The Structure of Potentially Semi-Stable Deformation Rings

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The Structure of Potentially Semi-Stable Deformation Rings

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Title: The Structure of Potentially Semi-Stable Deformation Rings
Author: Kisin, Mark
Citation: Kisin, Mark. Forthcoming. The structure of potentially semi-stable deformation rings. Proceedings of the International Congress of Mathematicians, Hyderabad, India, August 19-27, 2010.
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Abstract: Inside the universal deformation space of a local Galois representation one has the set of deformations which are potentially semi-stable of given p-adic Hodge and Galois type. It turns out these points cut out a closed subspace of the deformation space. A deep conjecture due to Breuil-M´ezard predicts that part of the structure of this space can be described in terms of the local Langlands correspondence. For 2-dimensional representations the conjecture can be made precise. We explain some of the progress in this case, which reveals that the conjecture is intimately connected to the p-adic local Langlands correspondence, as well as to the Fontaine-Mazur conjecture.
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