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be a unramified finite extension of
be an irreducible mod
two-dimensional representation of the absolute Galois group of
. The aim of this article is the explicit computation of the Kisin variety parameterizing the Breuil–Kisin modules associated to certain families of potentially Barsotti–Tate deformations of
. We prove that this variety is a finite union of products of
. Moreover, it appears as an explicit closed connected subvariety of
. We define a stratification of the Kisin variety by locally closed subschemes and explain how the Kisin variety equipped with its stratification may help in determining the ring of Barsotti–Tate deformations of
We present a new method to propagate
-adic precision in computations, which also applies to other ultrametric fields. We illustrate it with some examples and give a toy application to the stable computation of the SOMOS 4 sequence.
be a complete discrete valuation ring,
a positive integer. The aim of this paper is to explain how to efficiently compute usual operations such as sum and intersection of sub-
is not principal, it is not possible to have a uniform bound on the number of generators of the modules resulting from these operations. We explain how to mitigate this problem, following an idea of Iwasawa, by computing an approximation of the result of these operations up to a quasi-isomorphism. In the course of the analysis of the
-adic precisions of the computations, we have to introduce more general coefficient rings that may be interesting for their own sake. Being able to perform linear algebra operations modulo quasi-isomorphism with
-modules has applications in Iwasawa theory and
-adic Hodge theory. It is used in particular in Caruso and Lubicz (Preprint, 2013, arXiv:1309.4194) to compute the semi-simplified modulo
of a semi-stable representation.
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