no. 5
On compatibility of the ℓ-adic realisations of an abelian motive
Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2089-2120.

In this article we introduce the notion of quasi-compatible system of Galois representations. The quasi-compatibility condition is a mild relaxation of the classical compatibility condition in the sense of Serre. The main theorem that we prove is the following: Let M be an abelian motive in the sense of Yves André. Then the ℓ-adic realisations of M form a quasi-compatible system of Galois representations. (In Theorem 5.1 we actually prove something stronger.) As an application, we deduce that the absolute rank of the ℓ-adic monodromy groups of M does not depend on ℓ. In particular, the Mumford–Tate conjecture for M does not depend on ℓ.

Dans cet article, nous introduisons la notion de système quasi-compatible de représentations galoisiennes. La condition de quasi-compatibilité est un affaiblissement de la condition de compatibilité à la Serre. Le principal théorème que nous prouvons est le suivant : Soit M un motif abélien à la Yves André. Alors les réalisations ℓ-adiques de M forment un système quasi-compatible de représentations galoisiennes. Comme application, on en déduit que le rang absolu des groupes de monodromie ℓ-adiques de M ne dépend pas de ℓ. En particulier, la conjecture de Mumford–Tate pour M ne dépend pas de ℓ.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3290
Classification: 14F20
Keywords: Galois representations, ℓ-adic cohomology, abelian motives, compatability 
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Commelin, Johan M. On compatibility of the $\ell $-adic realisations of an abelian motive. Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2089-2120. doi : 10.5802/aif.3290. https://aif.centre-mersenne.org/articles/10.5802/aif.3290/

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