The general strategy of
p-adic Hodge theory, introduced by Fontaine, is to construct certain so-called
period rings such as
BdR,
Bst,
Bcris, and
BHT which have both an
action by
GK 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
GK-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}. 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 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 CHT). Specifically, let
CK be the
completion of an
algebraic closure of
K, let
CK(
i) denote
CK where the action of
GK 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
GK-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 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)) ⊗
Qp 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,
Qp) (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
BdR whose associated graded is
BHT and conjectured the following (called CdR) 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
GK-action. In this way,
BdR 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
BdR obtains its name of
ring of p-adic periods. Similarly, to formulate a conjecture explaining Grothendieck's mysterious functor, Fontaine introduced a ring
Bcris with
GK-action, a "Frobenius" φ, and a filtration after extending scalars from
K0 to
K. He conjectured the following (called Ccris) 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,
GK-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 CdR and the Ccris 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
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, Cst, this time allowing
X to have
semi-stable reduction. Fontaine constructed a ring
Bst with
GK-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. The conjecture then states that :B_{\mathrm{st}}\otimes_{K_0}H^\ast_{\mathrm{log-cris}}(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,
GK-action, filtration after extending scalars to
K, and monodromy operator
N. This conjecture was proved in the late nineties by Takeshi Tsuji. ==Notes==