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 ${\displaystyle K}$ be a local field with residue field ${\displaystyle k}$ of characteristic ${\displaystyle p}$. In this article, a ${\displaystyle p}$-adic representation of ${\displaystyle K}$ (or of ${\displaystyle G_{K}}$, the absolute Galois group of ${\displaystyle K}$) will be a continuous representation ${\displaystyle \rho :G_{K}\to {\text{GL}}(V)}$, where ${\displaystyle V}$ is a finite-dimensional vector space over ${\displaystyle \mathbb {Q} _{p}}$. The collection of all ${\displaystyle p}$-adic representations of ${\displaystyle K}$ form an abelian category denoted ${\displaystyle \mathrm {Rep} _{\mathbb {Q} _{p}}(K)}$ in this article. ${\displaystyle p}$-adic Hodge theory provides subcollections of ${\displaystyle p}$-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:

where each collection is a full subcategory properly contained in the next. In order, these are the categories of crystalline representations, semistable representations, de Rham representations, Hodge–Tate representations, and all p-adic representations. In addition, two other categories of representations can be introduced, the potentially crystalline representations ${\displaystyle \operatorname {Rep} _{\mathrm {pcrys} }(K)}$ and the potentially semistable representations ${\displaystyle \operatorname {Rep} _{\mathrm {pss} }(K)}$. The latter strictly contains the former which in turn generally strictly contains ${\displaystyle \operatorname {Rep} _{\mathrm {crys} }(K)}$; additionally, ${\displaystyle \operatorname {Rep} _{\mathrm {pss} }(K)}$ generally strictly contains ${\displaystyle \operatorname {Rep} _{\mathrm {ss} }(K)}$, and is contained in ${\displaystyle \operatorname {Rep} _{dR}(K)}$ (with equality when the residue field of ${\displaystyle K}$ is finite, a statement called the p-adic monodromy theorem).

The general strategy of p-adic Hodge theory, introduced by Fontaine, is to construct certain so-called period rings such as B_dR, B_st, B_cris, and B_HT which have both an action by G_K and some linear algebraic structure and to consider so-called Dieudonné modules

(where B is a period ring, and V is a p-adic representation) which no longer have a G_K-action, but are endowed with linear algebraic structures inherited from the ring B. In particular, they are vector spaces over the fixed field ${\displaystyle E:=B^{G_{K}}}$. This construction fits into the formalism of B-admissible representations introduced by Fontaine. For a period ring like the aforementioned ones B_∗ (for ∗ = HT, dR, st, cris), the category of p-adic representations Rep_∗(K) mentioned above is the category of B_∗-admissible ones, i.e. those p-adic representations V for which

or, equivalently, the comparison morphism

is an isomorphism.

This formalism (and the name period ring) grew out of a few results and conjectures regarding comparison isomorphisms in arithmetic and complex geometry: