On compatibility of the -adic realisations of an abelian motive
Annales de l'Institut Fourier, Volume 69 (2019) no. 5, p. 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 -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 -adic monodromy groups of M does not depend on . In particular, the Mumford–Tate conjecture for M does not depend on .

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 -adiques de M forment un système quasi-compatible de représentations galoisiennes. Comme application, on en déduit que le rang absolu des groupes de monodromie -adiques de M ne dépend pas de . En particulier, la conjecture de Mumford–Tate pour M ne dépend pas de .

Received : 2017-09-12
Revised : 2018-02-22
Accepted : 2018-04-26
Published online : 2019-09-16
DOI : https://doi.org/10.5802/aif.3290
Classification:  14F20
Keywords: Galois representations, -adic cohomology, abelian motives, compatability
@article{AIF_2019__69_5_2089_0,
     author = {Commelin, Johan M.},
     title = {On compatibility of the $\ell $-adic realisations of an abelian motive},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {5},
     year = {2019},
     pages = {2089-2120},
     doi = {10.5802/aif.3290},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_5_2089_0}
}
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/item/AIF_2019__69_5_2089_0/

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