Let be a smooth projective curve defined over a number field and let be a twist of . In this article we relate the -adic representations attached to the -adic Tate modules of the Jacobians of and through an Artin representation. This representation induces global relations between the local factors of the respective Hasse–Weil -functions. We make these relations explicit in a particularly illustrative situation. For all but a finite number of -isomorphism classes of genus 2 curves defined over with or , we find a representative curve such that, for every isomorphism satisfying some mild condition, we are able to determine either the local factor or the product from the local factor .
Cite this article
Francesc Fité, Joan-Carles Lario, The twisting representation of the <em>L</em>-function of a curve. Rev. Mat. Iberoam. 29 (2013), no. 3, pp. 749–764DOI 10.4171/RMI/738