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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.

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    $\begingroup$ I'd imagine this would work better if it was made communiti-wiki, and one application per answer. $\endgroup$ – Gjergji Zaimi Dec 22 '14 at 2:29
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    $\begingroup$ The notes by Brinon and Conrad are great, but I personally didn't get much motivation out of them. "An abelian variety has good reduction if and only if the associated Galois rep is crystalline" didn't seem to me like a good enough "application", as the definition of "crystalline" is complicated. I'd love to see here, say, new theorems about varieties over $\mathbb{C}$ proved using $p$-adic comparison theorems. $\endgroup$ – Piotr Achinger Dec 22 '14 at 6:18
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    $\begingroup$ @Piotr I guess you're right. I'm not a specialist in arithmetic geometry, and I don't think I can be of much help here. :-) $\endgroup$ – user62675 Dec 22 '14 at 15:28
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    $\begingroup$ One type of application of p-adic Hodge theory, including some integral Hodge theory, which gives crisp statements is to the study of smooth projective varieties over $\mathbb{Q}$ with everywhere good reduction (or very little ramification). The earliest result of this kind is Fontaine's theorem: there is no abelian variety over $\mathbb{Q}$ with everywhere good reduction, but there are generalizations due to Fontaine and Abrashkin. See e.g. arxiv.org/abs/1003.2905. $\endgroup$ – Simon Pepin Lehalleur Dec 22 '14 at 15:59
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    $\begingroup$ Some of the world experts in this area are Christophe Breuil, Frank Calegari, Pierre Colmez, Toby Gee, Mark Kisin, Peter Scholze, and Richard Taylor. Have you tried looking at their work (say by visiting their web-pages)? $\endgroup$ – tracing Dec 22 '14 at 17:20
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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.

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