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

Abstract

Let $C$ be a smooth projective curve defined over a number field and let $C^{\prime }$ 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 $C^{\prime }$ 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 $\mathrm{Aut}(C)\simeq D_8$ or $D_{12}$, we find a representative curve $C/\mathbb{Q}$ such that, for every isomorphism $\varphi :C^{\prime }\to C$ satisfying some mild condition, we are able to determine either the local factor $L_p(C^{\prime }/\mathbb{Q},T)$ or the product $L_p(C^{\prime }/\mathbb{Q},T)\cdot L_p(C^{\prime }/\mathbb{Q},-T)$ from the local factor $L_p(C/\mathbb{Q},T)$.