On some applications of integral p-adic Hodge theory to Galois representations

We explicitly construct an analytic family of n-dimensional crystalline representations by using integral p-adic Hodge theory. This is a generalization of results by Berger, Li, and Zhu and by Dousmanis. We show, by using Kisin's method, that the part of a universal deformation ring related to the above constructions is connected. From this we obtain an explicitly described subclass of potentially diagonalizable representations in the sense of Barnet-Lamb, Gee, Geraghty and Taylor. This yields automorphy lifting theorem and potential automorphy theorem, in which the condition at p is weakened.