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 .
Revised:
Accepted:
Published online:
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}, 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 TI - On compatibility of the $\ell $-adic realisations of an abelian motive JO - Annales de l'Institut Fourier PY - 2019 DA - 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/ UR - https://doi.org/10.5802/aif.3290 DO - 10.5802/aif.3290 LA - en ID - AIF_2019__69_5_2089_0 ER -
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] Pour une théorie inconditionnelle des motifs, Publ. Math., Inst. Hautes Étud. Sci. (1996) no. 83, pp. 5-49 | Article | MR: 1423019 | Zbl: 0874.14010
[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, Tome 305, Springer, 1973, vi+640 pages (avec la collaboration de P. Deligne et B. Saint-Donat) | MR: 0354654 | Zbl: 0245.00002
[3] Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. (1949) no. 23, pp. 200-221 | Article | Zbl: 0034.30701
[4] Éléments de mathématique. Algèbre. Chapitre 8. Modules et anneaux semi-simples, Springer, 2012, x+489 pages (Second revised edition of the 1958 edition) | Article | MR: 3027127 | Zbl: 1245.16001
[5] -adic and -adic representations associated to abelian varieties defined over number fields, Am. J. Math., Tome 114 (1992) no. 2, pp. 315-353 | Article | MR: 1156568 | Zbl: 0795.14024
[6] On -adic compatibility for abelian motives & the Mumford–Tate conjecture for products of K3 surfaces (2017) (Ph. D. Thesis)
[7] Théorie de Hodge. III, Publ. Math., Inst. Hautes Étud. Sci. (1974) no. 44, pp. 5-77 | Article | MR: 0498552 | Zbl: 0237.14003
[8] Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and -functions (Oregon State University, Corvallis, Oregon, 1977), Part 2 (Proceedings of Symposia in Pure Mathematics) Tome XXXIII, American Mathematical Society, 1979, pp. 247-289 | MR: 546620 | Zbl: 0437.14012
[9] Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., Tome 73 (1983) no. 3, pp. 349-366 | Article | MR: 718935 | Zbl: 0588.14026
[10] Complements to Mordell, Rational points (Bonn, 1983/1984) (Aspects of Mathematics) Tome E6, Vieweg & Sohn, 1984, pp. 203-227 | Article | MR: 766574 | Zbl: 0586.14012
[11] Un foncteur norme, Bull. Soc. Math. Fr., Tome 126 (1998) no. 1, pp. 1-49 | Article | MR: 1651380 | Zbl: 1017.13005
[12] Half twists of Hodge structures of cm-type, J. Math. Soc. Japan, Tome 53 (2001) no. 4, pp. 813-833 | Article | MR: 1852884 | Zbl: 1074.14509
[13] Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math., Tome 23 (1974), pp. 73-77 | Article | MR: 0332791 | Zbl: 0275.14011
[14] points on Shimura varieties of abelian type, J. Am. Math. Soc., Tome 30 (2017) no. 3, pp. 819-914 | Article | MR: 3630089 | Zbl: 1384.11075
[15] On -independence of algebraic monodromy groups in compatible systems of representations, Invent. Math., Tome 107 (1992) no. 3, pp. 603-636 | Article | MR: 1150604 | Zbl: 0778.11036
[16] Abelian varieties, -adic representations, and -independence, Math. Ann., Tome 302 (1995) no. 3, pp. 561-579 | Article | MR: 1339927 | Zbl: 0867.14019
[17] -independence for a system of motivic representations, Manuscr. Math., Tome 145 (2014) no. 1-2, pp. 125-142 | Article | MR: 3244729 | Zbl: 1301.11054
[18] On the Tate and Mumford–Tate conjectures in codimension for varieties with , Duke Math. J., Tome 166 (2017) no. 4, pp. 739-799 | Article | MR: 3619305 | Zbl: 1372.14008
[19] Models of Shimura varieties in mixed characteristic, J. Algebr. Geom., Tome 5 (1996) no. 1, pp. 187-207 | MR: 1358041 | Zbl: 0864.14015
[20] Classe de conjugaison du Frobenius d’une variété abélienne sur un corps de nombres, J. Lond. Math. Soc., Tome 79 (2009) no. 1, pp. 53-71 | Article | MR: 2472133 | Zbl: 1177.14084
[21] The system of representations of the Weil–Deligne group associated to an abelian variety, Algebra Number Theory, Tome 7 (2013) no. 2, pp. 243-281 | Article | MR: 3123639 | Zbl: 1319.11038
[22] Subgroups of maximal rank of reductive groups, Autour des schémas en groupes. Vol. III (Panoramas et Synthèses) Tome 47, Société Mathématique de France, 2015, pp. 147-172 | MR: 3525844 | Zbl: 1356.14036
[23] The Mumford–Tate conjecture for Drinfeld-modules, Publ. Res. Inst. Math. Sci., Tome 33 (1997) no. 3, pp. 393-425 | Article | MR: 1474696 | Zbl: 0895.11025
[24] Galois action on division points of Abelian varieties with real multiplications, Am. J. Math., Tome 98 (1976) no. 3, pp. 751-804 | Article | MR: 0457455 | Zbl: 0348.14022
[25] Periods of Hecke characters, Lecture Notes in Mathematics, Tome 1301, Springer, 1988, xvi+160 pages | Article | MR: 935127 | Zbl: 0659.14001
[26] Zeta and -functions, Arithmetical Algebraic Geometry (Proceedings of a Conference held at Purdue Univiversity, December 5–7, 1963), Harper & Row, 1965, pp. 82-92 | MR: 0194396 | Zbl: 0171.19602
[27] Abelian -adic representations and elliptic curves, Research Notes in Mathematics, Tome 7, A K Peters, 1998, 199 pages (With the collaboration of Willem Kuyk and John Labute, Revised reprint of the 1968 original) | MR: 1484415 | Zbl: 0902.14016
[28] Lectures on , CRC Research Notes in Mathematics, Tome 11, CRC Press, 2012, x+163 pages | MR: 2920749 | Zbl: 1238.11001
[29] Good reduction of abelian varieties, Ann. Math., Tome 88 (1968), pp. 492-517 | Article | MR: 0236190 | Zbl: 0172.46101
[30] Algebraic number fields and symplectic discontinuous groups, Ann. Math., Tome 86 (1967), pp. 503-592 | Article | MR: 0222048 | Zbl: 0205.50601
[31] Reductive groups over local fields, Automorphic forms, representations and -functions (Oregon State University, Corvallis, Oregon, 1977), Part 1 (Proceedings of Symposia in Pure Mathematics) Tome XXXIII, American Mathematical Society, 1979, pp. 29-69 | MR: 546588 | Zbl: 0415.20035
[32] Mumford–Tate and generalised Shafarevich conjectures, Ann. Math. Qué., Tome 37 (2013) no. 2, pp. 255-284 | Article | MR: 3117743 | Zbl: 1364.11115
[33] Some cases of the Mumford–Tate conjecture and Shimura varieties, Indiana Univ. Math. J., Tome 57 (2008) no. 1, pp. 1-75 | Article | MR: 2400251 | Zbl: 1173.11039
Cited by Sources: