On the Zeros of the Miller Basis of Cusp Forms

TL;DR

This paper investigates the distribution of zeros of cusp forms in the Miller basis with fixed vanishing order at infinity, demonstrating that for large weights, all zeros in the fundamental domain lie on the boundary circle.

Contribution

It establishes that for sufficiently large weights, zeros of these cusp forms are confined to the boundary circle, providing an explicit linear bound on the weight in terms of the vanishing order.

Findings

Zeros lie on the boundary circle for large weights

Effective linear bound on weight in terms of vanishing order

Zeros are confined to the boundary in the fundamental domain

Abstract

We study the zeros of cusp forms in the Miller basis whose vanishing order at infinity is a fixed number $m$. We show that for sufficiently large weights, the finite zeros of such forms in the fundamental domain, all lie on the circular part of the boundary of the fundamental domain. We further show and quantify an effective bound for the weight, which is linear in terms of $m$.