The Riemann Hypothesis for Period Polynomials of Modular and Hilbert Modular Forms

Abstract

We study the location of the zeros of period polynomials of modular forms. For an even weight $k\geq 4$ newform $f\in S_k^{\text{new}}(\Gamma_0(N))$, we show that the zeros of its period polynomial $r_f(z)$ lie on the circle $\lvert z\rvert=1/\sqrt{N}$. Moreover, we explore further generalizations to the case of Hilbert modular forms. In fact, we prove that the zeros of period polynomials of any parallel weight Hilbert modular eigenform on the full Hilbert modular group lie on the unit circle.