Artin representations attached to pairs of isogenous abelian varieties

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. Moreover, from this Artin representation, we are able to compute the moments of the Sato–Tate distributions of the traces of the local factors of the curve C′.