P-adic Hodge theory

In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p (such as Q_p). The theory has its beginnings in Jean-Pierre Serre and John Tate's study of Tate modules of abelian varieties and the notion of Hodge–Tate representation. Hodge–Tate representations are related to certain decompositions of p-adic cohomology theories analogous to the Hodge decomposition, hence the name p-adic Hodge theory. Further developments were inspired by properties of p-adic Galois representations arising from the étale cohomology of varieties. Jean-Marc Fontaine introduced many of the basic concepts of the field.

Let K be a local field of residue field k of characteristic p. In this article, a p-adic representation of K (or of G_K, the absolute Galois group of K) will be a continuous representation ρ : G_K→ GL(V) where V is a finite-dimensional vector space over Q_p. The collection of all p-adic representations of K form an abelian category denoted ${\displaystyle \mathrm {Rep} _{\mathbf {Q} _{p}}(K)}$ in this article. p-adic Hodge theory provides subcollections of p-adic representations based on how nice they are, and also provides faithful functors to categories of linear algebraic objects that are easier to study. The basic classification is as follows:

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