Zariski Density of Crystalline Representations for Any p-Adic Field

The aim of this article is to prove Zariski density of crystalline representations in the rigid analytic space associated to the universal deformation ring of a d-dimensional mod p representation of Gal(K̅/K) for any d and any p-adic field K. This is a generalization of the results of Colmezand Kisin for d = 2 and K = Qp, of the author for d = 2 and any K, and of Chenevier for any d and K = Qp. A key ingredient for the proof is to construct a p-adic family of trianguline representations which can be seen as a local analogue of eigenvarieties. In this article, we construct such a family by generalizing Kisin’s theory of finite slope subspace Xfs for any d and any K, and using Bellaïche- Chenevier’s idea of using exterior products in the study of trianguline deformations