Modularity of 2-adic Barsotti-Tate representations

By M. Kisin. 2009

182 citations recorded by the source corpus.

We prove a modularity lifting theorem for two dimensional, 2-adic, potentially Barsotti-Tate representations. This proves hypothesis (H) of Khare-Wintenberger, and completes the proof of Serre’s conjecture. The main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of ℚ2.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/08039d140886ff8355723504c8dd71e236f7cf64

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Serre’s modularity conjecture (II) (2009, background): “This statement is now a theorem of Kisin [20].”
A relation between Dieudonne displays and crystalline Dieudonne theory (2010, methodology): “By a well-known construction, the crystalline Dieudonné functor allows us to associate to a p-divisible group over R an F-window; this is explained in the proof of [Zink 2001b, Theorem 1.6] for the Dieudonné crystal of a nilpotent display, and in [Kisin 2006; 2009] for p-divisible groups.”
Finding meaning in error terms (2008, background): “31See [28], where the [LC(v)] condition for v|p > 2 is that the local representation is Barsotti-Tate; [29] deals with p = 2; and [30] covers the case where the local representations for v|p become semi-stable over an abelian extension of Qp.”
Galois theory of iterated endomorphisms (2007, result): “We remark that Serre’s conjecture has been proved owing to work of Khare and Wintenberger [17, 18], and Kisin [19].”
Even Galois representations and the Fontaine–Mazur conjecture (2009, result): “e representation ρλ|Dp is Hodge-Tate with Hodge-Tate weights mand n. Say that Ris irreducible if one (respectively, any) ρλ is irreducible. We prove the following result (contrast the result of Kisin [21], Corollary (0.5)). 7 5.2. Theorem. Let Rbe an irreducible weakly compatible family of two dimen”
Automorphic lifts of prescribed types (2008, background): “One can always replace U by a subgroup (satisfying the above assumptions) for which this holds (c.f. section 3.1.1 of [ 27 ]).”
Torsion p-adic Galois representations and a conjecture of Fontaine (2007, background): “3 in [14] and [4], if p 3, then Mod /S is anti-equivalent to the category of finite flat group schemes over OK (also see [17]).”