p-adic Hodge theory

In

denoted ${\displaystyle \mathrm {Rep} _{\mathbf {Q} _{p}}(K)}$ in this article. p-adic Hodge theory provides subcollections of p-adic representations based on how nice they are, and also provides

(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
  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

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
  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

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

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