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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
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