Real zeros of holomorphic Hecke cusp forms

TL;DR

This paper investigates the distribution of zeros of holomorphic Hecke cusp forms of large weight, focusing on the subset of zeros that lie on specific geodesic segments, and provides asymptotic estimates for their count.

Contribution

It offers new asymptotic estimates for the number of real zeros of holomorphic Hecke cusp forms as the weight increases.

Findings

Zeros are symmetric about three geodesic segments.

Number of real zeros grows asymptotically with the weight.

Provides estimates for the count of real zeros as weight tends to infinity.

Abstract

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.