Density of potentially crystalline representations of fixed weight

Abstract : Let K be a finite extension of Qp. We fix a continuous absolutely irreducible representation of the absolute Galois group of K over a finite dimensional vector space with coefficient in a finite field of characteristic p and consider its universal deformation ring R. If we fix a regular set of Hodge-Tate weights k, we prove, under some hypothesis, that the closed points of Spec(R[1/p]) corresponding to potentially crystalline representations of fixed Hodge-Tate weights k are dense in Spec(R[1/p]) for the Zariski topology. The main hypothesis we need is the existence of a potentially diagonalizable lift, so that in the two-dimensional case, the result is unconditional.
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Eugen Hellmann, Benjamin Schraen. Density of potentially crystalline representations of fixed weight. Compositio Mathematica, Foundation Compositio Mathematica, 2016, 152 (8), pp.1609-1647. ⟨https://doi.org/10.1112/S0010437X16007363⟩. ⟨10.1112/S0010437X16007363⟩. ⟨hal-00909045v2⟩

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