Artin representations attached to pairs of isogenous abelian varieties

TL;DR

This paper introduces Artin representations linked to pairs of isogenous abelian varieties over number fields, revealing relations between their L-functions and enabling explicit computations for genus 3 curve twists.

Contribution

It defines a new rational Artin representation associated with isogenous abelian varieties and demonstrates its application in computing L-functions of twisted curves.

Findings

Artin representations encode relations between L-functions of isogenous varieties.

Explicit computation of Artin representations for genus 3 curve Jacobians.

Application of the representations to determine L-functions of twisted curves.

Abstract

Given a pair of abelian varieties defined over a number field k and isogenous over a finite Galois extension L/k, we define a rational Artin representation of the group Gal(L/k) that shows a global relation between the L-functions of each variety and provides certain information about their decomposition up to isogeny over L. We study several properties of these Artin representations. As an application, for each curve C' in a family of twists of a certain genus 3 curve C, we explicitly compute the Artin representation attached to the Jacobians of C and C' and show how this Artin representation can be used to determine the L-function of the curve C' in terms of the L-function of C.