l- independence for a sytem of motivic representations.

Abstract : Let $X$ be an smooth projective algebraic variety over a number field $F \subset \mathbb{C}$. Suppose that the Absolute Hodge(AH) motive $M:= h^i(X)$ is contained in the Tannakian category generated by the AH motives of abelian varieties. For every prime number $\ell$ the Galois group $\Gamma_F:= Gal(\bar{F}/F)$ acts on $H_\ell(M)$, the $\ell$-adic realization of $M$. Over a finite extension of $F$ this action factorizes as $\rho_{M,\ell}:\Gamma_F\rightarrow G_M(\ql)$, where $G_M$ is the Mumford-Tate group of $M$. Fix a valuation $v$ of $F$ and suppose $v(\ell)=0$. The restriction $\rho_{M,\ell} \vert _{\Gamma_{F_v}}$ defines a representation ${}'W_v \rightarrow G_{M/\ql}$ of the Weil-Deligne group of $F_v$. J-P Serre and J-M Fontaine (independently) have made conjectures that indicates that ${}'W_v \rightarrow G_{M/\ql}$ should be defined over $\mathbb{Q}$ for $\ell$ fixed and that these representations form a compatible system for variable $\ell$. Under certain additional hypothesis , we answer these questions in affirmative, when $X$ has good reduction or Semi-Stable reduction at $v$.
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Submitted on : Friday, November 25, 2011 - 4:21:30 PM
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• HAL Id : tel-00644861, version 1

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Abhijit Laskar. l- independence for a sytem of motivic representations.. Number Theory [math.NT]. Université de Strasbourg, 2011. English. ⟨NNT : 2011STRA6187⟩. ⟨tel-00644861⟩

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