The twisting representation of the <em>L</em>-function of a curve

  • Francesc Fité

    Universität Bielefeld, Germany
  • Joan-Carles Lario

    Universitat Politècnica de Catalunya, Barcelona, Spain

Abstract

Let CC be a smooth projective curve defined over a number field and let CC' be a twist of CC. In this article we relate the \ell-adic representations attached to the \ell-adic Tate modules of the Jacobians of CC and CC' through an Artin representation. This representation induces global relations between the local factors of the respective Hasse–Weil LL-functions. We make these relations explicit in a particularly illustrative situation. For all but a finite number of Q\overline{\mathbb{Q}}-isomorphism classes of genus 2 curves defined over Q\mathbb{Q} with Aut(C)D8\operatorname{Aut}(C)\simeq D_8 or D12D_{12}, we find a representative curve C/QC/\mathbb{Q} such that, for every isomorphism ϕ ⁣:CC\phi\colon C'\rightarrow C satisfying some mild condition, we are able to determine either the local factor Lp(C/Q,T)L_{ p}(C'/\mathbb{Q},T) or the product Lp(C/Q,T)Lp(C/Q,T)L_{p}(C'/\mathbb{Q},T)\cdot L_{p}(C'/\mathbb{Q},-T) from the local factor Lp(C/Q,T)L_{p}(C/\mathbb{Q},T).

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. 749–764

DOI 10.4171/RMI/738