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Potentially semi-stable deformation rings. (English) Zbl 1205.11060
Let $$K$$ be a $$p$$-adic field in characteristic 0, $$F$$ a finite field in characteristic $$p$$ and $$V$$ a finite-dimensional $$F$$-vector space on which the absolute Galois group $$G_K$$ of $$K$$ operates linearly and continuously. Very often this representation admits a universal deformation ring $$R$$. Using a variant of Fontaine’s $$(\varphi,\Gamma)$$-modules, the author shows that for any finite extension $$L$$ of $$K$$ and integers $$a\leq b$$ there exists a quotient $$\tilde R$$ of $$R$$ with the following property: For every finite algebra $$B$$ over the quotient field of the ring of Witt-vectors of $$F$$, an algebra morphism from $$R$$ to $$B$$ factors over $$\tilde R$$ if and only if the associated $$B[G_L]$$-module is semistable and has Hodge-Tate weights between $$a$$ and $$b$$.
He also gives variants of this result which are closely related to the Fontaine-Mazur conjecture. Using properties of $$\tilde R$$ he then proves one part of the Breuil-Mézard conjecture [C. Breuil and A. Mézard, Duke Math. J. 115, No. 2, 205–310 (2002; Zbl 1042.11030)]. Furthermore, an application of his results to Galois representations attached to Hilbert modular forms is provided.

##### MSC:
 11F80 Galois representations 11S20 Galois theory
##### Keywords:
Galois representations; deformation theory
Full Text:
##### References:
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