A variational open image theorem in positive characteristic
Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 965-977.

We prove a variational open adelic image theorem for the Galois action on the cohomology of smooth proper S-schemes where S is a smooth variety over a finitely generated field of positive characteristic. A central tool is a recent result of Cadoret, Hui and Tamagawa.

Nous démontrons un théorème de l’image adélique ouverte variationnel pour l’action du groupe de Galois sur la cohomologie d’un S-schéma propre et lisse, où S est une variété lisse sur un corps de type fini sur 𝔽 p . Notre outil clé est un résultat récent de Cadoret, Hui et Tamagawa.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1059
Classification: 11G10,  14K15
Keywords: Compatible system, adelic openness, positive characteristic
Gebhard Böckle 1; Wojciech Gajda 2; Sebastian Petersen 3

1 Computational Arithmetic Geometry IWR (Interdisciplinary Center for Scientific Computing) University of Heidelberg Im Neuenheimer Feld 368 69120 Heidelberg, Germany
2 Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61614 Poznań, Poland
3 Universität Kassel Fachbereich 10 Wilhelmshöher Allee 73 34121 Kassel, Germany
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Gebhard Böckle; Wojciech Gajda; Sebastian Petersen. A variational open image theorem in positive characteristic. Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 965-977. doi : 10.5802/jtnb.1059. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1059/

[1] Pierre Berthelot Altération de variétés algébriques, Séminaire Bourbaki. Volume 1995/96 (Astérisque) Volume 241, Société Mathématique de France, 1997, pp. 273-311 | Zbl: 0924.14007

[2] Gebhard Böckle; Wojciech Gajda; Sebastian Petersen Independence of -adic representations of geometric Galois groups, J. Reine Angew. Math., Volume 736 (2015), pp. 69-93 | Article | Zbl: 0684.7718

[3] Anna Cadoret An open adelic image theorem for motivic representations over function fields (to appear in Math. Res. Lett., https://webusers.imj-prg.fr/~anna.cadoret/Cadoret_MRL.pdf)

[4] Anna Cadoret An open adelic image theorem for abelian schemes, Int. Math. Res. Not., Volume 2015 (2015) no. 20, pp. 10208-10242 | Zbl: 1352.14030

[5] Anna Cadoret; Chun-Yin Hui; Akio Tamagawa Geometric monodromy - semisimplicity and maximality, Ann. Math., Volume 186 (2017) no. 1, pp. 205-236

[6] Anna Cadoret; Arno Kret Galois generic points on Shimura varieties, Algebra Number Theory, Volume 10 (2016) no. 9, pp. 1893-1934 | Zbl: 06657570

[7] Anna Cadoret; Akio Tamagawa On the geometric image of 𝔽 -linear representations of étale fundamental groups, Int. Math. Res. Not. (2017), rnx193 (Art. ID rnx193) | Article

[8] Pierre Deligne La conjecture de Weil I, Publ. Math., Inst. Hautes Étud. Sci., Volume 43 (1974), pp. 273-307 | Zbl: 0287.14001

[9] Pierre Deligne La conjecture de Weil II, Publ. Math., Inst. Hautes Étud. Sci., Volume 52 (1980), pp. 137-252

[10] Lance Dixon; Marcus du Sautoy; Avinoam Mann; Dan Segal Analytic pro-p groups, Cambridge Studies in Advanced Mathematics, Volume 61, Cambridge University Press, 1999, xviii+368 pages | Zbl: 0934.20001

[11] Vladimir Drinfeld On a conjecture of Deligne, Mosc. Math. J., Volume 12 (2012) no. 3, pp. 515-542

[12] Chun-Yin Hui; Michael Larsen Adelic openness without the Mumford–Tate conjecture (2013) (https://arxiv.org/abs/1312.3812)

[13] Mortiz Kerz; Alexander Schmidt On different notions of tameness in arithmetic geometry, Math. Ann., Volume 346 (2010) no. 3, pp. 641-668 | Zbl: 1185.14019

[14] Michael Larsen; Richard Pink On -independence of algebraic monodromy groups in compatible systems of representations, Invent. Math., Volume 107 (1992) no. 3, pp. 603-636 | Zbl: 0778.11036

[15] Michael Larsen; Richard Pink Abelian varieties, -adic representations and -independence, Math. Ann., Volume 302 (1995) no. 3, pp. 561-579 | Zbl: 0867.14019

[16] James Milne Algebraic Groups – The theory of group schemes of finite type over a field (Lecture notes available at www.jmilne.org)

[17] James Milne Étale Cohomology, Princeton Mathematical Series, Volume 33, Princeton University Press, 1980, xiii+323 pages | Zbl: 0433.14012

[18] Richard Pink A Combination of the conjectures of Mordell-Lang and André-Oort, Math. Ann., Volume 302 (1995) no. 3, pp. 561-579

[19] Jean-Pierre Serre Sur les groupes de Galois attachés aux groups p-divisible, Proceedings on a conference on local fields, Springer, 1967

[20] Jean-Pierre Serre Lectures on the Mordell-Weil theorem, Aspects of Mathematics, Volume E15, Viehweg, 1989 | Zbl: 0676.14005

[21] Jean-Pierre Serre Lettre à Ken Ribet du 1/1/1981 et du 29/1/1981, Collected papers. Vol. IV: 1985–1998, Springer, 2000

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