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^[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 of 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 ${\mathrm {Rep}}_{{{\mathbf {Q}}_{p}}}(K)$ 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]

{\mathrm {Rep}}_{{cris}}(K)\subsetneq {\mathrm {Rep}}_{{st}}(K)\subsetneq {\mathrm {Rep}}_{{dR}}(K)\subsetneq {\mathrm {Rep}}_{{HT}}(K)\subsetneq {\mathrm {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

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 $E:=B^{{G_{K}}}$ .^[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

\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^[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 $B_{{{\mathrm {HT}}}}:=\oplus _{{i\in {\mathbf {Z}}}}{\mathbf {C}}_{K}(i)$ . 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^[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

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

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 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

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

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]

Notes

↑ In this article, a local field is complete discrete valuation field whose residue field is perfect.
↑ Fontaine 1994, p. 114
↑ These rings depend on the local field K in question, but this relation is usually dropped from the notation.
↑ For B = B_HT, B_dR, B_st, and B_cris, $B^{{G_{K}}}$ is K, K, K₀, and K₀, respectively, where K₀ = Frac(W(k)), the fraction field of the Witt vectors of k.
↑ See Serre 1967
↑ Faltings 1988
↑ Grothendieck 1971, p. 435
↑ Fontaine 1982
↑ Fontaine 1982, Conjecture A.6
↑ Fontaine 1982, Conjecture A.11
↑ Faltings 1989
↑ Fontaine 1994, Exposé II, section 3
↑ Hyodo 1991
↑ Tsuji 1999

References

Primary sources

Tate, John (1966), "p-Divisible Groups", in Proceedings of a Conference on Local Fields, Springer, 1967. doi:10.1007/978-3-642-87942-5
Faltings, Gerd (1988), "p-adic Hodge theory", Journal of the American Mathematical Society, 1 (1): 255–299, doi:10.2307/1990970, MR 0924705
Faltings, Gerd, "Crystalline cohomology and p-adic Galois representations", in Igusa, Jun-Ichi, Algebraic analysis, geometry, and number theory, Baltimore, MD: Johns Hopkins University Press, pp. 25–80, ISBN 978-0-8018-3841-5, MR 1463696
Fontaine, Jean-Marc (1982), "Sur certains types de représentations p-adiques du groupe de Galois d'un corps local; construction d'un anneau de Barsotti–Tate", Annals of Mathematics, 115 (3): 529–577, doi:10.2307/2007012, MR 0657238
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Secondary sources

Berger, Laurent (2004), "An introduction to the theory of p-adic representations", Geometric aspects of Dwork theory, I, Berlin: Walter de Gruyter GmbH & Co. KG, arXiv:math/0210184, ISBN 978-3-11-017478-6, MR 2023292
Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory (PDF), retrieved 2010-02-05
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Illusie, Luc (1990), "Cohomologie de de Rham et cohomologie étale p-adique (d'après G. Faltings, J.-M. Fontaine et al.) Exp. 726", Séminaire Bourbaki. Vol. 1989/90. Exposés 715–729, Astérisque, 189–190, Paris: Société Mathématique de France, pp. 325–374, MR 1099881

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