In the work we have considered Breuil-Kisin module over the ring of witt vectors $W(κ)$ over the residue field $κ$ of characteristic $p$ and a finite flat $\mathbb{Z}_p$-algebra $R$. Then considered Breuil-Kisin modules $M$ over the ring $W(κ)$ and taking the action of $R$ on $W(κ)$, we get again a Breuil-Kisin module $M$ over the ring $R \otimes_{\mathbb{Z}_p} W(κ)$. We have studied freeness and projectiveness of this module.
We prove that the category of (rigidified) Breuil-Kisin-Fargues modules up to isogeny is Tannakian. We then introduce and classify Breuil-Kisin-Fargues modules with complex multiplication mimicking the classical theory for rational Hodge structures. In particular, we compute an avatar of a "p-adic Serre group".
Let p be a rational prime and K/Q_p be an extension of complete discrete valuation fields. Let G be a truncated Barsotti-Tate group of level n, height h and dimension d over O_K with 0<d<h. In this paper, we prove the existence of higher canonical subgroups with expected properties for G if the Hodge height of G is less than 1/(p^{n-2}(p+1)), including the case of p=2.
For $K$ an extension of $\mathbb{Q}_{p}$ with ring of integers $R$ we show how Breuil-Kisin modules can be used to determine Hopf orders in $K$-Hopf algebras of $p$-power dimension. We find all cyclic Breuil-Kisin modules, and use them to compute all of the Hopf orders in the group ring $KΓ$ where $Γ$ is cyclic of order $p$ or $p^{2}.$ We also give a Laurent series interpretation of the Breuil-Kisin modules that give these Hopf orders.
In this article, we apply the methods of our work on Fontaine's theory in equal characteristics to the $φ/\mathfrak S$-modules of Breuil and Kisin. Thanks to a previous article of Kisin, this yields a new and rather elementary proof of the theorem "weakly admissible implies admissible" of Colmez-Fontaine.
Let p>2 be a rational prime and K/Q_p be an extension of complete discrete valuation fields. Let G be a truncated Barsotti-Tate group of level n, height h and dimension d over O_K with 0<d<h. In this paper, we show that an upper ramification subgroup G^j+ is free of rank d over Z/p^nZ if the Hasse invariant of G is less than 1/(2p^(n-1)). We also prove the usual properties as the canonical subgroup.
For a perfect field $k$ of characteristic $p>0$ and a smooth and proper formal scheme $\mathscr{X}$ over the ring of integers of a finite and totally ramified extension $K$ of $W(k)[1/p]$, we propose a cohomological construction of the Breuil-Kisin modules attached to the $p$-adic étale cohomology $H^i_{\mathrm{ét}}(\mathscr{X}_{\overline{K}},\mathbf{Z}_p)$. We then prove that our proposal works when $p>2$, $i < p-1$, and the crystalline cohomology of the special fiber of $\mathscr{X}$ is torsion-free in degrees $i$ and $i+1$.
Assume that $p>2$, and let $\mathscr{O}_K$ be a $p$-adic discrete valuation ring with residue field admitting a finite $p$-basis, and let $R$ be a formally smooth formally finite-type $\mathscr{O}_K$-algebra. (Indeed, we allow slightly more general rings $R$.) We construct an anti-equivalence of categories between the categories of $p$-divisible groups over $R$ and certain semi-linear algebra objects which generalise $(φ,\mathfrak{S})$-modules of height $\leqslant1$ (or Kisin modules). A similar classification result for $p$-power order finite flat group schemes is deduced from the classificat…
The Galois representation associated to a p-divisible group over a complete noetherian normal local ring with perfect residue field is described in terms of its Dieudonné display. As a corollary we deduce in arbitrary characteristic Kisin's description of the Galois representation associated to a commutative finite flat p-group scheme over a p-adic discrete valuation ring in terms of its Breuil-Kisin module. This was obtained earlier by W. Kim by a different method.
Let $K$ be a discrete valuation field with perfect residue field, we study the functor from weakly admissible filtered $(φ,N,G_K)$-modules over $K$ to the isogeny category of Breuil-Kisin-Fargues $G_K$-modules. This functor is the composition of a functor defined by Fargues-Fontaine from weakly admissible filtered $(φ,N,G_K)$-modules to $G_K$-equivariant modifications of vector bundles over the Fargues-Fontaine curve $X_{FF}$, with the functor of Fargues-Scholze that between the category of admissible modifications of vector bundles over $X_{FF}$ and the isogeny category of Breuil-Kisin-Fargue…
Let $K$ be a local function field of characteristic $l$, $\mathbb{F}$ be a finite field over $\mathbb{F}_p$ where $l \ne p$, and $\overlineρ: G_K \rightarrow \text{GL}_n (\mathbb{F})$ be a continuous representation. We apply the Taylor-Wiles-Kisin method over certain global function fields to construct a mod $p$ cycle map $\overline{\text{cyc}}$, from mod $p$ representations of $\text{GL}_n (\mathcal{O}_K)$ to the mod $p$ fibers of the framed universal deformation ring $R_{\overlineρ}^\square$. This allows us to obtain a function field analog of the Breuil--Mézard conjecture. Then we use the t…
Let $k$ be a perfect field of characteristic $p \geq 3$. We classify $p$-divisible groups over regular local rings of the form $W(k)[[t_1,...,t_r,u]]/(u^e+pb_{e-1}u^{e-1}+...+pb_1u+pb_0)$, where $b_0,...,b_{e-1}\in W(k)[[t_1,...,t_r]]$ and $b_0$ is an invertible element. This classification was in the case $r = 0$ conjectured by Breuil and proved by Kisin.
We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_K$ for a finite extension $K/\mathbb{Q}_p$. This is done by considering a moduli space of Breuil--Kisin modules, satisfying an additional Galois condition, over the universal deformation ring. For $K$ unramified over $\mathbb{Q}_p$ and Hodge--Tate weights in $[0,p]$, we study the geometry of this space. As a consequence we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_p$, with Hodge--Tate weights in $[0,p]$, are potentiall…
Let O_K be a discrete valuation ring of mixed characteristics (0,p), with residue field k. Using work of Sekiguchi and Suwa, we construct some finite flat O_K-models of the group scheme μ_{p^n,K} of p^n-th roots of unity, which we call Kummer group schemes. We set carefully the general framework and algebraic properties of this construction. When k is perfect and O_K is a complete totally ramified extension of the ring of Witt vectors W(k), we provide a parallel study of the Breuil-Kisin modules of finite flat models of μ_{p^n,K}, in such a way that the construction of Kummer groups and Breuil…
We define and study certain moduli stacks of modules equipped with a Frobenius semi-linear endomorphism. These stacks can be thought of as parametrizing the coefficients of a variable Galois representation and are global variants of the spaces of Kisin-Breuil $Φ$-modules used by Kisin in his study of deformation spaces of local Galois representations. We also define a version of a rigid analytic period map for these spaces, we show how their local structure can be described in terms of "local models", and we show how Bruhat-Tits buildings can be used to study their special fibers.
Let $k$ be a perfect field of characteristic $p>2$ and $K$ an extension of $F=\mathrm{Frac} W(k)$ contained in some $F(μ_{p^r})$. Using crystalline Dieudonné theory, we provide a classification of $p$-divisible groups over $\mathscr{O}_K$ in terms of finite height $(φ,Γ)$-modules over $\mathfrak{S}:=W(k)[[u]]$. Although such a classification is a consequence of (a special case of) the theory of Kisin--Ren, our construction gives an independent proof and allows us to recover the Dieudonné crystal of a $p$-divisible group from the Wach module associated to its Tate module by Berger--Breuil or by…
The Dieudonné crystal of a p-divisible group over a semiperfect ring R can be endowed with a window structure. If R satisfies a boundedness condition, this construction gives an equivalence of categories. As an application one obtains a classification of p-divisible groups and commutative finite locally free p-group schemes over perfectoid rings by Breuil-Kisin-Fargues modules if p>2.
Using $p$-adic local Langlands correspondence for $\operatorname{GL}_2(\mathbb{Q}_p)$, we prove that the support of patched modules constructed by Caraiani, Emerton, Gee, Geraghty, Paskunas, and Shin meet every irreducible component of the potentially semistable deformation ring. This gives a new proof of the Breuil-Mézard conjecture for 2-dimensional representations of the absolute Galois group of $\mathbb{Q}_p$ when $p > 2$, which is new in the case $p = 3$ and $\bar{r}$ a twist of an extension of the trivial character by the mod p cyclotomic character. As a consequence, a local restriction …
For a perfectoid ring $R$ and a natural number $n$ we investigate the essential image of the category of truncated by $n$ Barsotti-Tate groups under the anti-equivalence between commutative, finite, locally free, $R$-group schemes of $p$-power order and torsion Breuil-Kisin-Fargues modules over $R$. We describe the associated semi-liner algebra data and show as a consequence that every $\text{BT}_n$-group over $R$ is the $p^n$-torsion of some $\text{BT}$-group.
Let R be a discrete valuation ring with residue field of characteristic p>0. Let K be its fraction field. We prove that any finite and flat R-group scheme, isomorphic to μ_{p^2,K} on the generic fiber, is the kernel in a short exact sequence which generically coincides with the Kummer sequence. We will explicitly describe and classify such models. In the appendix X. Caruso shows how to classify models of μ_{p^2,K}, in the case of unequal characteristic, using the Breuil-Kisin theory.
