Derived Zeta Functions for Curves over Finite Fields

Abstract

For each $(m+1)$-tuple ${\mathbf{n}}_m=(n_0,n_1,\dots ,n_m)$ of positive integers, the ${\mathbf{n}}_m$-derived zeta function ${\hat{\zeta }}_{X,{\mathbb{F}}_q}^{({\mathbf{n}}_m)}(s)$ is defined for a curve $X$ over ${\mathbb{F}}_q$. This derived zeta function satisfies standard zeta properties. In particular, similar to the Artin Zeta function of $X/{\mathbb{F}}_q$, this ${\mathbf{n}}_m$-derived Zeta function of $X$ over ${\mathbb{F}}_q$ is a ratio of a degree $2g$ polynomial $P_{X,{\mathbb{F}}_q}^{({\mathbf{n}}_m)}$ in $T_{{\mathbf{n}}_m}=q^{-s{\prod }_{k=0}^m{n}_k}$ by $(1-T_{{\mathbf{n}}_m})(1-q_{{\mathbf{n}}_m}{T}_{{\mathbf{n}}_m})T_{{\mathbf{n}}_m}^{g-1}$ with $q_{{\mathbf{n}}_m}=q^{{\prod }_{k=0}^m{n}_k}$. Indeed, we have $$\begin{array}{rl}& {\hat{\zeta }}_{X,{\mathbb{F}}_q}^{({\mathbf{n}}_m)}(s)={\hat{Z}}_{X,{\mathbb{F}}_q}^{({\mathbf{n}}_m)}(T_{{\mathbf{n}}_m})\\ =& \left(\sum _{\ell =0}^{g-2}{\alpha }_{X,{\mathbb{F}}_q}^{({\mathbf{n}}_m)}(\ell )(T_{{\mathbf{n}}_m}^{\ell -(g-1)}+q_{{\mathbf{n}}_m}^{(g-1)-\ell }{T}_{{\mathbf{n}}_m}^{(g-1)-\ell })+{\alpha }_{X,{\mathbb{F}}_q}^{({\mathbf{n}}_m)}(g-1)))\right)+\frac{(q_{{\mathbf{n}}_m}-1)T_{{\mathbf{n}}_m}{\beta }_{X,{\mathbb{F}}_q}^{({\mathbf{n}}_m)}}{(1-T_{{\mathbf{n}}_m})(1-q_{{\mathbf{n}}_m}{T}_{{\mathbf{n}}_m})}\end{array}$$ for some ${\mathbf{n}}_m$-derived alpha and beta invariants of $X/{\mathbb{F}}_q$. Furthermore, when $X$ restrict to an elliptic curve, or when ${\mathbf{n}}_m=(2,2,\dots 2)$, established is the ${\mathbf{n}}_m$-derived Riemann hypothesis claiming that all zeros of ${\hat{\zeta }}_{X,{\mathbb{F}}_q}^{({\mathbf{n}}_m)}(s)$ lie on the central line $\Re (s)=\frac{1}{2}$. In addition, formulated is the Positivity Conjecture claiming that the above ${\mathbf{n}}_m$-derived alpha and beta invariants are all strict positivity.