Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.18024 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912210535579648 |
|---|---|
| author | Dicks, Robert Xue, Hui |
| author_facet | Dicks, Robert Xue, Hui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18024 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros of symmetric power period polynomials Dicks, Robert Xue, Hui Number Theory 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. |
| title | Zeros of symmetric power period polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.18024 |