0
$\begingroup$

If $p$ is a prime, $n\in\mathbb{N}$ is a natural number and $C$ is a nonsingular curve over $\mathbb{F}_{p^{n}}$, the $\zeta$ function associated to $C\mid_{\mathbb{F}_{p^{n}}}$ is defined as \begin{equation*} \zeta_{\mathbb{F}_{p^{n}},C}(s)=\prod_{\mathfrak{m}\in\operatorname{Spec}(K)\setminus\{0\}}\frac{1}{1-N_{\mathfrak{m}}^{-s}}\text{,} \end{equation*} where $N_{\mathfrak{m}}=\#(\mathcal{O}_{K}/\mathfrak{m})$ and $K$ is the function field of $C$.

Deligne has proven that the zeroes of $\zeta_{\mathbb{F}_{p^{n}},C}$ lie on the line $\mathfrak{Re}(s)=\frac{1}{2}$.

Question: Does the set \begin{equation*} \left\{s\in\mathbb{C}\mid\exists \text{ prime } p, n\in\mathbb{N}, C\text{ curve over }\mathbb{F}_{p^{n}}:\zeta_{\mathbb{F}_{p^{n}},C}(s)=0\right\} \end{equation*} lie dense in the line $\left\{s\in\mathbb{C}:\mathfrak{Re}(s)=\frac{1}{2}\right\}$?

Improve this question
$\endgroup$
3
  • 3
    $\begingroup$ Yes, though this is fairly trivial - zeta functions over $\mathbb F_q$ are $2\pi i/\log q$-periodic, so it is enough to pick for arbitrarily large $q$ some curves which have some zero - any curve of positive genus will work for that. $\endgroup$
    – Wojowu
    Commented May 20, 2022 at 17:19
  • 2
    $\begingroup$ For curves the Riemann hypothesis was proved by Weil 25 years before Deligne treated higher-dimensional varieties. And you have omitted conditions on the curve for RH to hold: it should be projective, not affine, or missing points will lead to extra zeros off the critical line. $\endgroup$
    – KConrad
    Commented May 20, 2022 at 17:21
  • 2
    $\begingroup$ Also, the product should be not over $Spec(K)$, but over closed points on $C$ - if $C$ is affine, then this coincides with $Spec(R)\setminus\{0\}$, where $R$ is the coordinate ring of $C$, not its function field. $\endgroup$
    – Wojowu
    Commented May 20, 2022 at 19:27

0

Reset to default

You must log in to answer this question.