#### Vol. 7, No. 2, 2013

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

### Rutger Noot

Vol. 7 (2013), No. 2, 243–281
##### 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 ${\Gamma }_{F}=Gal\left(\stackrel{̄}{F}∕F\right)$ on the étale cohomology group ${H}_{ét}^{1}\left({A}_{\stackrel{̄}{F}},{ℚ}_{\ell }\right)$ factors through a morphism ${\rho }_{\ell }:{\Gamma }_{F}\to {G}_{A}\left({ℚ}_{\ell }\right)$. 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 ${}^{\prime }{W}_{{F}_{v}}\to {G}_{A∕{ℚ}_{l}}$ of the Weil–Deligne group.

It is conjectured that, for every $\ell$, this representation of ${}^{\prime }{W}_{{F}_{v}}$ is defined over $ℚ$ 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.

##### Keywords
abelian variety, compatible system of Galois representations, Weil–Deligne group
##### Mathematical Subject Classification 2000
Primary: 11G10
Secondary: 14K15, 14F20