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

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 $\mathbb{F}_q$.