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 . 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 ′ .
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