# Ken Ono

Affiliation: Emory Unversity

Email: ono@mathcs.emory.edu

Title Of Talk: Riemann Hypothesis for Period Polynomials of Modular Forms

Abstract: The period polynomial $r_f(z)$ for an even weight $k\geq 4$ newform $f\in S_k(\Gamma_0(N))$ is the generating function for the critical values of $L(f,s)$. It has a functional equation relating $r_f(z)$ to $r_f\left(-\frac{1}{Nz}\right)$. We prove the Riemann Hypothesis for these polynomials: that the zeros of $r_f(z)$ lie on the circle $|z|=\frac{1}{\sqrt{N}}$. We prove that these zeros are equidistributed when either $k$ or $N$ is large. This is joint work with Seokho Jin, Wenjun Ma, and Kannan Soundararajan.