The twisting representation of the $L$-function of a curve

Abstract

Let C be a smooth projective curve defined over a number field k and let C 0 be a twist of C. For every isomorphism �: C 0 ! C defined over a finite Galois extension L/k, we introduce a rational Artin representation �� of the group Gal(L/k) such that the Tate module V`(C 0 ) becomes a sub-Q`[Gk]-module of �� V`(C). Besides, we define a rep