Skip to main content
×
×
Home

INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE

  • Anna Cadoret (a1) and Ben Moonen (a2)
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.

Copyright
References
Hide All
1. André, Y., On the Shafarevich and Tate conjectures for hyper-Kähler varieties, Math. Ann. 305(2) (1996), 205248.
2. Bogomolov, F., Sur l’algébricité des représentations -adiques, C. R. Acad. Sci. Paris A–B 290(15) (1980), A701A703.
3. Borovoi, M., Abelian Galois cohomology of reductive groups, Mem. Amer. Math. Soc. 132(626) (1998), p. viii+50.
4. 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).
5. Cadoret, A., An adelic open image theorem for Abelian schemes, Int. Math. Res. Not. IMRN (20) (2015), 1020810242.
6. Cadoret, A. and Kret, A., Galois-generic points on Shimura varieties, Algebra Number Theory 10(9) (2016), 18931934.
7. Deligne, P., Travaux de Shimura. Sém. Bourbaki (1970/71), Exp. No. 389, Lecture Notes in Mathematics, Volume 244, pp. 123165 (Springer, Berlin, 1971).
8. 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. 247289 (AMS, Providence, Rhode Island, 1979).
9. Hindry, M. and Ratazzi, N., Torsion pour les variétés abéliennes de type I et II, Algebra Number Theory 10(9) (2016), 18451891.
10. Hui, C. Y. and Larsen, M., Adelic openness without the Mumford–Tate conjecture, Preprint, 2013, arXiv:1312.3812.
11. Larsen, M., Maximality of Galois actions for compatible systems, Duke Math. J. 80 (1995), 601630.
12. Larsen, M. and Pink, R., On -independence of algebraic monodromy groups in compatible systems of representations, Invent. Math. 107(3) (1992), 603636.
13. Larsen, M. and Pink, R., Abelian varieties, -adic representations, and -independence, Math. Ann. 302(3) (1995), 561579.
14. Lombardo, D., On the -adic Galois representations attached to nonsimple abelian varieties, Ann. Inst. Fourier (Grenoble) 66(3) (2016), 12171245.
15. Milne, J. and Shih, K.-y., Conjugates of Shimura varieties, in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, Volume 900, pp. 280356 (Springer, Berlin – Heidelberg – New York – London – Paris – Tokyo – Hong Kong, 1982).
16. 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), 739799.
17. Pink, R., -adic algebraic monodromy groups, cocharacters, and the Mumford–Tate conjecture, J. reine angew. Math. 495 (1998), 187237.
18. Platonov, V. and Rapinchuk, A., Algebraic Groups and Number Theory, Pure and Applied Mathematics, Volume 139 (Academic Press, Inc., Boston, MA, 1994).
19. Pohlmann, H., Algebraic cycles on abelian varieties of complex multiplication type, Ann. of Math. (2) 88 (1968), 161180.
20. Rizov, J., Moduli stacks of polarized K3 surfaces in mixed characteristic, Serdica Math. J. 32(2–3) (2006), 131178.
21. Rizov, J., Kuga–Satake abelian varieties of K3 surfaces in mixed characteristic, J. reine angew. Math. 648 (2010), 1367.
22. Serre, J.-P., Abelian -adic Representations and Elliptic Curves (W.A. Benjamin, New York – Amsterdam, 1968).
23. Serre, J.-P., Représentations -adiques, in Algebraic Number Theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976), Japan Soc. Promotion Sci., Tokyo, 1977, pp. 177193. [=Œuvres 112.].
24. Serre, J.-P., Lettres à Ken Ribet du 1/1/1981 et du 29/1/1981, in Œuvres, Volume IV, number 133.
25. Serre, J.-P., Résumé des cours de 1985–86, in Œuvres, Volume IV, number 136.
26. Serre, J.-P., Lettre à Marie-France Vigneras du 10/2/1986, in Œuvres, Volume IV, number 137.
27. Serre, J.-P., Lettre à Ken Ribet du 7/3/1986, in Œuvres, Volume IV, number 138.
28. Serre, J.-P., Propriétés conjecturales des groupes de Galois motiviques et des représentations -adiques, in Motives, Proceedings of Symposia in Pure Mathematics, Volume 55, Part 1, pp. 377400 (AMS, 1994). [=Œuvres 161.].
29. Taelman, L., Complex multiplication and Shimura stacks, Preprint, 2017, arXiv:1707.01236.
30. 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), 846861; 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.
31. Tankeev, S., Cycles on abelian varieties, and exceptional numbers, Izv. Ross. Akad. Nauk Ser. Mat. 60(2) (1996), 159194; translation in Izv. Math. 60(2) (1996), 391–424.
32. Tits, J., Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics, Volume 9, pp. 3362 (AMS, Providence, Rhode Island, 1966).
33. Ullmo, E. and Yafaev, A., Mumford–Tate and generalised Shafarevich conjectures, Ann. Math. Qué. 37(2) (2013), 255284.
34. van Geemen, B., Real multiplication on K3 surfaces and Kuga–Satake varieties, Michigan Math. J. 56(2) (2008), 375399.
35. Wintenberger, J.-P., Relèvement selon une isogénie de systèmes -adiques de représentations galoisiennes associés aux motifs, Invent. Math. 120(2) (1995), 215240.
36. Wintenberger, J.-P., Une extension de la théorie de la multiplication complexe, J. reine angew. Math. 552 (2002), 114.
37. Wintenberger, J.-P., Démonstration d’une conjecture de Lang dans des cas particuliers, J. reine angew. Math. 553 (2002), 116.
38. Zarhin, Yu., Hodge groups of K3 surfaces, J. reine angew. Math. 341 (1983), 193220.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed