The nontrivial zeros of period polynomials of modular forms lie on the unit circle

TL;DR

This paper proves that most zeros of period polynomials for Hecke cusp forms are on the unit circle, with only five exceptions, advancing understanding of their zero distribution.

Contribution

It demonstrates that nearly all zeros of these period polynomials lie on the unit circle, revealing a significant geometric property of these polynomials.

Findings

All but 5 zeros are on the unit circle

Zeros exhibit a symmetric distribution

Supports conjectures about zeros of period polynomials

Abstract

We show that all but 5 of the zeros of the period polynomial associated to a Hecke cusp form are on the unit circle.