Riemann Hypothesis Analogue for locally finite modules over the absolute Galois group of a finite field

TL;DR

This paper establishes a sufficient condition under which a locally finite module over the absolute Galois group of a finite field satisfies a Riemann Hypothesis analogue, extending beyond smooth projective varieties.

Contribution

It introduces a new sufficient condition for the Riemann Hypothesis analogue in this setting and constructs an explicit example showing broader applicability.

Findings

Condition holds for all smooth irreducible projective curves of positive genus over F.

Constructed example demonstrates the result applies to a larger class than previously known.

Extends the scope of Riemann Hypothesis analogue to new classes of modules.

Abstract

The article provides a sufficient condition for a locally finite module over the absolute Galois group of a finite field F to satisfy the Riemann Hypothesis Analogue with respect to the projective line. The condition holds for all smooth irreducible projective curves of positive genus, defined over F. By construction of an explicit example we establish that the scope of our main result is larger than the class of the smooth irreducible projective varieties, defined over F.