Maps between curves and arithmetic obstructions

TL;DR

This paper investigates methods to determine the existence of rational maps between curves over finite fields using L-functions and covers, with applications to polynomial factorization.

Contribution

It introduces a new approach using L-functions of covers to detect rational maps and proposes a specific family of covers for testing isomorphism between curves.

Findings

L-functions can indicate the existence of rational maps between curves

A new family of covers helps determine curve isomorphism

Application to polynomial factorization over finite fields

Abstract

Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to address the special case of determining when X and Y are isomorphic. We also discuss an application to factoring polynomials over finite fields.