2022/03/22 by Lin Weng, Weng, Lin
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2203.11488
openalex publication_date 2022/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For each (m+1)-tuple \bf nm=(n0,n1,…,nm) of positive integers, the \bf nm-derived zeta function \widehatζX,\mathbb Fq^ (\bf nm)(s) is defined for a curve X over \mathbb Fq. This derived zeta function satisfies standard zeta properties. In particular, similar to the Artin Zeta function of X/\mathbb Fq, this \bf nm-derived Zeta function of X over \mathbb Fq is a ratio of a degree 2g polynomial PX,\mathbb Fq^(\bf nm) in T_\bf nm=q^-s∏k=0mnk by (1-T_\bf nm)(1-q_\bf nmT_\bf nm)T_\bf nmg-1 with q_\bf nm=q^∏k=0mnk. Indeed, we have \beginaligned amp;\widehat ζX,\mathbb Fq^ (\bf nm)(s)=\widehat ZX,\mathbb Fq^ (\bf nm)(T_\bf nm)
=amp; (∑ℓ=0g-2αX,\mathbb Fq^(\bf nm)(ℓ)(T_\bf nmℓ-(g-1)+q_\bf nm(g-1)-ℓT_\bf nm(g-1)-ℓ) +αX,\mathbb Fq^(\bf nm)(g-1))))+\frac(q_\bf nm-1)T_\bf nmβX,\mathbb Fq^(\bf nm)(1-T_\bf nm)(1-q_\bf nmT_\bf nm)
\endaligned for some \bf nm-derived alpha and beta invariants of X/\mathbb Fq. Furthermore, when X restrict to an elliptic curve, or when \bf nm=(2,2,… 2), established is the \bf nm-derived Riemann hypothesis claiming that all zeros of \widehat ζX,\mathbb Fq^ (\bf nm)(s) lie on the central line \Re(s)=(1)/(2). In addition, formulated is the Positivity Conjecture claiming that the above \bf nm-derived alpha and beta invariants are all strict positivity.