Derived Zeta Functions for Curves over Finite Fields

Abstract

For each $(m+1)$-tuple ${\bf n}_m=(n_0,n_1,\ldots,n_m)$ of positive integers, the ${\bf n}_m$-derived zeta function $\widehat\zeta_{X,\mathbb F_q}^{\,({\bf n}_m)}(s)$ 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 ${\bf n}_m$-derived Zeta function of $X$ over $\mathbb F_q$ is a ratio of a degree $2g$ polynomial $P_{X,\mathbb F_q}^{({\bf n}_m)}$ in $T_{{\bf n}_m}=q^{-s\prod_{k=0}^mn_k}$ by $(1-T_{{\bf n}_m})(1-q_{{\bf n}_m}T_{{\bf n}_m})T_{{\bf n}_m}^{g-1}$ with $q_{{\bf n}_m}=q^{\prod_{k=0}^mn_k}$. Indeed, we have $$\begin{aligned} &\widehat \zeta_{X,\mathbb F_q}^{\,({\bf n}_{m})}(s)=\widehat Z_{X,\mathbb F_q}^{\,({\bf n}_{m})}(T_{{\bf n}_{m}})\\ =& \left(\sum_{\ell=0}^{g-2}\alpha_{X,\mathbb F_q}^{({\bf n}_{m})}(\ell)\Big(T_{{\bf n}_{m}}^{\ell-(g-1)}+q_{{\bf n}_{m}}^{(g-1)-\ell}T_{{\bf n}_{m}}^{(g-1)-\ell}\Big) +\alpha_{X,\mathbb F_q}^{({\bf n}_{m})}(g-1))\Big)\right)+\frac{(q_{{\bf n}_{m}}-1)T_{{\bf n}_{m}}\beta_{X,\mathbb F_q}^{({\bf n}_{m})}}{(1-T_{{\bf n}_{m}})(1-q_{{\bf n}_{m}}T_{{\bf n}_{m}})}\\ \end{aligned}$$ for some ${\bf n}_m$-derived alpha and beta invariants of $X/\mathbb F_q$. Furthermore, when $X$ restrict to an elliptic curve, or when ${\bf n}_m=(2,2,\ldots 2)$, established is the ${\bf n}_m$-derived Riemann hypothesis claiming that all zeros of $\widehat \zeta_{X,\mathbb F_q}^{\,({\bf n}_{m})}(s)$ lie on the central line $\Re(s)=\frac{1}{2}$. In addition, formulated is the Positivity Conjecture claiming that the above ${\bf n}_m$-derived alpha and beta invariants are all strict positivity.