Let K be a local field with residue fieldk of characteristic p. In this article, a p-adic representation of K will be a continuous representation ρ : GK→ GL, where V is a finite-dimensional vector space over Qp. The collection of all p-adic representations of K form an abelian category denoted 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: 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 Reppcris and the potentially semistable representations Reppst. The latter strictly contains the former which in turn generally strictly contains Repcris; additionally, Reppst generally strictly contains Repst, and is contained in RepdR.
The general strategy of p-adic Hodge theory, introduced by Fontaine, is to construct certain so-called period rings such as BdR, Bst, Bcris, and BHT which have both an action by GK and some linear algebraic structure and to consider so-called Dieudonné modules which no longer have a GK-action, but are endowed with linear algebraic structures inherited from the ringB. In particular, they are vector spaces over the fixed field. This construction fits into the formalism of B-admissible representations introduced by Fontaine. For a period ring like the aforementioned ones B∗, the category of p-adic representations Rep∗ 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 grew out of a few results and conjectures regarding comparison isomorphisms in arithmetic and complex geometry:
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. Specifically, let CK be the completion of an algebraic closure of K, let CK denote CK where the action of GK is via g·z = χig·z, and let. Then there is a functorial isomorphism
For an abelian varietyX with good reduction over a p-adic field K, Alexander Grothendieck reformulated a theorem of Tate's to say that the crystalline cohomologyH1 ⊗ Qp of the special fiber and the p-adic étale cohomology H1 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, 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, Fontaine constructed a filtered ring BdR whose associated graded is BHT and conjectured the following for any smooth proper schemeX over K as filtered vector spaces with GK-action. In this way, BdR could be said to contain all 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 BdR obtains its name of ring of p-adic periods. Similarly, to formulate a conjecture explaining Grothendieck's mysterious functor, Fontaine introduced a ringBcris with GK-action, a "Frobenius" φ, and a filtration after extending scalars from K0 to K. He conjectured the following for any smooth proper schemeX over K with good reduction as vector spaces with φ-action, GK-action, and filtration after extending scalars to K. Both the CdR and the Ccris 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 and V is the p-adic Galois representation obtained as is its ith p-adic étale cohomology group, then 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, Cst, this time allowing X to have semi-stable reduction. Fontaine constructed a ring Bst with GK-action, a "Frobenius" φ, a filtration after extending scalars from K0 to K, 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 as vector spaces with φ-action, GK-action, filtration after extending scalars to K, and monodromy operator N. This conjecture was proved in the late nineties by Takeshi Tsuji.
Primary sources
Tate, John, "p-Divisible Groups", in Proceedings of a Conference on Local Fields, Springer, 1967. doi:10.1007/978-3-642-87942-5
