ON A CONJECTURE OF CONRAD, DIAMOND, AND TAYLOR

By David Savitt. 2004

74 citations recorded by the source corpus.

We prove a conjecture of Conrad, Diamond, and Taylor on the size of certain deformation rings parametrizing potentially Barsotti-Tate Ga- lois representations. To achieve this, we extend results of Breuil and M ezard (classifying Galois lattices in semistable representations in terms of \strongly divisible modules") to the potentially crystalline case in Hodge-Tate weights (0; 1). We then use these strongly divisible modules to compute the desired deformation rings. As a corollary, we obtain new results on the modularity of potentially Barsotti-Tate representations.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3384cbdf7111ed80b0465f460241c0d5414e06de

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Lattices in the cohomology of Shimura curves (2013, background): “13 of [40] and the remarks immediately preceding it (together with the remarks in the previous paragraph) show that if R = O, the category of strongly divisible modules with tame descent data and R-coefficients is equivalent to the category of GL -stableO-lattices in E-representations of GL which be”
A geometric perspective on the Breuil–Mézard conjecture (2011, unknown): “22 of [36] (noting that the framed deformation ring is formally smooth over the deformation ring, since r̄ is irreducible), we have R ,ψ (0, 1, ω̃m+n ⊕ ω̃m, r̄)/π ∼= F[[U,V,W,X,Y]]/(XY), so Cm,n 6= Cm+n,p−1−n, as required.”
Serre’s modularity conjecture (II) (2009, background): “15 (1) of Savitt’s paper [46] (see the similar argument below).”