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

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

2010-12-15

arxiv.org/abs/1012.3393v1

Mathematics

Number Theory

20 pages

Scientific paper

Let $C$ be a smooth projective curve defined over a number field $k$ and let $C'$ be a twist of $C$. For every isomorphism $\phi\colon C'\rightarrow C$ defined over a finite Galois extension $L/k$, we introduce a rational Artin representation $\theta_\phi$ of the group $\Gal(L/k)$ such that the Tate module $V_\ell(C')$ becomes a sub-$\Q_\ell[G_k]$-module of $\theta_\phi\otimes V_\ell(C)$. Besides, we define a representation $\theta_C$, the twisting representation of the $L$-function of $C$, which only depends on $C$ and such that $\theta_\phi$ factors through $\theta_C$ whenever $L$ satisfies that ${L}^0(J(C))=K^0(J(C))$, where $K$ is the field of definition of $\Aut(C)$. For curves of genus 2, under some hypothesis, we are able to compute the decomposition of $\theta_\phi\otimes V_\ell(C)$ into simple $\Q_\ell[G_k]$-modules. Thanks to this decomposition, from the local factor $L_{\mathfrak p}(C/k,T)$ and the representation $\theta_\phi$, we determine either $L_{\mathfrak p}(C'/k,T)$ or $L_{\mathfrak p}(C'/k,T)\cdot L_{\mathfrak p}(C'/k,-T)$.

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