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Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Potentially semi-stable deformation rings

Author: Mark Kisin
Journal: J. Amer. Math. Soc. 21 (2008), 513-546
MSC (2000): Primary 11S20
Published electronically: September 20, 2007
MathSciNet review: 2373358
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Abstract: Let $ K/\Q_p$ be a finite extension and $ G_K$ the absolute Galois group of $ K$. For $ (A^{\circ}, \mathfrak{m})$ a complete local ring with finite residue and $ V_{A^{\circ}}$ a finite free $ A^{\circ}$-module equipped with an action of $ G_K$ , we show that $ A^{\circ}[1/p]$ has a maximal quotient over which the representation $ V_{A^{\circ}}$ is semi-stable with Hodge-Tate weights in a given range. We show an analogous result for representations which are potentially semi-stable of fixed Galois type and $ p$-adic Hodge type.

If $ V_{A^{\circ}}$ is the universal deformation of $ V_{A^{\circ}}\otimes_{A^{\circ}} A^{\circ}/\mathfrak{m}$, then we compute the dimension of $ A^{\circ}[1/p]$ and we show that these rings are sometimes smooth.

Finally we apply this theory to show, in some new cases, the compatibility of the $ p$-adic Galois representation attached to a Hilbert modular form with the local Langlands correspondence at $ p$.

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Additional Information

Mark Kisin
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Received by editor(s): April 13, 2006
Published electronically: September 20, 2007
Additional Notes: The author was partially supported by NSF grant DMS-0400666 and a Sloan Research Fellowship.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.