Revista Matemática Iberoamericana

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Volume 29, Issue 3, 2013, pp. 749–764
DOI: 10.4171/RMI/738

Published online: 2013-08-04

The twisting representation of the L-function of a curve

Francesc Fité[1] and Joan-Carles Lario[2]

(1) Universität Bielefeld, Germany
(2) Universitat Politècnica de Catalunya, Barcelona, Spain

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 $\ell$-adic representations attached to the $\ell$-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 illustrative situation. For all but a finite number of $\overline{\mathbb{Q}}$-isomorphism classes of genus 2 curves defined over $\mathbb{Q}$ with $\operatorname{Aut}(C)\simeq D_8$ or $D_{12}$, we find a representative curve $C/\mathbb{Q}$ such that, for every isomorphism $\phi\colon C'\rightarrow C$ satisfying some mild condition, we are able to determine either the local factor $L_{ p}(C'/\mathbb{Q},T)$ or the product $L_{p}(C'/\mathbb{Q},T)\cdot L_{p}(C'/\mathbb{Q},-T)$ from the local factor $L_{p}(C/\mathbb{Q},T)$.

Keywords: Abelian varieties, genus 2 curves, L-functions, Artin representations

Fité Francesc, Lario Joan-Carles: The twisting representation of the L-function of a curve. Rev. Mat. Iberoamericana 29 (2013), 749-764. doi: 10.4171/RMI/738