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Potentially semi-stable deformation rings for discrete series extended types. (Anneaux de déformations potentiellement semi-stables pour les types étendus de la série discrète.) (English. French summary) Zbl 1330.11039
Let $$K/\mathbb{Q}_p$$ be a finite extension, and $$\overline{\rho}$$ be a mod $$p$$ Galois representation of the absolute Galois group $$G_K$$. Potentially semi-stable deformation rings for $$\overline{\rho}$$ with fixed determinant, Hodge-Tate weights and inertial type have been previously constructed by M. Kisin [J. Am. Math. Soc. 21, No. 2, 513–546 (2008; Zbl 1205.11060)]. An inertial type is a representation of the inertia group $$I_K$$ with $$p$$-adic coefficients, which extends to a representation of the Weil group $$W_K$$. In this article, the author analyzes potentially semi-stable deformations of $$\overline{\rho}$$ with fixed determinant, Hodge-Tate weights and extended inertial type. The author defines the latter as a representation of $$W_K$$, and in the article the results are concentrated in the case where this extended inertial type is of discrete series type. She shows that the corresponding deformation ring is a $$p$$-torsion free, reduced quotient of the Kisin deformation ring, it is supported on a set of irreducible components of it, and it is equidimensional after inverting $$p$$.
The author then proceeds to express the Hilbert-Samuel multiplicity of the special fibre of these deformation rings, in terms of representations of a certain group $$\mathcal{G}$$ which is a quotient of $$\mathcal{O}_D$$, where $$D$$ is the unique non-split quaternion algebra over $$K$$. The formula is proved when $$K=\mathbb{Q}_p$$, or for general K when $$\overline{\rho}$$ satisfies some additional assumptions, including that the Hodge-Tate weights are $$(0, 1)$$. The result is obtained from a combination of a reformulation by T. Gee and D. Geraghty of the usual Breuil-Mézard conjecture and modularity lifting theorems on a quaternion algebra ramified at infinity and at primes dividing $$p$$ [“The Breuil-Mezard conjecture for quaternion algebras”, Preprint, arXiv:1309.0019]. A consequence of the main formula is that when the choice of extended inertial type divides the deformation ring in two parts, then each of them has the same multiplicity. Finally, the author ends with an application of the results to the existence of congruences mod $$p$$ of certain modular forms.

##### MSC:
 11F80 Galois representations 11F33 Congruences for modular and $$p$$-adic modular forms
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