A variational open image theorem in positive characteristic
Gebhard Böckle  ; Wojciech Gajda  ; Sebastian Petersen
Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 965-977.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.1059
Classification : 11G10,  14K15
Mots clés : Compatible system, adelic openness, positive characteristic 
@article{JTNB_2018__30_3_965_0,
     author = {Gebhard B\"ockle and Wojciech Gajda and Sebastian Petersen},
     title = {A variational open image theorem in positive characteristic},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {965--977},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1059},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1059/}
}
TY  - JOUR
AU  - Gebhard Böckle
AU  - Wojciech Gajda
AU  - Sebastian Petersen
TI  - A variational open image theorem in positive characteristic
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2018
DA  - 2018///
SP  - 965
EP  - 977
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1059/
UR  - https://doi.org/10.5802/jtnb.1059
DO  - 10.5802/jtnb.1059
LA  - en
ID  - JTNB_2018__30_3_965_0
ER  - 
Gebhard Böckle; Wojciech Gajda; Sebastian Petersen. A variational open image theorem in positive characteristic. Journal de Théorie des Nombres de Bordeaux, Tome 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 ℓ-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 ℓ -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 ℓ-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, ℓ-adic representations and ℓ-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

Cité par Sources :