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 ${\displaystyle K}$ be a local field with residue field ${\displaystyle k}$ of characteristic ${\displaystyle p}$. In this article, a ${\displaystyle p}$-adic representation of ${\displaystyle K}$ (or of ${\displaystyle G_{K}}$, the absolute Galois group of ${\displaystyle K}$) will be a continuous representation ${\displaystyle \rho :G_{K}\to {\text{GL}}(V)}$, where ${\displaystyle V}$ is a finite-dimensional vector space over ${\displaystyle \mathbb {Q} _{p}}$. The collection of all ${\displaystyle p}$-adic representations of ${\displaystyle K}$ form an abelian category denoted ${\displaystyle \mathrm {Rep} _{\mathbb {Q} _{p}}(K)}$ in this article. ${\displaystyle p}$-adic Hodge theory provides subcollections of ${\displaystyle p}$-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]

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

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 ${\displaystyle \operatorname {Rep} _{\mathrm {pcrys} }(K)}$ and the potentially semistable representations ${\displaystyle \operatorname {Rep} _{\mathrm {pss} }(K)}$. The latter strictly contains the former which in turn generally strictly contains ${\displaystyle \operatorname {Rep} _{\mathrm {crys} }(K)}$; additionally, ${\displaystyle \operatorname {Rep} _{\mathrm {pss} }(K)}$ generally strictly contains ${\displaystyle \operatorname {Rep} _{\mathrm {ss} }(K)}$, and is contained in ${\displaystyle \operatorname {Rep} _{dR}(K)}$ (with equality when the residue field of ${\displaystyle K}$ 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

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

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

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

See also

• Hodge theory
• Arakelov theory
• Hodge-Arakelov theory
• p-adic Teichmüller theory

References

Primary sources

• Tate, John (1967), "p-Divisible Groups"", Proceedings of a Conference on Local Fields, Springer, pp. 158–183, doi:10.1007/978-3-642-87942-5_12, ISBN   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, JSTOR   1990970, MR   0924705
• Faltings, Gerd (1989), "Crystalline cohomology and p-adic Galois representations", in Igusa, Jun-Ichi (ed.), 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, JSTOR   2007012, MR   0657238
• Grothendieck, Alexander (1971), "Groupes de Barsotti–Tate et cristaux", Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 1, pp. 431–436, MR   0578496
• Hyodo, Osamu (1991), "On the de Rham–Witt complex attached to a semi-stable family", Compositio Mathematica , 78 (3): 241–260, MR   1106296
• Serre, Jean-Pierre (1967), "Résumé des cours, 1965–66", Annuaire du Collège de France, Paris, pp. 49–58
• Tsuji, Takeshi (1999), "p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case", Inventiones Mathematicae , 137 (2): 233–411, Bibcode:1999InMat.137..233T, doi:10.1007/s002220050330, MR   1705837, S2CID   121547567

Secondary sources

• Berger, Laurent (2004), "An introduction to the theory of p-adic representations", Geometric aspects of Dwork theory, vol. I, Berlin: Walter de Gruyter GmbH & Co. KG, arXiv: math/0210184 , Bibcode:2002math.....10184B, 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
• Fontaine, Jean-Marc, ed. (1994), Périodes p-adiques, Astérisque, vol. 223, Paris: Société Mathématique de France, MR   1293969
• 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, vol. 189–190, Paris: Société Mathématique de France, pp. 325–374, MR   1099881