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]

Rep_cris(K) ⊊ Rep_st(K) ⊊ Rep_dR(K) ⊊ Rep_HT(K) ⊊ Rep_Qp(K)

where each collection is a full subcategory properly contained in the next. In order, these are the categories of cristalline 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 cristalline 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 edit] 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 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

$\dim_ED_{B_\ast}(V)=\dim_{\mathbf{Q}_p}V$

or, equivalently, the comparison morphism

$\alpha_V:B_\ast\otimes_ED_{B_\ast}(V)\longrightarrow B_\ast\otimes_{\mathbf{Q}_p}V$

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)

$H^\ast_{\mathrm{dR}}(X/\mathbf{C})\cong H^\ast(X(\mathbf{C}),\mathbf{Q})\otimes_\mathbf{Q}\mathbf{C}.$

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

$B_{\mathrm{HT}}\otimes_K\mathrm{gr}H^\ast_{\mathrm{dR}}(X/K)\cong B_{\mathrm{HT}}\otimes_{\mathbf{Q}_p}H^\ast_{\mathrm{\acute{e}t}}(X\times_K\overline{K},\mathbf{Q}_p)$

of graded vector spaces with G_K-action (the de Rham cohomology is equipped with the Hodge filtration, and $\mathrm{gr}H^\ast_{\mathrm{dR}}$ 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).

• By combining the comparison of crystalline cohomology of an X over K which has good reduction with both algebraic de Rham cohomology and p-adic étale cohomology, a comparison isomorphism is obtained between algebraic de Rham cohomology and p-adic étale cohomology. Alexander Grothendieck conjectured that there should be a way to go directly from algebraic de Rham cohomology to p-adic étale cohomology without going through crystalline cohomology. He termed this suggested relation the mysterious functor.

To improve the Hodge–Tate conjecture to one involving the de Rham cohomology (not just its associated graded), Fontaine constructed^[7] a filtered ring B_dR whose associated graded is B_HT and conjectured^[8] the following (called C_dR) for any smooth proper scheme X over K

$B_{\mathrm{dR}}\otimes_KH^\ast_{\mathrm{dR}}(X/K)\cong B_{\mathrm{dR}}\otimes_{\mathbf{Q}_p}H^\ast_{\mathrm{\acute{e}t}}(X\times_K\overline{K},\mathbf{Q}_p)$

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^[9] the following (called C_cris) for any smooth proper scheme X over K with good reduction

$B_{\mathrm{cris}}\otimes_{K_0}H^\ast_{\mathrm{dR}}(X/K)\cong B_{\mathrm{cris}}\otimes_{\mathbf{Q}_p}H^\ast_{\mathrm{\acute{e}t}}(X\times_K\overline{K},\mathbf{Q}_p)$

as vector spaces with φ-action, G_K-action, and filtration after extending scalars to K (here $H^\ast_{\mathrm{dR}}(X/K)$ 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.^[10]

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

$D_{B_\ast}(V)=H^i_{\mathrm{dR}}(X/K).$

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^[11] 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.^[12] The conjecture then states that

$B_{\mathrm{st}}\otimes_{K_0}H^\ast_{\mathrm{dR}}(X/K)\cong B_{\mathrm{st}}\otimes_{\mathbf{Q}_p}H^\ast_{\mathrm{\acute{e}t}}(X\times_K\overline{K},\mathbf{Q}_p)$

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.^[13]

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, $B^{G_K}$ 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. ^ Fontaine 1982
8. ^ Fontaine 1982, Conjecture A.6
9. ^ Fontaine 1982, Conjecture A.11
10. ^ Faltings 1989
11. ^ Fontaine 1994, Exposé II, section 3
12. ^ Hyodo 1991
13. ^ Tsuji 1999

References

Primary sources

• Faltings, Gerd (1988), "p-adic Hodge theory", Journal of the American Mathematical Society 1 (1): 255–299, MR0924705 
• 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 978-0-801-83841-5, MR1463696 
• 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, MR0657238 
• Hyodo, Osamu (1991), "On the de Rham–Witt complex attached to a semi-stable family", Compositio Mathematica 78 (3): 241–260, MR1106296 
• 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, MR1705837 

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 978-3-110-17478-6, MR2023292 
• Brinon, Olivier; Conrad, Brian (2009), CMI Summer School notes on p-adic Hodge theory, http://math.stanford.edu/~conrad/papers/notes.pdf, retrieved 2010-02-05 
• Fontaine, Jean-Marc, ed. (1994), Périodes p-adiques, Astérisque, 223, Paris: Société Mathématique de France, MR1293969 
• 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, MR1099881