Zeros of symmetric power period polynomials

TL;DR

This paper extends the study of zeros of period polynomials associated with modular forms to symmetric power L-functions, proving a Riemann hypothesis analogue for large weights and levels.

Contribution

It introduces a new analogue of period polynomials for symmetric power L-functions and proves the Riemann hypothesis for their zeros under certain conditions.

Findings

Zeros lie on a specific circle for large weights.

Riemann hypothesis analogue holds for large weights and levels.

Results extend previous work on classical period polynomials.

Abstract

Suppose that $k$ and $N$ are positive integers. Let $f$ be a newform on $\Gamma_0(N)$ of weight $k$ with $L$-function $L_f(s)$. Previous works have studied the zeros of the period polynomial $r_f(z)$, which is a generating function for the critical values of $L_f(s)$ and has a functional equation relating $z$ and $-1/Nz$. In particular, $r_f(z)$ satisfies a version of the Riemann hypothesis: all of its zeros are on the circle of symmetry $\{z \in \C \ : \ |z|=1/\sqrt{N}\}$. In this paper, for a positive integer $m$, we define a natural analogue of $r_f(z)$ for the $m^{\operatorname{th}}$ symmetric power $L$-function of $f$ when $N$ is squarefree. Our analogue also has a functional equation relating $z$ and $-1/Nz$. We prove the corresponding version of the Riemann hypothesis when $k$ is large enough. Moreover, when $k>2(\operatorname{log}_2(13e^{2\pi}/9)+m)+1$, we prove our result when $N$ is large enough.