no. 5
On compatibility of the ℓ-adic realisations of an abelian motive
Commelin, Johan M.1
1 Albert–Ludwigs-Universität Freiburg Mathematisches Institut Ernst-Zermelo-Straße 1 79104 Freiburg im Breisgau (Germany)
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, $\ell $-adic cohomology, abelian motives, compatability
Mots-clés : Représentation galoisienne, cohomologie $\ell $-adique, motifs abéliens, compatibilité

Commelin, Johan M. 1

1 Albert–Ludwigs-Universität Freiburg Mathematisches Institut Ernst-Zermelo-Straße 1 79104 Freiburg im Breisgau (Germany)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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/

[1] André, Yves Pour une théorie inconditionnelle des motifs, Publ. Math., Inst. Hautes Étud. Sci. (1996) no. 83, pp. 5-49 | DOI | MR | Zbl

[2] Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Théorie des topos et cohomologie étale des schémas. Tome 3 (Artin, Michael; Grothendieck, Alexander; Verdier, Jean-Louis, eds.), Lecture Notes in Mathematics, 305, Springer, 1973, vi+640 pages (avec la collaboration de P. Deligne et B. Saint-Donat) | MR | Zbl

[3] Borel, Armand; de Siebenthal, Jean Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. (1949) no. 23, pp. 200-221 | DOI | Zbl

[4] Bourbaki, Nicolas Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples, Springer, 2012, x+489 pages (Second revised edition of the 1958 edition) | DOI | MR | Zbl

[5] Chi, Wên Chên ℓ-adic and λ-adic representations associated to abelian varieties defined over number fields, Am. J. Math., Volume 114 (1992) no. 2, pp. 315-353 | DOI | MR | Zbl

[6] Commelin, Johan On ℓ-adic compatibility for abelian motives & the Mumford–Tate conjecture for products of K3 surfaces, Radboud Universiteit Nijmegen (2017) (Ph. D. Thesis) | Zbl

[7] Deligne, Pierre Théorie de Hodge. III, Publ. Math., Inst. Hautes Étud. Sci. (1974) no. 44, pp. 5-77 | DOI | MR | Zbl

[8] Deligne, Pierre Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and L-functions (Oregon State University, Corvallis, Oregon, 1977), Part 2 (Proceedings of Symposia in Pure Mathematics), Volume XXXIII, American Mathematical Society, 1979, pp. 247-289 | MR | Zbl

[9] Faltings, Gerd Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., Volume 73 (1983) no. 3, pp. 349-366 | DOI | MR | Zbl

[10] Faltings, Gerd Complements to Mordell, Rational points (Bonn, 1983/1984) (Aspects of Mathematics), Volume E6, Vieweg & Sohn, 1984, pp. 203-227 | DOI | MR | Zbl

[11] Ferrand, Daniel Un foncteur norme, Bull. Soc. Math. Fr., Volume 126 (1998) no. 1, pp. 1-49 | DOI | MR | Zbl

[12] van Geemen, Bert Half twists of Hodge structures of cm-type, J. Math. Soc. Japan, Volume 53 (2001) no. 4, pp. 813-833 | DOI | MR | Zbl

[13] Katz, Nicholas; Messing, William Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math., Volume 23 (1974), pp. 73-77 | DOI | MR | Zbl

[14] Kisin, Mark mod p points on Shimura varieties of abelian type, J. Am. Math. Soc., Volume 30 (2017) no. 3, pp. 819-914 | DOI | MR | Zbl

[15] Larsen, Michael; Pink, Richard On ℓ-independence of algebraic monodromy groups in compatible systems of representations, Invent. Math., Volume 107 (1992) no. 3, pp. 603-636 | DOI | MR | Zbl

[16] Larsen, Michael; Pink, Richard Abelian varieties, ℓ-adic representations, and ℓ-independence, Math. Ann., Volume 302 (1995) no. 3, pp. 561-579 | DOI | MR | Zbl

[17] Laskar, Abhijit ℓ-independence for a system of motivic representations, Manuscr. Math., Volume 145 (2014) no. 1-2, pp. 125-142 | DOI | MR | Zbl

[18] Moonen, Ben On the Tate and Mumford–Tate conjectures in codimension 1 for varieties with h 2,0 =1, Duke Math. J., Volume 166 (2017) no. 4, pp. 739-799 | DOI | MR | Zbl

[19] Noot, Rutger Models of Shimura varieties in mixed characteristic, J. Algebr. Geom., Volume 5 (1996) no. 1, pp. 187-207 | MR | Zbl

[20] Noot, Rutger Classe de conjugaison du Frobenius d’une variété abélienne sur un corps de nombres, J. Lond. Math. Soc., Volume 79 (2009) no. 1, pp. 53-71 | DOI | MR | Zbl

[21] Noot, Rutger The system of representations of the Weil–Deligne group associated to an abelian variety, Algebra Number Theory, Volume 7 (2013) no. 2, pp. 243-281 | DOI | MR | Zbl

[22] Pepin Lehalleur, Simon Subgroups of maximal rank of reductive groups, Autour des schémas en groupes. Vol. III (Panoramas et Synthèses), Volume 47, Société Mathématique de France, 2015, pp. 147-172 | MR | Zbl

[23] Pink, Richard The Mumford–Tate conjecture for Drinfeld-modules, Publ. Res. Inst. Math. Sci., Volume 33 (1997) no. 3, pp. 393-425 | DOI | MR | Zbl

[24] Ribet, Kenneth Galois action on division points of Abelian varieties with real multiplications, Am. J. Math., Volume 98 (1976) no. 3, pp. 751-804 | DOI | MR | Zbl

[25] Schappacher, Norbert Periods of Hecke characters, Lecture Notes in Mathematics, 1301, Springer, 1988, xvi+160 pages | DOI | MR | Zbl

[26] Serre, Jean-Pierre Zeta and L-functions, Arithmetical Algebraic Geometry (Proceedings of a Conference held at Purdue Univiversity, December 5–7, 1963), Harper & Row, 1965, pp. 82-92 | MR | Zbl

[27] Serre, Jean-Pierre Abelian l-adic representations and elliptic curves, Research Notes in Mathematics, 7, A K Peters, 1998, 199 pages (With the collaboration of Willem Kuyk and John Labute, Revised reprint of the 1968 original) | MR | Zbl

[28] Serre, Jean-Pierre Lectures on N X (p), CRC Research Notes in Mathematics, 11, CRC Press, 2012, x+163 pages | MR | Zbl

[29] Serre, Jean-Pierre; Tate, John Good reduction of abelian varieties, Ann. Math., Volume 88 (1968), pp. 492-517 | DOI | MR | Zbl

[30] Shimura, Goro Algebraic number fields and symplectic discontinuous groups, Ann. Math., Volume 86 (1967), pp. 503-592 | DOI | MR | Zbl

[31] Tits, Jacques Reductive groups over local fields, Automorphic forms, representations and L-functions (Oregon State University, Corvallis, Oregon, 1977), Part 1 (Proceedings of Symposia in Pure Mathematics), Volume XXXIII, American Mathematical Society, 1979, pp. 29-69 | MR | Zbl

[32] Ullmo, Emmanuel; Yafaev, Andrei Mumford–Tate and generalised Shafarevich conjectures, Ann. Math. Qué., Volume 37 (2013) no. 2, pp. 255-284 | DOI | MR | Zbl

[33] Vasiu, Adrian Some cases of the Mumford–Tate conjecture and Shimura varieties, Indiana Univ. Math. J., Volume 57 (2008) no. 1, pp. 1-75 | DOI | MR | Zbl

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