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 be an abelian motive in the sense of Yves André. Then the -adic realisations of 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 does not depend on . In particular, the Mumford–Tate conjecture for 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 un motif abélien à la Yves André. Alors les réalisations -adiques de forment un système quasi-compatible de représentations galoisiennes. Comme application, on en déduit que le rang absolu des groupes de monodromie -adiques de ne dépend pas de . En particulier, la conjecture de Mumford–Tate pour ne dépend pas de .
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Keywords: Galois representations, $\ell $-adic cohomology, abelian motives, compatability
Mot clés : Représentation galoisienne, cohomologie $\ell $-adique, motifs abéliens, compatibilité
CC-BY-ND 4.0
@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},
pages = {2089--2120},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {69},
number = {5},
year = {2019},
doi = {10.5802/aif.3290},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3290/}
}
TY - JOUR AU - Commelin, Johan M. TI - On compatibility of the $\ell $-adic realisations of an abelian motive JO - Annales de l'Institut Fourier PY - 2019 SP - 2089 EP - 2120 VL - 69 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3290/ DO - 10.5802/aif.3290 LA - en ID - AIF_2019__69_5_2089_0 ER -
%0 Journal Article %A Commelin, Johan M. %T On compatibility of the $\ell $-adic realisations of an abelian motive %J Annales de l'Institut Fourier %D 2019 %P 2089-2120 %V 69 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3290/ %R 10.5802/aif.3290 %G en %F 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/articles/10.5802/aif.3290/
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