The twisting representation of the \(L\)-function of a curve

Let C be a smooth projective curve defined over a number field and let C' be a twist of C. In this article we relate the l-adic representations attached to the l-adic Tate modules of the Jacobians of C and C' through an Artin representation. This representation induces global relations bet...

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Bibliographic Details
Published in:arXiv.org 2012-12
Main Authors: Fité, Francesc, Joan-C Lario
Format: Article
Language:English
Subjects:
Automorphisms
Isomorphism
Jacobians
Number theory
Representations
Twisting
Online Access:Get full text
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Summary:Let C be a smooth projective curve defined over a number field and let C' be a twist of C. In this article we relate the l-adic representations attached to the l-adic Tate modules of the Jacobians of C and C' through an Artin representation. This representation induces global relations between the local factors of the respective Hasse-Weil L-functions. We make these relations explicit in a particularly illustrating situation. For every Qbar-isomorphism class of genus 2 curves defined over Q with automorphism group isomorphic to D_8 or D_{12}, except for a finite number, we choose a representative curve C/Q such that, for every twist C' of C satisfying some mild condition, we are able to determine either the local factor L_p(C'/Q,T) or the product L_p(C'/Q,T)L_p(C'/Q,-T) from the local factor L_p(C/Q,T).
ISSN:2331-8422