Derived Zeta Functions for Curves over Finite Fields (2203.11488v1)

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.