  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.

Number Theory