changes and extends these results to cusp forms under spectral moment bounds.

Findings

Unconditional sharp lower bounds for Eisenstein series sign changes.

Extension of sign change results to cusp forms under spectral assumptions.

Utilization of mean square bounds and removal of Lindelöf hypothesis assumptions.

Abstract

Given a compact segment, $\beta$, of a cuspidal geodesic on the modular surface, we study the number of sign changes of cusp forms and Eisenstein series along $\beta$. We prove unconditionally a sharp lower bound for Eisenstein series along a full density set of spectral parameters. Conditioned on certain moment bounds, we extend this to all spectral parameters, and prove similar theorems for cusp forms. The arguments rely in part on the authors' mean square bounds [KKL24], and on removing the assumption of the Lindel\"of hypothesis from recent work of Ki [Ki23].