L-functions of Genus Two Abelian Coverings of Elliptic Curves over Finite Fields

TL;DR

This paper investigates the zeta-functions of genus two abelian coverings of elliptic curves over finite fields, aiming to comprehensively classify these functions with potential implications for number theory and algebraic geometry.

Contribution

It provides a complete description of the zeta-functions associated with genus two abelian coverings of elliptic curves over finite fields.

Findings

Classification of zeta-functions for genus two abelian coverings

Connections established between L-functions and Galois representations

Insights into the structure of coverings over finite fields

Abstract

Initially motivated by the relations between Anabelian Geometry and Artin's L-functions of the associated Galois-representations, here we study the list of zeta-functions of genus two abelian coverings of elliptic curves over finite fields. Our goal is to provide a complete description of such a list.