# The twisting representation of the $L$-function of a curve

### Francesc Fité

Universität Bielefeld, Germany### Joan-Carles Lario

Universitat Politècnica de Catalunya, Barcelona, Spain

## Abstract

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 illustrative situation. For all but a finite number of $Q $-isomorphism classes of genus 2 curves defined over $Q$ with $Aut(C)≃D_{8}$ or $D_{12}$, we find a representative curve $C/Q$ such that, for every isomorphism $ϕ:C_{′}→C$ 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 $L_{p}(C/Q,T)$.

## Cite this article

Francesc Fité, Joan-Carles Lario, The twisting representation of the $L$-function of a curve. Rev. Mat. Iberoam. 29 (2013), no. 3, pp. 749–764

DOI 10.4171/RMI/738