Zeros of modular forms and Faber polynomials

TL;DR

This paper investigates the distribution of zeros of large-weight cusp forms for the modular group, revealing they cluster near specific vertical lines, contrasting with known zero distributions of Eisenstein series and Hecke eigenforms.

Contribution

It introduces a novel analysis of zeros of cusp forms with large weight using Faber polynomials, showing their convergence to truncated exponential polynomials.

Findings

Zeros cluster near D vertical lines at height log(k) for large weight k

Faber polynomials associated with these cusp forms converge to truncated exponential polynomials

Contrasts zero distribution with Eisenstein series and Hecke eigenforms

Abstract

We study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number D of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near D vertical lines, with the zeros of a weight k form lying at height approximately log(k). This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain. Our method uses the Faber polynomials. We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree D.