# 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.

## General classification of p-adic representations

Let $K$ be a local field with 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 $\rho :G_{K}\to {\text{GL}}(V)$, where $V$ is a finite-dimensional vector space over $\mathbb {Q} _{p}$. The collection of all $p$-adic representations of $K$ form an abelian category denoted $\mathrm {Rep} _{\mathbb {Q} _{p}}(K)$ 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:
    $\operatorname {Rep} _{\mathrm {crys} }(K)\subsetneq \operatorname {Rep} _{ss}(K)\subsetneq \operatorname {Rep} _{dR}(K)\subsetneq \operatorname {Rep} _{HT}(K)\subsetneq \operatorname {Rep} _{\mathbb {Q} _{p}}(K)$
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 $\operatorname {Rep} _{\mathrm {pcrys} }(K)$ and the potentially semistable representations $\operatorname {Rep} _{\mathrm {pss} }(K)$. The latter strictly contains the former which in turn generally strictly contains $\operatorname {Rep} _{\mathrm {crys} }(K)$; additionally, $\operatorname {Rep} _{\mathrm {pss} }(K)$ generally strictly contains $\operatorname {Rep} _{\mathrm {ss} }(K)$, and is contained in $\operatorname {Rep} _{dR}(K)$ (with equality when the residue field of $K$ is finite, a statement called the p-adic monodromy theorem).

## Period rings and comparison isomorphisms in arithmetic geometry

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
    $D_{B}(V)=(B\otimes _{\mathbf {Q} _{p}}V)^{G_{K}}$
(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 $E:=B^{G_{K}}$. 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
    $\dim _{E}D_{B_{\ast }}(V)=\dim _{\mathbf {Q} _{p}}V$
or, equivalently, the comparison morphism
    $\alpha _{V}:B_{\ast }\otimes _{E}D_{B_{\ast }}(V)\longrightarrow B_{\ast }\otimes _{\mathbf {Q} _{p}}V$
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:
* If X is a proper smooth scheme over C, there is a classical comparison isomorphism between the algebraic de Rham cohomology of X over C and the singular cohomology of X(C)
       $H_{\mathrm {dR} }^{\ast }(X/\mathbf {C} )\cong H^{\ast }(X(\mathbf {C} ),\mathbf {Q} )\otimes _{\mathbf {Q} }\mathbf {C} .$
    This isomorphism can be obtained by considering a pairing obtained by integrating differential forms in the algebraic de Rham cohomology over cycles in the singular cohomology. The result of such an integration is called a period and is generally a complex number. This explains why the singular cohomology must be tensored to C, and from this point of view, C can be said to contain all the periods necessary to compare algebraic de Rham cohomology with singular cohomology, and could hence be called a period ring in this situation.
* In the mid sixties, Tate conjectured that a similar isomorphism should hold for proper smooth schemes X over K between algebraic de Rham cohomology and p-adic étale cohomology (the Hodge–Tate conjecture, also called C$_{HT}$). Specifically, let C$_K$ be the completion of an algebraic closure of K, let C$_K$(i) denote C$_K$ where the action of G$_K$ is via g·z = χ(g)$^i$g·z (where χ is the p-adic cyclotomic character, and i is an integer), and let $B_{\mathrm {HT} }:=\oplus _{i\in \mathbf {Z} }\mathbf {C} _{K}(i)$. Then there is a functorial isomorphism
       $B_{\mathrm {HT} }\otimes _{K}\mathrm {gr} H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {HT} }\otimes _{\mathbf {Q} _{p}}H_{\mathrm {{\acute {e}}t} }^{\ast }(X\times _{K}{\overline {K}},\mathbf {Q} _{p})$
    of graded vector spaces with G$_K$-action (the de Rham cohomology is equipped with the Hodge filtration, and $\mathrm {gr} H_{\mathrm {dR} }^{\ast }$ is its associated graded). This conjecture was proved by Gerd Faltings in the late eighties after partial results by several other mathematicians (including Tate himself).
* For an abelian variety X with good reduction over a p-adic field K, Alexander Grothendieck reformulated a theorem of Tate's to say that the crystalline cohomology H$^1$(X/W(k)) ⊗ Q$_p$ of the special fiber (with the Frobenius endomorphism on this group and the Hodge filtration on this group tensored with K) and the p-adic étale cohomology H$^1$(X,Q$_p$) (with the action of the Galois group of K) contained the same information. Both are equivalent to the p-divisible group associated to X, up to isogeny. Grothendieck conjectured that there should be a way to go directly from p-adic étale cohomology to crystalline cohomology (and back), for all varieties with good reduction over p-adic fields. This suggested relation became known as the mysterious functor.
To improve the Hodge–Tate conjecture to one involving the de Rham cohomology (not just its associated graded), Fontaine constructed a filtered ring B$_{dR}$ whose associated graded is B$_{HT}$ and conjectured the following (called C$_{dR}$) for any smooth proper scheme X over K
    $B_{\mathrm {dR} }\otimes _{K}H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {dR} }\otimes _{\mathbf {Q} _{p}}H_{\mathrm {{\acute {e}}t} }^{\ast }(X\times _{K}{\overline {K}},\mathbf {Q} _{p})$
as filtered vector spaces with G$_K$-action. In this way, B$_{dR}$ could be said to contain all (p-adic) periods required to compare algebraic de Rham cohomology with p-adic étale cohomology, just as the complex numbers above were used with the comparison with singular cohomology. This is where B$_{dR}$ obtains its name of ring of p-adic periods.
Similarly, to formulate a conjecture explaining Grothendieck's mysterious functor, Fontaine introduced a ring B$_{cris}$ with G$_K$-action, a "Frobenius" φ, and a filtration after extending scalars from K$_0$ to K. He conjectured the following (called C$_{cris}$) for any smooth proper scheme X over K with good reduction
    $B_{\mathrm {cris} }\otimes _{K_{0}}H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {cris} }\otimes _{\mathbf {Q} _{p}}H_{\mathrm {{\acute {e}}t} }^{\ast }(X\times _{K}{\overline {K}},\mathbf {Q} _{p})$
as vector spaces with φ-action, G$_K$-action, and filtration after extending scalars to K (here $H_{\mathrm {dR} }^{\ast }(X/K)$ is given its structure as a K$_0$-vector space with φ-action given by its comparison with crystalline cohomology). Both the C$_{dR}$ and the C$_{cris}$ conjectures were proved by Faltings.
Upon comparing these two conjectures with the notion of B$_∗$-admissible representations above, it is seen that if X is a proper smooth scheme over K (with good reduction) and V is the p-adic Galois representation obtained as is its ith p-adic étale cohomology group, then
    $D_{B_{\ast }}(V)=H_{\mathrm {dR} }^{i}(X/K).$
In other words, the Dieudonné modules should be thought of as giving the other cohomologies related to V.
In the late eighties, Fontaine and Uwe Jannsen formulated another comparison isomorphism conjecture, C$_{st}$, this time allowing X to have semi-stable reduction. Fontaine constructed a ring B$_{st}$ with G$_K$-action, a "Frobenius" φ, a filtration after extending scalars from K$_0$ to K (and fixing an extension of the p-adic logarithm), and a "monodromy operator" N. When X has semi-stable reduction, the de Rham cohomology can be equipped with the φ-action and a monodromy operator by its comparison with the log-crystalline cohomology first introduced by Osamu Hyodo. The conjecture then states that
    $B_{\mathrm {st} }\otimes _{K_{0}}H_{\mathrm {log-cris} }^{\ast }(X/K)\cong B_{\mathrm {st} }\otimes _{\mathbf {Q} _{p}}H_{\mathrm {{\acute {e}}t} }^{\ast }(X\times _{K}{\overline {K}},\mathbf {Q} _{p})$
as vector spaces with φ-action, G$_K$-action, filtration after extending scalars to K, and monodromy operator N. This conjecture was proved in the late nineties by Takeshi Tsuji.