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 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 $\mathrm_(K)$ 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:
:$\operatorname_\mathrm(K)\subsetneq\operatorname_(K) \subsetneq \operatorname_(K)\subsetneq \operatorname_(K) \subsetneq \operatorname_(K)$
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 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
:$D_B(V)=(B\otimes_V)^$
(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 $E:=B^$. 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
:$\dim_ED_(V)=\dim_V$
or, equivalently, the comparison morphism
:$\alpha_V:B_\ast\otimes_ED_(V)\longrightarrow B_\ast \otimes_V$
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)
::$H^\ast_(X/\mathbf)\cong H^\ast(X(\mathbf),\mathbf)\otimes_\mathbf\mathbf.$
: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 $B_:=\oplus_\mathbf_K(i)$. Then there is a functorial isomorphism
::$B_\otimes_K\mathrmH^\ast_(X/K)\cong B_\otimes_H^\ast_(X\times_K\overline,\mathbf_p)$
:of graded vector spaces with ''G_K''-action (the de Rham cohomology is equipped with the Hodge filtration, and $\mathrmH^\ast_$ 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''
:$B_\otimes_KH^\ast_(X/K)\cong B_\otimes_H^\ast_(X\times_K\overline,\mathbf_p)$
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
:$B_\otimes_H^\ast_(X/K)\cong B_\otimes_H^\ast_(X\times_K\overline,\mathbf_p)$
as vector spaces with φ-action, ''G_K''-action, and filtration after extending scalars to ''K'' (here $H^\ast_(X/K)$ 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 ''i''th ''p''-adic étale cohomology group, then
:$D_(V)=H^i_(X/K).$
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
:$B_\otimes_H^\ast_(X/K)\cong B_\otimes_H^\ast_(X\times_K\overline,\mathbf_p)$
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.

Notes

See also

* Hodge theory
* Arakelov theory
* Hodge-Arakelov theory
* p-adic Teichmüller theory

References

Primary sources

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Secondary sources

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*{{Citation
, last=Illusie
, first=Luc
, contribution=Cohomologie de de Rham et cohomologie étale ''p''-adique (d'après G. Faltings, J.-M. Fontaine et al.) Exp. 726
, title=Séminaire Bourbaki. Vol. 1989/90. Exposés 715–729
, publisher=Société Mathématique de France
, location=Paris
, year=1990
, pages=325–374
, mr=1099881
, series=Astérisque
, volume=189–190

Algebraic number theory
Galois theory
Representation theory of groups
Hodge theory
Arithmetic geometry