The correspondence between Barsotti-Tate groups and Kisin modules when $p=2$

Liu, Tong

Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 661-676.

Soit $K$ une extension finie de ${\mathbb{Q}}_2$ d’anneau des entiers ${𝒪}_K$. Dans cet article, on construit une équivalence de catégories entre la catégorie des modules de Kisin de hauteur $1$ et la catégorie des groupes de Barsotti-Tate sur ${𝒪}_K$.

Let $K$ be a finite extension over ${\mathbb{Q}}_2$ and ${𝒪}_K$ the ring of integers. We prove the equivalence of categories between the category of Kisin modules of height 1 and the category of Barsotti-Tate groups over ${𝒪}_K$.

[1] C. Breuil, Schémas en groupes et corps des normes. Unpublished.

[2] C. Breuil, Représentations $p$-adiques semi-stables et transversalité de Griffiths. Math. Ann., 307(2) (1997), 191–224. | MR | Zbl

[3] C. Breuil, Groupes $p$-divisibles, groupes finis et modules filtrés. Ann. of Math.(2) 152(2) (2000), 489–549. | MR | Zbl

[4] C. Breuil, Integral $p$-adic Hodge theory. In Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. 36 (2002), Pure Math., Math. Soc. Japan, Tokyo, 51–80. | MR | Zbl

[5] X. Caruso and T. Liu, Some bounds for ramification of $p^n$-torsion semi-stable representations. J. Algebra, 325 (2011), 70–96. | MR | Zbl

[6] J.-M. Fontaine, Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate. In Journées de Géométrie Algébrique de Rennes, (Rennes, 1978), Vol.III, volume 65 of Astérisque, . Soc. Math. France Paris (1979), 3–80. | Numdam | MR | Zbl

[7] J.-M. Fontaine, Le corps des périodes $p$-adiques. Astérisque, 223 (1994), 59–111. | Numdam | MR | Zbl

[8] J.-M. Fontaine, Représentations $p$-adiques semi-stables. Astérisque, 223 (1994), 113–184. With an appendix by Pierre Colmez, Périodes $p$-adiques (Bures-sur-Yvette, 1988). | Numdam | MR | Zbl

[9] W. Kim, The classification of $p$-divisible groups over 2-adic discrete valuation rings. Math. Res. Lett., 19(1) (2012), 121–141. | MR

[10] M. Kisin, Crystalline representations and $F$-crystals. In Algebraic geometry and number theory,Progr. Math. 253, Birkhäuser Boston, Boston, MA, (2006), 459–496. | MR | Zbl

[11] M. Kisin Modularity of 2-adic Barsotti-Tate representations. Invent. Math., 178(3) (2009), 587–634. | MR

[12] M. Kisin Moduli of finite flat group schemes, and modularity. Ann. of Math.(2), 170(3) (2009), 1085–1180. | MR | Zbl

[13] E. Lau, A relation between dieudonne displays and crystalline dieudonne theory. arXiv:1006.2720.

[14] T. Liu, Torsion $p$-adic Galois representations and a conjecture of fontaine. Ann. Sci. École Norm. Sup. (4), 40(4) (2007), 633–674. | MR | Zbl

[15] T. Liu, On lattices in semi-stable representations: a proof of a conjecture of Breuil. Compos. Math., 144(1) (2008), 61–88. | MR | Zbl

[16] T. Liu, A note on lattices in semi-stable representations. Mathematische Annalen, 346(1) (2010), 117–138. | MR | Zbl

[17] M. Raynaud, Schémas en groupes de type $(p,\cdots ,p)$. Bull. Soc. Math. France, 102 (1974), 241–280. | Numdam | MR | Zbl

[18] J. T. Tate, $p$-divisible groups. In Proc. Conf. Local Fields (Driebergen, 1966), Springer, Berlin, (1967), 158–183. | MR | Zbl