Potentially semi-stable deformation rings

By M. Kisin. 2007

227 citations recorded by the source corpus.

Let K/Qp be a finite extension and Gk = Gal(K/K) the Galois group of an alge braic closure K. Let F be a finite field of characteristic p, and Vf a finite dimensional F-vector space equipped with a continuous action of GkThe study of the defor mation theory of Galois representations was initiated by Mazur [Ma], who showed that if Vf has no non-trivial endomorphisms, then it admits a universal deformation ring Ryw. After the work of Wiles [Wi] and the conjectures of Fontaine-Mazur [FM] it became clear that for arithmetic applications it was important to understand cer tain quotients of Ry? corresponding to deformations satisfying certain conditions. For example Wiles uses deformations which arise from finite flat group schemes, and the corresponding quotient of Ryw was constructed by Ramakrishna [Ra]. Suppose that L/K is a finite extension and let a ^ b be integers. It seems to be a kind of folklore conjecture that there should be a quotient of Ry? whose points in finite extensions of W(F)[l/p] correspond to deformations of Vf which become semi-stable over L and have Hodge-Tate weights in the interval [a, b]. This is closely related to the final conjecture of [Fo 3]. Special cases of this, when Vf is 2-dimensional, are conjectured in the papers of Fontaine-Mazur [FM, p. 191], Breuil Conrad-Diamond-Taylor [BCDT, Conj. 1.1.1] and Breuil-M?zard [BM, 2.2.2.4]. The purpose of this paper is to prove such a result.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7b482f6a0c566e73827f5c32a79d295753e970e1

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Citations in the corpus

Modularity of 2-adic Barsotti-Tate representations (2009, background): “is trivial, ρf � |GFv arises from a p-divisible group C. This follows from [ 15 ], but can also be deduced using the argument of [25].”
Serre’s modularity conjecture (II) (2009, methodology): “(3), computation of k(ρ̄q): This follows from the corollary in the introduction of [39] and Corollary 6.”
On the Breuil–Mézard conjecture (2012, result): “Moreover, the existence of potentially semi-stable deformation rings has been established by Kisin in [38] in much greater generality, however our arguments give a new proof of this for ρ as above.”
Automorphic lifts of prescribed types (2008, methodology): “In [ 25 ] one considers deformations of weakly admissible modules, whereas in our case we consider deformations of Weil–Deligne representations; the main difference is that in our case one has a linear (rather than a semi-linear) Frobenius, and we have no analogue of the Hodge filtration.”
A geometric perspective on the Breuil–Mézard conjecture (2011, background): “It is a theorem of [28] that each of these closed subschemes Z is equidimensional of dimension 5, and so we may define the associated cycles Z ( Spec R (m, n, τ, r̄)/π ) .”