Zeros of even and odd period polynomials

TL;DR

This paper proves that for large level and weight, the zeros of the even and odd period polynomials associated with newforms lie on a specific circle in the complex plane.

Contribution

It establishes a geometric location for zeros of period polynomials of newforms at high levels and weights, extending understanding of their zero distributions.

Findings

Zeros of period polynomials lie on the circle |X|=1/√N for large level and weight.

The result applies to both even and odd period polynomials.

Provides insight into the symmetry and distribution of zeros in modular form theory.

Abstract

Let $f \in S_k(\Gamma_0(N))$ be a newform, and let $r_f^{\pm}(X)$ denote its corresponding even and odd period polynomials. For sufficiently large level and weight, we show that the zeros of $r_f^{\pm}(X)$ all lie on the circle $|X| = \frac{1}{\sqrt N}$.