Zeros of symmetric power period polynomials
Robert Dicks, Hui Xue
TL;DR
This work generalizes the classical period polynomial of a newform $f$ by defining the $m$-th symmetric power analogue $R_{m,f}(z)$ using the completed symmetric power $L$-function $L_{m,f}^*(s)$ and its functional equation. The main result proves that all zeros of $R_{m,f}(z)$ lie on the symmetry circle $|z|=1/\sqrt{N}$ in the squarefree level case under either large weight $k$ or large level $N$, with an explicit large-$N$ condition $k>2(\log_2(13e^{2\pi}/9)+m)+1$. The proof reduces the problem to zeros of self-inversive polynomials by constructing auxiliary polynomials $P_{m,f}$, $Q_{m,f}$ (odd $m$) or $p_{m,f}$, $q_{m,f}$ (even $m$) and applying the Lalín–Smyth ElG,R criterion together with Rouché’s theorem to control zeros inside the unit disk, thereby deducing unit-circle zeros for the transformed polynomials. The results extend the $m=1$ case of Ono–Soundararajan and relate to broader questions about the zeros of motivic-like polynomials, with implications for special values of symmetric power $L$-functions and period-type generating functions.
Abstract
Suppose that $k$ and $N$ are positive integers. Let $f$ be a newform on $Γ_0(N)$ 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 $k>2(\operatorname{log}_2(13e^{2π}/9)+m)+1$, we prove our result when $N$ is large enough.