Density of potentially crystalline representations of fixed weight

Eugen Hellmann (1) , Benjamin Schraen (2)

1 Mathematisches Institut
2 LMV - Laboratoire de Mathématiques de Versailles

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.

Domaines: Théorie des nombres [math.NT]

Eugen Hellmann, Benjamin Schraen. Density of potentially crystalline representations of fixed weight. 2013. ⟨hal-00909045v1⟩