Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T08:14:06.541Z Has data issue: false hasContentIssue false
  • English
  • Français

INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  07 June 2018

Anna Cadoret  and
Ben Moonen Open the ORCID record for Ben Moonen [Opens in a new window]
Anna Cadoret
Affiliation:
IMJ-PRG – Sorbonne Université, Paris, France (anna.cadoret@imj-prg.fr)
Ben Moonen
Affiliation:
Radboud University, IMAPP, Nijmegen, The Netherlands (b.moonen@science.ru.nl)
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre are true for $Y$. In particular, if a certain Hodge-maximality condition is satisfied, we obtain an adelic open image theorem for the Galois representation on the (full) Tate module of $Y$. We also obtain an (unconditional) adelic open image theorem for K3 surfaces. These results are special cases of a more general statement for the image of a natural adelic representation of the fundamental group of a Shimura variety.

Keywords

Galois representationsMumford-Tate conjectureShimura varietiesAbelian varietiesK3 surfaces

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties 14G35: Modular and Shimura varieties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 3 , May 2020 , pp. 869 - 890
Copyright
© Cambridge University Press 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

André, Y., On the Shafarevich and Tate conjectures for hyper-Kähler varieties, Math. Ann. 305(2) (1996), 205–248.Google Scholar
Bogomolov, F., Sur l’algébricité des représentations ℓ-adiques, C. R. Acad. Sci. Paris A–B 290(15) (1980), A701–A703.Google Scholar
Borovoi, M., Abelian Galois cohomology of reductive groups, Mem. Amer. Math. Soc. 132(626) (1998), p. viii+50.Google Scholar
Bourbaki, N., Éléments de mathématique, Groupes et Algèbres de Lie Chap. 4, 5 et 6, (Masson, Paris – New York – Barcelone – Milan – Mexico – Rio de Janeiro, 1981).Google Scholar
Cadoret, A., An adelic open image theorem for Abelian schemes, Int. Math. Res. Not. IMRN (20) (2015), 10208–10242.10.1093/imrn/rnu259Google Scholar
Cadoret, A. and Kret, A., Galois-generic points on Shimura varieties, Algebra Number Theory 10(9) (2016), 1893–1934.Google Scholar
Deligne, P., Travaux de Shimura. Sém. Bourbaki (1970/71), Exp. No. 389, Lecture Notes in Mathematics, Volume 244, pp. 123–165 (Springer, Berlin, 1971).Google Scholar
Deligne, P., Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, in Automorphic Forms, Representations and L-functions, Proceedings of Symposia in Pure Mathematics, Volume 33, Part 2, pp. 247–289 (AMS, Providence, Rhode Island, 1979).Google Scholar
Hindry, M. and Ratazzi, N., Torsion pour les variétés abéliennes de type I et II, Algebra Number Theory 10(9) (2016), 1845–1891.Google Scholar
Hui, C. Y. and Larsen, M., Adelic openness without the Mumford–Tate conjecture, Preprint, 2013, arXiv:1312.3812.Google Scholar
Larsen, M., Maximality of Galois actions for compatible systems, Duke Math. J. 80 (1995), 601–630.Google Scholar
Larsen, M. and Pink, R., On ℓ-independence of algebraic monodromy groups in compatible systems of representations, Invent. Math. 107(3) (1992), 603–636.Google Scholar
Larsen, M. and Pink, R., Abelian varieties, ℓ-adic representations, and ℓ-independence, Math. Ann. 302(3) (1995), 561–579.Google Scholar
Lombardo, D., On the ℓ-adic Galois representations attached to nonsimple abelian varieties, Ann. Inst. Fourier (Grenoble) 66(3) (2016), 1217–1245.Google Scholar
Milne, J. and Shih, K.-y., Conjugates of Shimura varieties, in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, Volume 900, pp. 280–356 (Springer, Berlin – Heidelberg – New York – London – Paris – Tokyo – Hong Kong, 1982).10.1007/978-3-540-38955-2_7Google Scholar
Moonen, B., On the Tate and Mumford–Tate conjectures in codimension 1 for varieties with h 2, 0 = 1, Duke Math. J. 166(4) (2017), 739–799.Google Scholar
Pink, R., ℓ-adic algebraic monodromy groups, cocharacters, and the Mumford–Tate conjecture, J. reine angew. Math. 495 (1998), 187–237.Google Scholar
Platonov, V. and Rapinchuk, A., Algebraic Groups and Number Theory, Pure and Applied Mathematics, Volume 139 (Academic Press, Inc., Boston, MA, 1994).Google Scholar
Pohlmann, H., Algebraic cycles on abelian varieties of complex multiplication type, Ann. of Math. (2) 88 (1968), 161–180.Google Scholar
Rizov, J., Moduli stacks of polarized K3 surfaces in mixed characteristic, Serdica Math. J. 32(2–3) (2006), 131–178.Google Scholar
Rizov, J., Kuga–Satake abelian varieties of K3 surfaces in mixed characteristic, J. reine angew. Math. 648 (2010), 13–67.Google Scholar
Serre, J.-P., Abelian ℓ-adic Representations and Elliptic Curves (W.A. Benjamin, New York – Amsterdam, 1968).Google Scholar
Serre, J.-P., Représentations ℓ-adiques, in Algebraic Number Theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976), Japan Soc. Promotion Sci., Tokyo, 1977, pp. 177–193. [=Œuvres 112.].Google Scholar
Serre, J.-P., Lettres à Ken Ribet du 1/1/1981 et du 29/1/1981, in Œuvres, Volume IV, number 133.Google Scholar
Serre, J.-P., Résumé des cours de 1985–86, in Œuvres, Volume IV, number 136.Google Scholar
Serre, J.-P., Lettre à Marie-France Vigneras du 10/2/1986, in Œuvres, Volume IV, number 137.Google Scholar
Serre, J.-P., Lettre à Ken Ribet du 7/3/1986, in Œuvres, Volume IV, number 138.Google Scholar
Serre, J.-P., Propriétés conjecturales des groupes de Galois motiviques et des représentations ℓ-adiques, in Motives, Proceedings of Symposia in Pure Mathematics, Volume 55, Part 1, pp. 377–400 (AMS, 1994). [=Œuvres 161.].Google Scholar
Taelman, L., Complex multiplication and Shimura stacks, Preprint, 2017, arXiv:1707.01236.Google Scholar
Tankeev, S., Surfaces of K3 type over number fields and the Mumford–Tate conjecture. I, II, Izv. Akad. Nauk SSSR Ser. Mat. 54(4) (1990), 846–861; Izv. Ross. Akad. Nauk Ser. Mat. 59(3) (1995), 179–206; translations in Math. USSR-Izv. 37(1) (1991) 191–208; Izv. Math. 59(3) (1995) 619–646.Google Scholar
Tankeev, S., Cycles on abelian varieties, and exceptional numbers, Izv. Ross. Akad. Nauk Ser. Mat. 60(2) (1996), 159–194; translation in Izv. Math. 60(2) (1996), 391–424.Google Scholar
Tits, J., Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics, Volume 9, pp. 33–62 (AMS, Providence, Rhode Island, 1966).Google Scholar
Ullmo, E. and Yafaev, A., Mumford–Tate and generalised Shafarevich conjectures, Ann. Math. Qué. 37(2) (2013), 255–284.10.1007/s40316-013-0009-4Google Scholar
van Geemen, B., Real multiplication on K3 surfaces and Kuga–Satake varieties, Michigan Math. J. 56(2) (2008), 375–399.Google Scholar
Wintenberger, J.-P., Relèvement selon une isogénie de systèmes ℓ-adiques de représentations galoisiennes associés aux motifs, Invent. Math. 120(2) (1995), 215–240.Google Scholar
Wintenberger, J.-P., Une extension de la théorie de la multiplication complexe, J. reine angew. Math. 552 (2002), 1–14.Google Scholar
Wintenberger, J.-P., Démonstration d’une conjecture de Lang dans des cas particuliers, J. reine angew. Math. 553 (2002), 1–16.Google Scholar
Zarhin, Yu., Hodge groups of K3 surfaces, J. reine angew. Math. 341 (1983), 193–220.Google Scholar