RIEMANN HYPOTHESIS ANALOGUE FOR LOCALLY FINITE MODULES OVER THE ABSOLUTE GALOIS GROUP OF A FINITE FIELD

Abstract

The article provides a sufficient condition for a locally finite module $M$ over the absolute Galois group $\mathfrak{G}=Gal(\overline{{\mathbb{F}}_q}/{\mathbb{F}}_q)$ of a finite field ${\mathbb{F}}_q$ to satisfy the Riemann Hypothesis Analogue with respect to the projective line ${\mathbb{P}}^1(\overline{{\mathbb{F}}_q})$. The condition holds for all smooth irreducible projective curves of positive genus, defined over ${\mathbb{F}}_q$. We give an explicit example of a locally finite module, subject to the assumptions of our main theorem and, therefore, satisfying the Riemann Hypothesis Analogue with respect to ${\mathbb{P}}^1(\overline{{\mathbb{F}}_q})$, which is not isomorphic to a smooth irreducible projective curve, defined over ${\mathbb{F}}_q$.