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# 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

^{[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. Iberoam.* 29 (2013), 749-764. doi: 10.4171/RMI/738