Breuil-Kisin module

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breuil-kisin module

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We prove that, under some mild conditions, a two dimensional p-adic Galois representation which is residually modular and potentially Barsotti-Tate at p is modular. This provides a more conceptual way of establishing the Shimura-Taniyama-Weil conjecture, especially for elliptic curves which acquire good reduction over a wildly ramified extension of Q{double-struck}3. The main ingredient is a new technique for analyzing flat deformation rings. It involves resolving them by spaces which parametrize finite flat group scheme models of Galois representations.
Let k be a perfect field of characteristic p>0. When p>2, Fontaine and Laffaille have classified p-divisibles groups and finite flat p-groups over the Witt vectors W(k) in terms of filtered modules. Still assuming p>2, we extend these classifications over an arbitrary complete discrete valuation ring A with unequal characteristic (0,p) and residue field k by using "generalized" filtered modules. In particular, there is no restriction on the ramification index. In the case k is included in \bar{F}_p (and p>2), we then use this new classification to prove that any crystalline representation of the Galois group of Frac(A) with Hodge-Tate weights in {0,1} contains as a lattice the Tate module of a p-divisible group over A.
Breuil-Kisin module and integral p-adic Hodge theory
  • H Gao
H. Gao, Breuil-Kisin module and integral p-adic Hodge theory, 2019.
  • J Tate
J. Tate, p-divisible groups, Proceedings of a conference on local fields, NUFFIC Summer School held at Driebergen, Netherlands, 1966.
E-mail address: 1,
  • M Morrow
M. Morrow, p-divisible groups, Masters lecture course, Universität-Bonn, 2015/16. Department of Mathematics,, The University of Burdwan,, Burdwan-713101, West Bengal, India. E-mail address: 1, Department of Mathematics, The University of Burdwan, Burdwan-713101, India. E-mail address: 2