Semistable abelian varieties and maximal torsion 1-crystalline submodules

Cody Gunton

doi:10.5802/jtnb.1151

Cody Gunton ¹

¹ Department of Mathematical Sciences Universitetsparken 5 University of Copenhagen 2100 Copenhagen Ø, Denmark

Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 39-81.

Abstract
Résumé

Let $p$ be a prime, let $K$ be a discretely valued extension of $ℚ_{p}$ , and let $A_{K}$ be an abelian $K$ -variety with semistable reduction. Extending work by Kim and Marshall from the case where $p > 2$ and $K / ℚ_{p}$ is unramified, we prove an $l = p$ complement of a Galois cohomological formula of Grothendieck for the $l$ -primary part of the Néron component group of $A_{K}$ . Our proof involves constructing, for each $m \in ℤ_{\geq 0}$ , a finite flat $𝒪_{K}$ -group scheme with generic fiber equal to the maximal $1$ -crystalline submodule of $A_{K} [p^{m}]$ . As a corollary, we have a new proof of the Coleman–Iovita monodromy criterion for good reduction of abelian $K$ -varieties.

Soient $p$ un nombre premier, $K$ une extension de $ℚ_{p}$ de valuation discrète, et $A_{K}$ une $K$ -variété abélienne à réduction semistable. En étendant les travaux de Kim et Marshall portant sur le cas $p > 2$ et $K / ℚ_{p}$ non ramifié, nous prouvons un complément pour $l = p$ à la formule cohomologique de Grothendieck pour la partie $l$ -primaire du groupe des composantes connexes du modèle de Néron de $A_{K}$ . Notre démonstration consiste à construire, pour chaque $m \in ℤ_{\geq 0}$ , un schéma en groupes plat et fini sur $𝒪_{K}$ dont la fibre générique est isomorphe au sous-module $1$ -cristallin maximal de $A_{K} [p^{m}]$ . Comme corollaire, on obtient une nouvelle preuve du critère monodromique de bonne réduction de Coleman–Iovita.

Received: 2020-01-15
Revised: 2020-08-27
Accepted: 2021-02-01
Published online: 2021-05-21

DOI: 10.5802/jtnb.1151

Classification: 11R33, 11R34, 14K15
Keywords: Néron component group, log 1-motive, torsion 1-crystalline representation

Author's affiliations:

Cody Gunton ¹

¹ Department of Mathematical Sciences Universitetsparken 5 University of Copenhagen 2100 Copenhagen Ø, Denmark

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{JTNB_2021__33_1_39_0,
     author = {Cody Gunton},
     title = {Semistable abelian varieties and maximal torsion 1-crystalline submodules},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {39--81},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {1},
     year = {2021},
     doi = {10.5802/jtnb.1151},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1151/}
}

TY  - JOUR
AU  - Cody Gunton
TI  - Semistable abelian varieties and maximal torsion 1-crystalline submodules
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 39
EP  - 81
VL  - 33
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1151/
UR  - https://doi.org/10.5802/jtnb.1151
DO  - 10.5802/jtnb.1151
LA  - en
ID  - JTNB_2021__33_1_39_0
ER  -

%0 Journal Article
%A Cody Gunton
%T Semistable abelian varieties and maximal torsion 1-crystalline submodules
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 39-81
%V 33
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1151
%R 10.5802/jtnb.1151
%G en
%F JTNB_2021__33_1_39_0

Cody Gunton. Semistable abelian varieties and maximal torsion 1-crystalline submodules. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 39-81. doi : 10.5802/jtnb.1151. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1151/

References
Cited by

[1] Fabrizio Andreatta; Luca Barbieri-Viale Crystalline realizations of 1-motives, Math. Ann., Volume 331 (2005) no. 1, pp. 111-172 | DOI | MR | Zbl

[2] Alexander Beilinson; Floric Tavares Ribeiro On a theorem of Kisin, C. R. Math. Acad. Sci. Paris, Volume 351 (2013) no. 13-14, pp. 505-506 | DOI | MR | Zbl

[3] Alessandra Bertapelle; Maurizio Candilera; Valentino Cristante Monodromy of logarithmic Barsotti–Tate groups attached to 1-motives, J. Reine Angew. Math., Volume 573 (2004), pp. 211-234 | DOI | MR | Zbl

[4] Pierre Berthelot; Lawrence Breen; William Messing Théorie de Dieudonné cristalline. II, Lecture Notes in Mathematics, 930, Springer, 1982, x+261 pages | Zbl

[5] Mikhail V. Bondarko Finite flat commutative group schemes over complete discrete valuation fields: classification, structural results; application to reduction of abelian varieties, Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005, Universitätsdrucke Göttingen, 2005, pp. 99-108 | MR | Zbl

[6] Siegfried Bosch; Werner Lütkebohmert Degenerating abelian varieties, Topology, Volume 30 (1991) no. 4, pp. 653-698 | DOI | MR | Zbl

[7] Siegfried Bosch; Werner Lütkebohmert; Michel Raynaud Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 21, Springer, 1990, x+325 pages | Zbl

[8] Christophe Breuil Groupes $p$ -divisibles, groupes finis et modules filtrés, Ann. Math., Volume 152 (2000) no. 2, pp. 489-549 | DOI | MR | Zbl

[9] Christophe Breuil Integral $p$ -adic Hodge theory, Algebraic geometry 2000, Azumino (Hotaka) (Advanced Studies in Pure Mathematics), Volume 36, Mathematical Society of Japan, 2002, pp. 51-80 | DOI | MR | Zbl

[10] Olivier Brinon; Brian Conrad CMI Summer School Notes on $p$ -adic Hodge Theory (2009) (available at http://math.stanford.edu/~conrad/papers/notes.pdf)

[11] Robert Coleman; Adrian Iovita The Frobenius and monodromy operators for curves and abelian varieties, Duke Math. J., Volume 97 (1999) no. 1, pp. 171-215 | DOI | MR | Zbl

[12] Tim Dokchitser; Vladimir Dokchitser Local invariants of isogenous elliptic curves, Trans. Am. Math. Soc., Volume 367 (2015) no. 6, pp. 4339-4358 | DOI | MR | Zbl

[13] Gerd Faltings; Ching-Li Chai Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 22, Springer, 1990, xii+316 pages (with an appendix by David Mumford) | DOI | MR | Zbl

[14] Jean-Marc Fontaine Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate, Journées de Géométrie Algébrique de Rennes. (Rennes, 1978), Vol. III (Astérisque), Volume 65, Société Mathématique de France, 1979, pp. 3-80 | Numdam | MR | Zbl

[15] Alexander Grothendieck SGA7, Exposé IX, Lecture Notes in Mathematics, 288, Springer, 1967-1969

[16] Alexander Grothendieck Groupes de Barsotti–Tate et cristaux de Dieudonné, Séminaire de Mathématiques Supérieures, 45, Presses de l’Université de Montréal, 1974, 155 pages | MR | Zbl

[17] Uwe Jannsen Continuous étale cohomology, Math. Ann., Volume 280 (1988) no. 2, pp. 207-245 | DOI | MR | Zbl

[18] Takeshi Kajiwara; Kazuya Kato; Chikara Nakayama Logarithmic abelian varieties, Part IV: Proper models, Nagoya Math. J., Volume 219 (2015), pp. 9-63 | DOI | MR | Zbl

[19] Kazuya Kato Logarithmic Dieudonné Theory (unpublished preprint) | Zbl

[20] Minhyong Kim; Susan H. Marshall Crystalline subrepresentations and Néron models, Math. Res. Lett., Volume 7 (2000) no. 5-6, pp. 605-614 | DOI | MR | Zbl

[21] Wansu Kim The classification of $p$ -divisible groups over 2-adic discrete valuation rings, Math. Res. Lett., Volume 19 (2012) no. 1, pp. 121-141 | DOI | MR | Zbl

[22] Mark Kisin Crystalline representations and $F$ -crystals, Algebraic geometry and number theory (Progress in Mathematics), Volume 253, Birkhäuser, 2006, pp. 459-496 | DOI | MR | Zbl

[23] Mark Kisin Integral models for Shimura varieties of abelian type, J. Am. Math. Soc., Volume 23 (2010) no. 4, pp. 967-1012 | DOI | MR | Zbl

[24] Kai-Wen Lan Arithmetic compactifications of PEL-type Shimuravarieties, London Mathematical Society Monographs, 36, Princeton University Press, 2013, xxvi+561 pages | DOI | MR | Zbl

[25] Christopher Lazda; Ambrus Pál Rigid Cohomology over Laurent Series Fields, Algebra and Applications, 21, Springer, 2016 | DOI | MR | Zbl

[26] Tong Liu Torsion $p$ -adic Galois representations and a conjecture of Fontaine, Ann. Sci. Éc. Norm. Supér., Volume 40 (2007) no. 4, pp. 633-674 | Numdam | MR | Zbl

[27] Keerthi Madapusi-Pera Log p-divisible groups and semi-stable representations (2014) (unpublished preprint)

[28] Susan H. Marshall Crystalline representations and Néron models, Ph. D. Thesis, University of Arizona (USA) (2001) | Zbl

[29] James S. Milne Algebraic groups, Cambridge Studies in Advanced Mathematics, 170, Cambridge University Press, 2017, xvi+644 pages | DOI | MR

[30] David Mumford An analytic construction of degenerating abelian varieties over complete rings, Compos. Math., Volume 24 (1972), pp. 239-272 | Numdam | MR | Zbl

[31] Yoshiyasu Ozeki On Galois equivariance of homomorphisms between torsion crystalline representations, Nagoya Math. J., Volume 229 (2018), pp. 169-214 | DOI | MR | Zbl

[32] Michel Raynaud Variétés abéliennes et géométrie rigide, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, 1971, pp. 473-477 | Zbl

[33] Michel Raynaud Schémas en groupes de type $(p, \dots, p)$ , Bull. Soc. Math. Fr., Volume 102 (1974), pp. 241-280 | DOI | Numdam | MR | Zbl

[34] Michel Raynaud 1-motifs et monodromie géométrique, Périodes $p$ -adiques (Bures-sur-Yvette, 1988) (Astérisque), Volume 223, Société Mathématique de France, 1994, pp. 295-319 | Numdam | MR | Zbl

[35] The Stacks Project Authors Stacks Project, 2018 (http://stacks.math.columbia.edu)

[36] John T. Tate $p$ -divisible groups, Proc. Conf. Local Fields (Driebergen, 1966), Springer, 1967, pp. 158-183 | DOI | MR | Zbl

Cited by Sources: