The Riemann Hypothesis For Period Polynomials Of Modular Forms

Abstract

The period polynomial $r_f(z)$ for an even weight $k\ge 4$ 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.

arXiv comment11 pages. We added the one sentence explanation for the $2n^{(k-1)/2}$ bound for ${\Lambda }_f(n)$. This paper will be appearing in Proceedings of the National Academy of Sciences, USA

References in corpus (2)

• Modular forms and period polynomials
• Local zeta factors and geometries under Spec Z

Cited by in corpus (5)

• An algebraic characterization of the Kronecker function
• Homotopy types and geometries below Spec Z
• On the distribution of periods of holomorphic cusp forms and zeroes of period polynomials
• Special Values of Motivic $L$-Functions and Zeta-Polynomials for Symmetric Powers of Elliptic Curves
• Zeta-polynomials, Hilbert polynomials, and the Eichler-Shimura identities