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^[1] 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 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:^[2]

${\displaystyle \operatorname {Rep} _{\mathrm {crys} }(K)\subsetneq \operatorname {Rep} _{ss}(K)\subsetneq \operatorname {Rep} _{dR}(K)\subsetneq \operatorname {Rep} _{HT}(K)\subsetneq \operatorname {Rep} _{\mathbf {Q} _{p))(K)}$

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 Rep_pcris(K) and the potentially semistable representations Rep_pst(K). The latter strictly contains the former which in turn generally strictly contains Rep_cris(K); additionally, Rep_pst(K) generally strictly contains Rep_st(K), and is contained in 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^[3] 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

${\displaystyle D_{B}(V)=(B\otimes _{\mathbf {Q} _{p))V)^{G_{K))}$

(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 ${\displaystyle E:=B^{G_{K))}$.^[4] 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

${\displaystyle \dim _{E}D_{B_{\ast ))(V)=\dim _{\mathbf {Q} _{p))V}$

or, equivalently, the comparison morphism

${\displaystyle \alpha _{V}:B_{\ast }\otimes _{E}D_{B_{\ast ))(V)\longrightarrow B_{\ast }\otimes _{\mathbf {Q} _{p))V}$

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)

${\displaystyle H_{\mathrm {dR} }^{\ast }(X/\mathbf {C} )\cong H^{\ast }(X(\mathbf {C} ),\mathbf {Q} )\otimes _{\mathbf {Q} }\mathbf {C} .}$

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^[5] 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)ⁱg·z (where χ is the p-adic cyclotomic character, and i is an integer), and let ${\displaystyle B_{\mathrm {HT} }:=\oplus _{i\in \mathbf {Z} }\mathbf {C} _{K}(i)}$. Then there is a functorial isomorphism

${\displaystyle B_{\mathrm {HT} }\otimes _{K}\mathrm {gr} H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {HT} }\otimes _{\mathbf {Q} _{p))H_{\mathrm ((\acute {e))t} }^{\ast }(X\times _{K}{\overline {K)),\mathbf {Q} _{p})}$

of graded vector spaces with G_K-action (the de Rham cohomology is equipped with the Hodge filtration, and ${\displaystyle \mathrm {gr} H_{\mathrm {dR} }^{\ast ))$ is its associated graded). This conjecture was proved by Gerd Faltings in the late eighties^[6] 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 H¹(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 H¹(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.^[7] 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^[8] a filtered ring B_dR whose associated graded is B_HT and conjectured^[9] the following (called C_dR) for any smooth proper scheme X over K

${\displaystyle B_{\mathrm {dR} }\otimes _{K}H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {dR} }\otimes _{\mathbf {Q} _{p))H_{\mathrm ((\acute {e))t} }^{\ast }(X\times _{K}{\overline {K)),\mathbf {Q} _{p})}$

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 K₀ to K. He conjectured^[10] the following (called C_cris) for any smooth proper scheme X over K with good reduction

${\displaystyle B_{\mathrm {cris} }\otimes _{K_{0))H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {cris} }\otimes _{\mathbf {Q} _{p))H_{\mathrm ((\acute {e))t} }^{\ast }(X\times _{K}{\overline {K)),\mathbf {Q} _{p})}$

as vector spaces with φ-action, G_K-action, and filtration after extending scalars to K (here ${\displaystyle H_{\mathrm {dR} }^{\ast }(X/K)}$ is given its structure as a K₀-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.^[11]

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

${\displaystyle D_{B_{\ast ))(V)=H_{\mathrm {dR} }^{i}(X/K).}$

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^[12] a ring B_st with G_K-action, a "Frobenius" φ, a filtration after extending scalars from K₀ 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.^[13] The conjecture then states that

${\displaystyle B_{\mathrm {st} }\otimes _{K_{0))H_{\mathrm {dR} }^{\ast }(X/K)\cong B_{\mathrm {st} }\otimes _{\mathbf {Q} _{p))H_{\mathrm ((\acute {e))t} }^{\ast }(X\times _{K}{\overline {K)),\mathbf {Q} _{p})}$

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.^[14]