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 "GK", the absolute Galois group of "K") will be a continuous representation ρ : "GK"→ 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 formula_1 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("K") and the potentially semistable representations Reppst("K"). The latter strictly contains the former which in turn generally strictly contains Repcris("K"); additionally, Reppst("K") generally strictly contains Repst("K"), and is contained in RepdR("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 "GK" 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 "GK"-action, but are endowed with linear algebraic structures inherited from the ring "B". In particular, they are vector spaces over the fixed field formula_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
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:
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 CdR) for any smooth proper scheme "X" over "K"
as filtered vector spaces with "GK"-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 "GK"-action, a "Frobenius" φ, and a filtration after extending scalars from "K"0 to "K". He conjectured the following (called Ccris) for any smooth proper scheme "X" over "K" with good reduction
as vector spaces with φ-action, "GK"-action, and filtration after extending scalars to "K" (here formula_13 is given its structure as a "K"0-vector space with φ-action given by its comparison with crystalline cohomology). 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" (with good reduction) and "V" is the "p"-adic Galois representation obtained as is its "i"th "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 "B"st with "GK"-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
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.