TITLE: Applications of $p$-adic Hodge theory
QUESTION [16 upvotes]: I am trying to learn $p$-adic Hodge theory. I found some materials explaining main theorems (or aspects) of the theory. However, I could not find references which explaining how to use the theory. Especially, I would like to know geometric/arithmetic applications, if there are, of $p$-adic Hodge theory in number theory. Since I did not specify meaning of the words 'geometric/arithmetic', any reference would be appreciated.

REPLY [7 votes]: For an example of an application of $p$-adic Hodge theory in a geometric setting, I thoroughly recommend reading the beautiful paper
P. Berthelot, H. Esnault, K. Rulling, Rational points over finite fields for regular models of algebraic varieties of Hodge type $\geq 1$, Ann. of Math. (2) $\bf{176}$ 2012, no. 1, 413-508.
The result is nice and concrete: they prove congruences for the number of rational points on such varieties. As you will see, the proof makes use of a nice range of big theorems from $p$-adic Hodge theory and is very clearly explained.