The system of representations of the Weil–Deligne group associated to an abelian variety

Abstract

Fix a number field $F\subset ℂ$, an abelian variety $A∕F$ and let $G_A$ be the Mumford–Tate group of $A_{∕ℂ}$. After replacing $F$ by finite extension one can assume that, for every prime number $\ell$, the action of the absolute Galois group ${\Gamma }_F=Gal\left(\bar{F}∕F\right)$ on the étale cohomology group $H_{ét}^1\left(A_{\bar{F}},{\mathbb{Q}}_{\ell }\right)$ factors through a morphism ${\rho }_{\ell }:{\Gamma }_F\to G_A\left({\mathbb{Q}}_{\ell }\right)$. Let $v$ be a valuation of $F$ and write ${\Gamma }_{F_v}$ for the absolute Galois group of the completion $F_v$. For every $\ell$ with $v\left(\ell \right)=0$, the restriction of ${\rho }_{\ell }$ to ${\Gamma }_{F_v}$ defines a representation ${}^{\prime }{W}_{F_v}\to G_{A∕{\mathbb{Q}}_l}$ of the Weil–Deligne group.

It is conjectured that, for every $\ell$, this representation of ${}^{\prime }{W}_{F_v}$ is defined over $\mathbb{Q}$ as a representation with values in $G_A$ and that the system above, for variable $\ell$, forms a compatible system of representations of ${}^{\prime }{W}_{F_v}$ with values in $G_A$. A somewhat weaker version of this conjecture is proved for the valuations of $F$, where $A$ has semistable reduction and for which ${\rho }_{\ell }\left({Fr}_v\right)$ is neat.