On a Conjecture of Conrad, Diamond, and Taylor

Details On a Conjecture of Conrad, Diamond, and Taylor On a Conjecture of Conrad, Diamond, and Taylor

2004-04-19

arxiv.org/abs/math/0404327v3

Duke Math. Journal 128 (2005), no. 1, 141-197

Mathematics

Number Theory

This version seamlessly folds in the erratum that can be found separately as Appendix A of http://arxiv.org/pdf/0909.1278. Mai

Scientific paper

We prove a conjecture of Conrad, Diamond, and Taylor on the size of certain deformation rings parametrizing potentially Barsotti-Tate Galois representations. To achieve this, we extend results of Breuil and Mezard (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.

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