A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic

Urs Hartl ^{∗}

July 7, 2006

Abstract

In this survey we explain the main ingredients and results of the analogue of Fontaine-Theory in equal positive characteristic which was recently developed by Genestier-Lafforgue and the author. Mathematics Subject Classification (2000): 11S20, (11G09, 14L05)

Introduction

The aim of Fontaine-Theory is to classify p-adic Galois representations of p-adic fields and to attach various invariants to them. To achieve this, Fontaine constructed for a p-adic field K several ℚ_{p}-algebras with G_{K} := Gal(K^{alg}/K)-action and additional structure. Then for a representation G_{K} → GL(V) in a finite dimensional ℚ_{p}-vector space V and any such ℚ_{p}-algebra B the inherited additional structure on (B ⊗_{ℚ_{p}} V)^{G_{K}} provides these invariants. This approach was extremely successful. In its application to geometry it allowed to recover all the cohomological invariants attached to a smooth proper variety X over K solely from the étale cohomology H_{ét}^{•}(X ×_{K} K^{alg}, ℚ_{p}) of X (which is a G_{K}-representation). After contributions by Grothendieck, Tate, Fontaine, Lafaille, Messing, an many others the later was finally accomplished by Faltings [16, 17] and Tsuji [43]. See [28] for a survey.

Inspired by the close parallel between number fields and function fields, an analogue for Fontaine-Theory in equal characteristic was developed by Genestier-Lafforgue [19] and the author [24]. There the local field ℚ_{p} is replaced by the Laurent series field 𝔽_{q}((z)) over the finite field 𝔽_{q}. But the theory of Galois representations G_{L} := Gal(L^{sep}/L) → GL_{n}(𝔽_{q}((z))) for finite extensions L of 𝔽_{q}((z)) is spoiled by the following two facts. Firstly by the Ax-Sen-Tate Theorem the fixed field of G_{L} inside the completion C of an algebraic closure of L is much larger than L; see 1.4 below. And secondly all the Tate twists C(n) are isomorphic as G_{L}-modules as was observed by Anderson [2]; see 1.5. Therefore it was proposed in [19, 24] to discard G_{L}-representations and to view in equal characteristic so called local shtuka as the appropriate analogue of (crystalline) Galois representations; see 5.4.

This is consistent with the following facts. The link between crystalline Galois representations and other structures like filtered isocrystals first appeared via the mediator p-divisible groups. Fontaine-Theory then provided the direct link. Now the category of local shtuka over a formal 𝔽_{q}[[z]]-scheme S is anti-equivalent to the category of divisible local Anderson modules over S, see 3.1, and the analogues of a special class of these modules in mixed characteristic are p-divisible groups. Also with a divisible local Anderson module or a local shtuka over O_{L} one can associate a Galois representation G_{L} → GL_{n}(𝔽_{q}((z))) like one associates with a p-divisible group over O_{K} the crystalline Galois representation on its rational Tate module; see 3.3. So a priory it seems natural to use local shtuka as a mediator between Galois representations G_{L} → GL_{n}(𝔽_{q}((z))) and other structures. But then it turns out that the notion of local shtuka is so general (in contrast to the notion of p-divisible group) that one can dispense at all with Galois representations in mixed characteristic and only work with local shtuka. Striking evidence for this statement is given in 5.6 and 6.2.

∗The author acknowledges support of the Deutsche Forschungsgemeinschaft in form of DFG-grant HA3006/2-1

This article contains no new results. It is rather a survey of the analogue of Fontaine-Theory in equal characteristic [19, 24]. It has various purposes. First of all, a dictionary between the arithmetic of number fields and function fields was begun by Goss [21]. This article is meant to be a sequel to Goss’ dictionary. Secondly for those readers familiar with Fontaine-Theory it should serve as an introduction to the equal characteristic theory which is somewhat simpler but not less fascinating. Thirdly we reveal a hidden geometric interpretation of the rings in Fontaine-Theory as functions on the “unit disc with coordinate p”. This interpretation is inspired by the equal characteristic counterparts of these rings. And finally we hope that this interpretation might serve as an Ariadne’s thread for those who want to learn Fontaine-Theory and experience all its rings as a kind of maze.

As with most dictionaries the reader will hardly want to read it from first to last page. We suggest to look at Section 2, which contains the definitions of the various rings of Fontaine-Theory and their analogues, only when these rings are needed. Instead the reader should focus on the remaining sections in which we explain the analogue of Fontaine-Theory in equal characteristic.

Mixed Characteristic

Equal Characteristic

1 The Arithmetic Ground Field

ℚ_{p} the p-adic numbers and ℤ_{p} the p-adic integers 𝔽_{q}((z)) the Laurent series field and 𝔽_{q}[[z]] the power series ring in the variable z over the finite field 𝔽_{q} with q elements

These rings carry the p-adic topology.

These rings are equipped with the z-adic topology.

1.1 Witt Vectors

The functor R ↦ W (R) which assigns to a perfect 𝔽_{p}-algebra R the ring of p-typical Witt vectors over R, see [22, 41], corresponds to the functor R ↦ R[[z]] which assigns to an 𝔽_{q}-algebra R the power series ring over R in the variable z.

The Frobenius lift φ = W (Frob_{p}) ∈ End(W (R)) corresponds to σ : R[[z]] → R[[z]] , ∑_{i} b_{i}z^{i} ↦ ∑_{i} b_{i}^{q}z^{i} (where b_{i} ∈ R), also called the Frobenius lift.

For an element b ∈ R let [b] ∈ W (R) be the Teichmüller representative of b, see [22, 41]. The map b ↦ [b] is multiplicative but not compatible with addition.

For an element b ∈ R the element b · z^{0} ∈ R[[z]] is called the Teichmüller representative of b. The map b ↦ b · z^{0} is a ring homomorphism.

If Spec R is connected W (R)^{φ=1} = ℤ_{p}.

If Spec R is connected R[[z]]^{σ=1} = 𝔽_{q}[[z]].

1.2 The Two Roles of p

The number p enters in Fontaine-Theory in a twofold way

• as uniformizing parameter of ℤ_{p} and

• as element of the base rings or fields (which are ℤ_{p}-algebras) over which the arithmetic objects like p-divisible groups or Galois representations are defined.

The necessity to separate the two roles of p in equal characteristic and to work with two indeterminants z and ζ was first pointed out by Anderson [1]. So we let

• z be the uniformizing parameter of 𝔽_{q}[[z]],

• ζ be the element of the base rings (which are 𝔽_{q}[[z]]-algebras, ζ is the image of z).

Note that the natural number p never can act on a module or vector space as anything else than the scalar p, whereas there is no such restriction on z. This makes the distinction between z and ζ possible. Strictly speaking a distinction between the two roles of p is also searched for in Fontaine-Theory where an object, called [p̃] by Colmez [12], is constructed that behaves like ζ, whereas p behaves like z; see 2.1 and 2.9.

1.3 The Cyclotomic Character

For n ∈ ℕ_{0} let ε^{(n)} ∈ ℚ_{p}^{alg} be a primitive p^{n}-th root of unity with (ε^{(n+1)})^{p} = ε^{(n)}. Let ℚ_{p,∞} := ℚ_{p}(ε^{(n)} : n ∈ ℕ). Then there is an isomorphism of topological groups

χ : Gal(ℚ_{p,∞}/ℚ_{p}) ⟶^{∼} ℤp^{×}

with γ(ε^{(n)}) = (ε^{(n)})^{χ(γ) mod p^{n}} for γ in the Galois group. χ is called the cyclotomic character.

For n ∈ ℕ_{0} let t_{n} ∈ 𝔽_{q}((ζ))^{sep} be solutions of the equations t_{0}^{q−1} = −ζ and t_{n}^{q} + ζt_{n} = t_{n−1}. Let 𝔽_{q}((ζ))_{∞} := 𝔽_{q}((ζ))(t_{n} : n ∈ ℕ_{0}) and define t_{+} := ∑_{n=0}^{∞} t_{n}z^{n} ∈ 𝔽_{q}((ζ))_{∞}[[z]]. Then there is an isomorphism of topological groups

χ : Gal(𝔽_{q}((ζ))_{∞}/𝔽_{q}((ζ))) ⟶^{∼} 𝔽_{q}[[z]]^{×}

satisfying γ(t_{+}) := ∑_{n=0}^{∞} γ(t_{n})z^{n} = χ(γ) · t_{+} in 𝔽_{q}((ζ))_{∞}[[z]] for γ in the Galois group. In view of 3.4 below, χ is called the cyclotomic character. (The existence of χ follows from the fact that by construction σ(t_{+}) = (z − ζ) · t_{+}. Hence χ(γ) := \frac{γ(t_{+})}{t_{+}} is σ-invariant, that is χ(γ) ∈ 𝔽_{q}[[z]]^{×}. Furthermore, χ is an isomorphism because t_{n−1} is a uniformizing parameter of 𝔽_{q}((ζ))(t_{0}, …, t_{n−1}) and so the equations defining the t_{n} are irreducible by Eisenstein.)

1.4 The Ax-Sen-Tate Theorem

Let K be a field which is complete with respect to a valuation of rank one and let C be the completion of an algebraic closure of K. By continuity the absolute Galois group G_{K} := Gal(K^{sep}/K) of K acts on C. The Ax-Sen-Tate Theorem [5] states that the fixed field in C of this action equals the completion of the perfection of K. If K is an extension of ℚ_{p} the fixed field is C^{G_{K}} = K.

If K is an extension of 𝔽_{q}((ζ)) the fixed field is much larger than K unless K is already perfect.

1.5 The Tate Twists

Let K_{∞} := K(ε^{(n)} : n ∈ ℕ) in the notation of 1.3 and 1.4. Then Gal(K_{∞}/K) is a quotient of G_{K} and a subgroup of Gal(ℚ_{p,∞}/ℚ_{p}).

Let K_{∞} := K(t_{n} : n ∈ ℕ_{0}) in the notation of 1.3 and 1.4. Then Gal(K_{∞}/K) is a quotient of G_{K} and a subgroup of Gal(𝔽_{q}((ζ))_{∞}/𝔽_{q}((ζ))).

For m ∈ ℤ let C(m) be the field C on which γ ∈ G_{K} acts via

x ↦ χ(γ)^{m} · γ(x) for x ∈ C .

The C(m) are called the Tate twists of C.

For m ∈ ℤ let C(m) be the field C on which γ ∈ G_{K} acts via

x ↦ (χ(γ)|_{z=ζ})^{m} · γ(x) for x ∈ C .

The C(m) are called the Tate twists of C.

If K is discretely valued the Tate twists are mutually non-isomorphic by [42, Theorem 2, p. 176].

Anderson [2] observed that all Tate twists are isomorphic! Namely, multiplication with (t_{+}|_{z=ζ})^{m} = (∑_{n≥0} t_{n}ζ^{n})^{m} ∈ C defines an isomorphism C(m) ⟶^{∼} C(0) of G_{K}-modules.

2 Fontaine’s Rings

We describe most of the rings encountered in Fontaine Theory and their equal characteristic analogues. Many of these analogues are Laurent series rings in the variable z over complete extensions of 𝔽_{q}((ζ)) with varying convergence conditions. Therefore we view them as rings of (rigid) analytic functions on suitable subsets of the unit disc with coordinate z. We want to advertise the point of view that the corresponding rings from Fontaine Theory have a geometric interpretation as rings of analytic functions in the “variable p” on certain subsets of the “unit disc with coordinate p”; see 2.3 – 2.9. We use the notation of Colmez [12] where proofs and further references can be found. Let K be a complete discretely valued field extension of ℚ_{p} with perfect residue field k and let C be the completion of an algebraic closure of K. With the notation of 1.3 let K_{n} := K(ε^{(n)}) and K_{∞} := K(ε^{(n)} : n ∈ ℕ_{0}).

We use the notation from [24]. Let L be a field extension of 𝔽_{q}((ζ)) which is complete with respect to an absolute value | . | : L → ℝ_{≥0} extending the absolute value on 𝔽_{q}((ζ)). There is no assumption on the residue field ℓ or on the value group of L. Let C be the completion of an algebraic closure of L and let R be an affinoid L-algebra with L-Banach norm | . |; see [6, 7]. For instance the absolute case R = L is allowed.

2.1 The Field of Norms

One defines the following rings

and their equal characteristic analogues.

Ẽ^{+} := { x = (x^{(n)})_{n∈ℕ_{0}} : x^{(n)} ∈ O_{C},

(x^{(n+1)})^{p} = x^{(n)} }

Ẽ := Frac(Ẽ^{+}) is an algebraically closed field of characteristic p complete with respect to the valuation v_{E} : Ẽ → ℝ ∪ {∞} , v_{E}(x) := v_{p}(x^{(0)})

O_{C} ≅ { x = (x^{(n)})_{n∈ℕ_{0}} : x^{(n)} ∈ O_{C},

(x^{(n+1)})^{q} = x^{(n)} }

C = Frac(O_{C}) v_{C} : C → ℝ ∪ {∞} the valuation with v_{C}(ζ) = 1. Since we want to extend the theory to arbitrary affinoid L-algebras R we prefer to work with the associated absolute value | . | : C → ℝ_{≥0} , |x| = |ζ|^{v_{C}(x)} on C.

The absolute Galois group G_{K} of K acts component-wise on Ẽ^{+} and hence on Ẽ.

The absolute Galois group G_{L} of L acts on C but this action is not well behaved; see 1.4, 1.5, and Section 5.

E_{K}^{+} := { x ∈ Ẽ^{+} : for n ≫ 0 there exists

x̂_{n} ∈ O_{K_{n}} with v_{E}(x^{(n)} − x̂_{n}) ≥ 1 }

E_{K} := Frac(E_{K}^{+}) is called the Field of Norms of K. It is a complete discretely valued field of characteristic p with perfect residue field.

Since the analogue of Ẽ is C we may take as the analogue of E_{K} the field L in the absolute case or even the ring R in the relative case.

The Theory of the field of norms states that

H_{K} := Gal(K^{sep}/K_{∞}) = Gal(E_{K}^{sep}/E_{K}) .

The reader should note that there is also a relative version of this theory by Andreatta and Brinon [3, 9, 4] for certain affinoid K-algebras.

In equal characteristic L (or even R) appears as the analogue of both E_{K} and K_{∞}. So the Theory of the field of norms is trivial in the absolute case and in the relative case for arbitrary affinoid Lalgebras. The analogue of H_{K} is H_{L} := G_{L}.

The Theory of the field of norms is fundamental to Fontaine’s theory of p-adic Galois representations of K. It allows to break up a representation of G_{K} into a representation of H_{K} which corresponds to a φ-module over B_{K}, see 5.1, on which the remaining piece Γ_{K} := G_{K}/H_{K} ≅ Gal(K_{∞}/K) of G_{K} acts. On the other hand this splitting up along K_{∞} is unavoidable since it is impossible to identify the whole Galois group G_{K} with the absolute Galois group of an appropriate field of characteristic p.

Strictly speaking L_{∞} = L(t_{n} : n ∈ ℕ_{0}) is the proper analogue of both E_{K} and K_{∞} and the Theory of the field of norms is trivial for L_{∞}. But since this is even true for L there is no need to split up Galois representations along L_{∞}. Moreover, the notion of z-adic Galois representation of L is not well behaved; see Section 5. Therefore it is preferable to work with different structures (as local shtuka, see 3.2) from the start. And for those structures again it is possible and better to define them over L instead of L_{∞}. For these reasons we view L as the analogue of E_{K}.

Ẽ_{K} the completion of the perfection of E_{K} also equals the fixed field of H_{K} in Ẽ.

L^{p} := { \atop L^{perf}} the completion of the perfection of L also equals the fixed field of G_{L} in C.

Ẽ_{K}^{+} the ring of integers of Ẽ_{K}.

O_{L^{p}} the ring of integers of L^{p}.

Fix an element p̃ ∈ Ẽ^{+} with p̃^{(0)} = p. Let ε := (ε^{(n)})_{n} ∈ E_{ℚ_{p}}^{+} be the sequence defined in 1.3 and set π̄ := ε − 1.

The analogue of p̃ is ζ ∈ L.

2.2 The Witt and Power Series Rings

Consider the rings

Their equal characteristic analogues are

Ã^{+} := W (Ẽ^{+}) the ring of Witt vectors,

O_{C}[[z]] the ring of power series in z,

B̃^{+} := Ã^{+}[\frac{1}{p}],  O_{C}[[z]][\frac{1}{z}],
Ã := W (Ẽ),  C[[z]],

B̃ := Ã[\frac{1}{p}] the fraction field of Ã.

C((z)) = C[[z]][\frac{1}{z}] the fraction field of C[[z]].

For example π := [ε] − 1 ∈ Ã^{+} where [ .] denotes the Teichmüller representative, see 1.1.

There are functions f_{i} : B̃ → Ẽ defined by the

equation x = ∑_{i≫−∞}^{∞} p^{i}[f_{i}(x)] in B̃.

There are functions f_{i} : C((z)) → C defined by

the equation x = ∑_{i≫−∞}^{∞} f_{i}(x) z^{i} in C((z)).

Ã_{K}^{+} := W (Ẽ_{K}^{+}) the ring of Witt vectors,

O_{L^{p}}[[z]] the ring of power series in z,

B̃_{K}^{+} := Ã_{K}^{+}[\frac{1}{p}],  O_{L^{p}}[[z]][\frac{1}{z}],
Ã_{K} := W (Ẽ_{K}),  L^{p}[[z]],

B̃_{K} := Ã_{K}[\frac{1}{p}] the fraction field of Ã_{K}.

L^{p}((z)) the fraction field of L^{p}[[z]].

By Witt vector functoriality these rings inherit a G_{K}-action commuting with the Frobenius lift φ.

By functoriality these rings inherit a G_{L}-action which is also flawed; see Section 5.

2.3 The Non-Perfect Rings

Set K_{0} := W (k)[\frac{1}{p}] and define

A_{K_{0}} := { ∑_{n∈ℤ}a_{n}π^{n} : a_{n} ∈ W (k), v_{p}(a_{n}) → ∞

for n → +∞ },

B_{K_{0}} := A_{K_{0}}[\frac{1}{p}] is naturally a subfield of B̃ with residue field E_{K_{0}}. Let B_{K} be the unique finite extension of B_{K_{0}} contained in B̃ whose residue field is E_{K} and let Its analogues are L((z)) or even R[[z]][\frac{1}{z}] in the relative situation,

A_{K} be the ring of integers of B_{K} and

respectivelyL[[z]] or R[[z]].

B_{K}^{+} := B̃^{+} ∩ B_{K}, O_{L}[[z]][\frac{1}{z}] or R^{∘}[[z]][\frac{1}{z}] where R^{∘} is an admissible formal O_{L}-algebra in the sense of Raynaud [8] with R^{∘} ⊗_{O_{L}} L ≅ R. When considering R^{∘} we always assume that the L-Banach norm | . | on R satisfies R^{∘} = {b ∈ R : |b| ≤ 1}.

A_{K}^{+} := Ã^{+} ∩ A_{K}.

O_{L}[[z]] or R^{∘}[[z]]. The elements of these rings converge on the whole (relative) open unit disc {|z| < 1} and are bounded by 1.

Let B ⊂ B̃ be the completion of the maximal unramified extension of B_{K} and

Their analogues are L^{sep}((z)) and

A := Ã ∩ B.  L^{sep}[[z]].

By continuity H_{K} = Gal(E_{K}^{sep}/E_{K}) acts on B with B^{H_{K}} = B_{K}.

By continuity G_{L} = Gal(L^{sep}/L) acts on L^{sep}((z)) with L^{sep}((z))^{G_{L}} = L((z)).

The φ-invariants of these rings are (A_{K}^{+})^{φ=1} = (A_{K})^{φ=1} = (Ã^{+})^{φ=1} = Ã^{φ=1} = ℤ_{p}, The σ-invariants of these rings are R^{∘}[[z]]^{σ=1} = R[[z]]^{σ=1} = 𝔽_{q}[[z]] if Spec R is connected and

(B_{K}^{+})^{φ=1} = (B_{K})^{φ=1} = (B̃^{+})^{φ=1} = B̃^{φ=1} = ℚ_{p}.

R^{∘}[[z]][\frac{1}{z}]^{σ=1} = R[[z]][\frac{1}{z}]^{σ=1} = 𝔽_{q}((z)) if Spec R is connected (in particular for R = L or L^{sep} or C).

2.4 The Overconvergent Rings

Let r̄ ∈ ℝ_{>0}, set r = 1/r̄ and define

Let r ∈ ℝ_{>0}, set r̄ = 1/r and define

Ã^{(0,r̄]} := { x ∈ Ã : lim_{i→+∞}v_{E}(f_{i}(x)) + i/r̄ = ∞ }, C⟨\frac{z}{ζ^{r}} ⟩ := \{ ∑_{i=0}^{∞} b_{i}z^{i} ∈ C[[z]] : lim_{i→+∞}|b_{i}| |ζ|^{ri} = 0 \}.

Its elements converge on the disc {|z| ≤ |ζ|^{r}}.

B̃^{(0,r̄]} := Ã^{(0,r̄]}[\frac{1}{p}], and  In C⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] we allow a pole at z = 0.

their K-rational versions A_{K}^{(0,r̄]} := A_{K} ∩ Ã^{(0,r̄]} and

B_{K}^{(0,r̄]} := B_{K} ∩ B̃^{(0,r̄]} = A_{K}^{(0,r̄]}[\frac{1}{p}].

The L- (or R)-rational versions are L⟨\frac{z}{ζ^{r}} ⟩ or even R⟨\frac{z}{ζ^{r}} ⟩, and

L⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] or even R⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}], which are defined as the preceding rings except that one must take | . | to be the L-Banach norm on R fixed in the beginning. (Since the topology of R does not depend on the choice of the norm also R⟨\frac{z}{ζ^{r}} ⟩ and R⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] do not depend on this choice.)

The rings Ã^{(0,r̄]}, A_{K}^{(0,r̄]}, B̃^{(0,r̄]} and B_{K}^{(0,r̄]} are principal ideal domains.

The rings C⟨\frac{z}{ζ^{r}} ⟩, L⟨\frac{z}{ζ^{r}} ⟩, C⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] and L⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] are principal ideal domains.

There is a natural valuation v^{(0,r̄]}(x) := inf{ v_{E}(f_{i}(x)) + i/r̄: i ∈ ℕ_{0} }

on B̃^{(0,r̄]} with respect to which Ã^{(0,r̄]} and A_{K}^{(0,r̄]} are complete. We also consider There is a natural norm ∥ ∑_{i∈ℕ_{0}} b_{i}z^{i}∥_{r} := sup{ |b_{i}| |ζ|^{ri} : i ∈ ℕ_{0} } on R⟨\frac{z}{ζ^{r}} ⟩ with respect to which it is complete. We also consider

Ã^{†} := ⋃_{r̄→0} Ã^{(0,r̄]},  ⋃_{r→∞} C⟨\frac{z}{ζ^{r}} ⟩,

B̃^{†} := ⋃_{r̄→0} B̃^{(0,r̄]} the subfield of B̃ of overconver-

gent elements, ⋃_{r→∞} C⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] the subfield of C((z)) of overconver-

gent elements which converge on some disc with small enough radius and have a pole at z = 0,

A_{K}^{†} := ⋃_{r̄→0} A_{K}^{(0,r̄]}, and  ⋃_{r→∞} R⟨\frac{z}{ζ^{r}} ⟩, and

B_{K}^{†} := ⋃_{r̄→0} B_{K}^{(0,r̄]} the subfield of B_{K} of overcon-

vergent elements,

⋃_{r→∞} R⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}].

B^{†} := B̃^{†} ∩ B.

φ : Ã^{(0,r̄]} ⟶^{∼} Ã^{(0,r̄/p]} is a bicontinuous isomorphism σ : C⟨\frac{z}{ζ^{r}} ⟩ ⟶^{∼} C⟨\frac{z}{ζ^{qr}} ⟩ is a bicontinuous isomorphism

φ : A_{K}^{(0,r̄]} ⟶^{∼} A_{K}^{(0,r̄/p]} is a continuous homomorphism σ : R⟨\frac{z}{ζ^{r}} ⟩ ⟶^{∼} R⟨\frac{z}{ζ^{qr}} ⟩ is a continuous homomorphism

Note the inclusions A_{K}^{+} ⊂ A_{K}^{(0,r̄]} ⊂ A_{K} and the corresponding inclusions for the B’s or for the “tilde versions”.

Note the inclusions R^{∘}[[z]] ⊂ R⟨\frac{z}{ζ^{r}} ⟩ ⊂ R[[z]] and the corresponding inclusions after adjoining \frac{1}{z} or replacing R by C.

There are natural inclusions A_{K}^{+} ⊂ A_{K}^{(0,r̄]} ⊂ A_{K}^{†} ⊂ A_{K} identifying A_{K} with the p-adic completion of A_{K}^{(0,r̄]} and A_{K}^{†}. The same holds for the B’s and the “tilde versions”.

There are natural inclusions R^{∘}[[z]] ⊂ R⟨\frac{z}{ζ^{r}} ⟩ ⊂ R[[z]] identifying R[[z]] with the z-adic completion of R⟨\frac{z}{ζ^{r}} ⟩. The same holds after adjoining \frac{1}{z} or replacing R by C.

2.5 The Rings B̃_{max} and B̃_{rig}

Let r̄ ∈ ℝ_{>0} and set r = 1/r̄.

Let r ∈ ℝ_{>0} and set r̄ = 1/r.

Consider the semi-valuation v^{(0,r̄]} on B̃^{+}.

Consider the norm ∥ . ∥_{r} on O_{C}[[z]][\frac{1}{z}].

Let Ã_{max}^{+} be the p-adic completion of Ã^{+}[\frac{[p̃]}{p}] or equivalently its completion with respect to v^{(0,1]}. (This ring is usually denoted A_{max}.)

Let O_{C}[[z, \frac{ζ}{z} ⟩ be the ζ-adic completion of O_{C}[[z]][\frac{ζ}{z}] or equivalently the completion with respect to ∥ . ∥_{1}. Its elements converge on the half open annulus {|ζ| ≤ |z| < 1} and are bounded by 1.

Set B̃_{max}^{+} := Ã_{max}^{+}[\frac{1}{p}]. It equals the completion of B̃^{+} with respect to v^{(0,1]} (and is usually denoted B_{max}^{+}.)

Then O_{C}[[z, \frac{ζ}{z} ⟩[\frac{1}{z}] equals the completion

{ ∑_{i=−∞}^{∞} b_{i}z^{i} : b_{i} ∈ O_{C}, lim_{i→−∞} |b_{i}| |ζ|^{i} = 0 }

of O_{C}[[z]][\frac{1}{z}] with respect to ∥ . ∥_{1}. Its elements converge on the half open annulus {|ζ| ≤ |z| < 1} and are bounded by 1 as |z| → 1.

Set B̃_{rig}^{+} := ⋂_{n∈ℕ_{0}} φ^{n}(B̃_{max}^{+}). Set O_{C}[[z, \frac{1}{z}\} := ⋂_{n∈ℕ_{0}} σ^{n}(O_{C}[[z, \frac{ζ}{z} ⟩[\frac{1}{z}]) =

{ ∑_{i=−∞}^{∞} b_{i}z^{i} : b_{i} ∈ O_{C}, lim_{i→−∞} |b_{i}| |ζ|^{ri} = 0 ∀r > 0 }.

Its elements converge on the punctured open unit disc {0 < |z| < 1} and are bounded by 1 as |z| approaches 1.

B̃_{rig}^{+} also equals the Fréchet completion of B̃^{+} with respect to the family of semi-valuations v^{(0,r̄]} for 0 < r̄ ≤ 1.

O_{C}[[z, \frac{1}{z}\} also equals the Fréchet completion of O_{C}[[z]][\frac{1}{z}] with respect to the family of norms ∥ . ∥_{r} for 1 ≤ r.

The φ-invariants are

The σ-invariants are

(B̃_{max}^{+})^{φ=1} = (B̃_{rig}^{+})^{φ=1} = ℚ_{p}. O_{C}[[z, \frac{ζ}{z} ⟩[\frac{1}{z}]^{σ=1} = O_{C}[[z, \frac{1}{z}\}^{σ=1} = 𝔽_{q}((z)).

The Galois invariants are (B̃_{rig}^{+})^{G_{K}} = K_{0} := W (k)[\frac{1}{p}].

In contrast, the Galois invariants are O_{C}[[z, \frac{1}{z}\}^{G_{L}} = O_{L^{p}}[[z, \frac{1}{z}\} ≠ ℓ((z)) making the notion of crystalline Galois representation problematic, see 5.3.

2.6 The Rings B_{K,max}^{+} and B_{K,rig}^{+}

Let R^{∘} be an admissible formal O_{L}-algebra with R^{∘} ⊗_{O_{L}} L ≅ R as in 2.3.

Let B_{K,max}^{+} be the completion of B_{K}^{+} with respect to the semi-valuation v^{(0,1]}.

Let R^{∘}[[z, \frac{ζ}{z} ⟩[\frac{1}{z}] be the completion of R^{∘}[[z]][\frac{1}{z}] with respect to the norm ∥ . ∥_{1}. It equals

{ ∑_{i=−∞}^{∞} b_{i}z^{i} : b_{i} ∈ R^{∘}, lim_{i→−∞} |b_{i}| |ζ|^{i} = 0 }. Its ele-

ments converge on the relative half open annulus {|ζ| ≤ |z| < 1} over Sp R and are bounded by 1 as |z| → 1.

A_{K,max}^{+} := \{ x ∈ B_{K,max}^{+} : v^{(0,1]}(x) ≥ 0 \}. R^{∘}[[z, \frac{ζ}{z} ⟩ := \{ x ∈ R^{∘}[[z, \frac{ζ}{z} ⟩[\frac{1}{z}] : ∥x∥_{1} ≤ 1 \} equals the ζ-adic completion of R^{∘}[[z]][\frac{ζ}{z}] or equivalently the completion with respect to ∥ . ∥_{1}.

Let B_{K,rig}^{+} be the Fréchet completion of B_{K}^{+} with respect to the family of semi-valuations v^{(0,r̄]} for 0 < r̄ ≤ 1 Let R^{∘}[[z, \frac{1}{z}\} be the Fréchet completion of R^{∘}[[z]][\frac{1}{z}] with respect to the family of norms ∥ . ∥_{r} for 1 ≤ r. It equals

{ ∑_{i=−∞}^{∞} b_{i}z^{i} : b_{i} ∈ R^{∘}, lim_{i→−∞} |b_{i}| |ζ|^{ri} = 0 ∀r > 0 }.

Its elements converge on the relative punctured open unit disc {0 < |z| < 1} over Sp R and are bounded by 1 as |z| → 1.

The φ-invariants are

If Spec R^{∘} is connected the σ-invariants are

(B_{K,max}^{+})^{φ=1} = (B_{K,rig}^{+})^{φ=1} = ℚ_{p}. R^{∘}[[z, \frac{ζ}{z} ⟩[\frac{1}{z}]^{σ=1} = R^{∘}[[z, \frac{1}{z}\}^{σ=1} = 𝔽_{q}((z)).

2.7 The Analogues of 2πi

The series t := log[ε] := ∑_{n∈ℕ} \frac{(−1)^{n−1}}{n} π^{n} converges

in B_{ℚ_{p},rig}^{+} and appears as the p-adic analogue of 2πi. It satisfies The product t_{−} := ∏_{n∈ℕ_{0}}(1 − \frac{ζ^{q^{n}}}{z} ) converges in

𝔽_{q}[[ζ]][[z, \frac{1}{z}\}. Using the notation of 1.3 we let 𝔽_{q}[[ζ]]_{∞} := 𝔽_{q}[[ζ]][t_{n} : n ∈ ℕ_{0}] be the valuation ring of 𝔽_{q}((ζ))_{∞}. Then we define

t := t_{+} · t_{−} ∈ 𝔽_{q}[[ζ]]_{∞}[[z, \frac{1}{z}\}. It satisfies

φ(t) = p t and γ(t) = χ(γ) · t for all γ ∈ Gal(ℚ_{p}^{alg}/ℚ_{p}) where χ is the cyclotomic character from 1.3. σ(t) = z t and γ(t) = χ(γ) · t for all γ ∈ Gal(𝔽_{q}((ζ))^{sep}/𝔽_{q}((ζ))) where χ is the cyclotomic character from 1.3.

Set B̃_{max} := B̃_{max}^{+}[\frac{1}{t}] , B̃_{rig} := B̃_{rig}^{+}[\frac{1}{t}] ,

B_{K,max} := B_{K,max}^{+}[\frac{1}{t}] , and B_{K,rig} := B_{K,rig}^{+}[\frac{1}{t}] For the convergence behavior of the analogues of these rings note that t has simple zeroes precisely at z = ζ^{q^{i}} for i ∈ ℤ.

2.8 The Robba Ring

Let r̄, s̄ ∈ ℝ_{>0} with s̄ ≤ r̄ and r = 1/r̄, s = 1/s̄. Let r, s ∈ ℝ_{>0} with s ≥ r and r̄ = 1/r, s̄ = 1/s.

Consider the semi-valuation on B̃^{(0,r]} v^{[s̄,r̄]}(x) := inf{ v^{(0,s̄]}(x), v^{(0,r̄]}(x) } and let Consider the norm on C⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] or on R⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] ∥x∥_{[r,s]} := sup{ ∥x∥_{r}, ∥x∥_{s} } and let

B̃^{[s̄,r̄]} (respectively B_{K}^{[s̄,r̄]}) be the completion of B̃^{(0,r]} (respectively B_{K}^{(0,r]}) with respect to v^{[s̄,r̄]}.

C⟨\frac{z}{ζ^{r}} , \frac{ζ^{s}}{z} ⟩ (respectively R⟨\frac{z}{ζ^{r}} , \frac{ζ^{s}}{z} ⟩) be the completion of C⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] (respectively R⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}]) with respect to ∥ . ∥_{[r,s]}. It equals

{ ∑_{i=−∞}^{∞} b_{i}z^{i} : lim_{i→∞} |b_{i}| |ζ|^{ri} = lim_{i→−∞} |b_{i}| |ζ|^{si} = 0 }

the ring of Laurent series with coefficients b_{i} ∈ C (respectively b_{i} ∈ R), which converge on the (relative) annulus {|ζ|^{s} ≤ |z| ≤ |ζ|^{r}}.

Let B̃^{]0,r̄]} (respectively B_{K}^{]0,r̄]}) be the Fréchet completion of B̃^{(0,r]} (respectively B_{K}^{(0,r]}) with respect to the family of semi-valuations v^{[s̄,r̄]} for all 0 < s̄ ≤ r̄. Then Let C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} (respectively R⟨\frac{z}{ζ^{r}} , \frac{1}{z}\}) be the Fréchet completion of C⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] (respectively R⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}]) with respect to the family of norms ∥ . ∥_{[r,s]} for all s ≥ r. Then

B̃^{]0,r̄]} = ⋂_{s̄→0} B̃^{[s̄,r̄]} and B_{K}^{]0,r̄]} = ⋂_{s̄→0} B_{K}^{[s̄,r̄]}.  C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} = ⋂_{s→∞} C⟨\frac{z}{ζ^{r}} , \frac{ζ^{s}}{z} ⟩ =

The functions f_{i} from 2.2 extend by continuity to B̃^{[s̄,r̄]} and B̃^{]0,r̄]}, and the sum ∑_{i=−∞}^{∞} p^{i}[f_{i}(x)] converges to x in B̃^{[s̄,r̄]}, respectively in B̃^{]0,r̄]}. This is not true for B_{K}^{[s̄,r̄]} and B_{K}^{]0,r̄]} since E_{K} is not perfect and so there is no Teichmüller map E_{K} → A_{K}.

{ ∑_{i=−∞}^{∞} b_{i}z^{i} : lim_{i→±∞}|b_{i}| |ζ|^{si} = 0 for all s ≥ r }

equals the ring of Laurent series with coefficients b_{i} ∈ C, which converge on the punctured disc {0 < |z| ≤ |ζ|^{r}}. This is also true if C is replaced by R.

The rings B̃^{[s̄,r̄]} and B_{K}^{[s̄,r̄]} are principal ideal domains, the rings B̃^{]0,r̄]} and B_{K}^{]0,r̄]} are Bezout domains (that is, every finitely generated ideal is principal).

The rings C⟨\frac{z}{ζ^{r}} , \frac{ζ^{s}}{z} ⟩ and L⟨\frac{z}{ζ^{r}} , \frac{ζ^{s}}{z} ⟩ are principal ideal domains, the rings C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} and L⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} are Bezout domains.

For [s̄′, r̄′] ⊂ [s̄, r̄] one has inclusions with dense image B̃^{(0,r̄]} ⊂ B̃^{]0,r̄]} ⊂ B̃^{[s̄,r̄]} ⊂ B̃^{[s̄′,r̄′]} and the same for the B_{K}-versions.

For [r′, s′] ⊂ [r, s] one has the dense inclusions C⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] ⊂ C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} ⊂ C⟨\frac{z}{ζ^{r}} , \frac{ζ^{s}}{z} ⟩ ⊂ C⟨\frac{z}{ζ^{r′}} , \frac{ζ^{s′}}{z} ⟩ and the same if C is replaced by L or R.

Set B̃_{rig}^{†} := ⋃_{r̄→0} B̃^{]0,r̄]} and

B_{K,rig}^{†} := ⋃_{r̄→0} B_{K}^{]0,r̄]}. The later ring is called the

Robba ring associated with K.

⋃_{r→∞} C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} (respectively ⋃_{r→∞} R⟨\frac{z}{ζ^{r}} , \frac{1}{z}\}) is the

ring of Laurent series which converge on some (relative) punctured disc {0 < |z| ≤ |ζ|^{r}} with small enough radius |ζ|^{r}.

φ induces isomorphisms of topological rings

φ : B̃^{[s̄,r̄]} ⟶^{∼} B̃^{[s̄/p , r̄/p]},

σ induces isomorphisms of topological rings

σ : C⟨\frac{z}{ζ^{r}} , \frac{ζ^{s}}{z} ⟩ ⟶^{∼} C⟨\frac{z}{ζ^{qr}} , \frac{ζ^{qs}}{z} ⟩,

φ : B̃^{]0,r̄]} ⟶^{∼} B̃^{]0 , r̄/p]},  σ : C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} ⟶^{∼} C⟨\frac{z}{ζ^{qr}} , \frac{1}{z}\},

φ : B̃_{rig}^{†} ⟶^{∼} B̃_{rig}^{†}, σ : ⋃_{r→∞} C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} ⟶^{∼} ⋃_{r→∞} C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\},

and homomorphisms of topological rings

φ : B_{K}^{[s̄,r̄]} ⟶ B_{K}^{[s̄/p , r̄/p]}, and homomorphisms of topological rings

σ : R⟨\frac{z}{ζ^{r}} , \frac{ζ^{s}}{z} ⟩ ⟶ R⟨\frac{z}{ζ^{qr}} , \frac{ζ^{qs}}{z} ⟩,

φ : B_{K}^{]0,r̄]} ⟶ B_{K}^{]0 , r̄/p]},  σ : R⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} ⟶ R⟨\frac{z}{ζ^{qr}} , \frac{1}{z}\},

φ : B_{K,rig}^{†} ⟶ B_{K,rig}^{†}. σ : ⋃_{r→∞} R⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} ⟶ ⋃_{r→∞} R⟨\frac{z}{ζ^{r}} , \frac{1}{z}\}.

There are exact sequences of rings

0 → B̃^{+} → B̃_{rig}^{+} ⊕ B̃^{(0,r̄]} → B̃^{]0,r̄]} → 0,
0 → B̃^{+} → B̃_{max}^{+} ⊕ B̃^{(0,1]} → B̃^{[1,1]} → 0,

and the same for the B_{K}-versions.

There are exact sequences of rings

\begin{matrix}O_{C}[[z]][\frac{1}{z}] ↪ & O_{C}[[z, \frac{1}{z}\} ⊕ C⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] → → C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\}, \\ O_{C}[[z]][\frac{1}{z}] ↪ & O_{C}[[z, \frac{ζ}{z} ⟩[\frac{1}{z}] ⊕ C⟨\frac{z}{ζ} ⟩[\frac{1}{z}] → → C⟨\frac{z}{ζ} , \frac{ζ}{z} ⟩, \end{matrix}

and the same if O_{C} and C are replaced by R^{∘} and R.

2.9 The Field B_{dR}

The surjective homomorphism

θ : B̃^{+} → C , x ↦ ∑_{i∈ℤ}p^{i}f_{i}(x)^{(0)} extends by

continuity to θ : B̃^{[1,1]} → C. It has

Let θ : C⟨\frac{z}{ζ} , \frac{ζ}{z} ⟩ → C , ∑_{i=−∞}^{∞} b_{i}z^{i} ↦ ∑_{i=−∞}^{∞} b_{i}ζ^{i}.

It is a surjective homomorphism with

ker θ = (p − [p̃])B̃^{[1,1]} = t B̃^{[1,1]}. However, note that the ideal t B̃_{rig}^{+} of B̃_{rig}^{+} is not maximal as opposed to t B̃^{[1,1]} ⊂ B̃^{[1,1]}. ker θ = (z − ζ)C⟨\frac{z}{ζ} , \frac{ζ}{z} ⟩ = t C⟨\frac{z}{ζ} , \frac{ζ}{z} ⟩. However, note that t also has other zeroes outside {|z| = |ζ|}.

Set B_{dR}^{+} := lim_{⟵_{n}} B̃^{[1,1]}/(ker θ)^{n}. It is a complete

discrete valuation ring with residue field C and uniformizing parameter p − [p̃] or t. There is no continuous section C → B_{dR}^{+}.

Let C[[z − ζ]] = lim_{⟵_{n}} C⟨\frac{z}{ζ} , \frac{ζ}{z} ⟩/(ker θ)^{n} be the power

series ring over C in the “variable” z − ζ. It is a complete discrete valuation ring with residue field C and uniformizing parameter z −ζ or t with canonical section C → C[[z−ζ]]. It is the complete local ring at the point {z = ζ} = ker θ.

B_{dR} := B_{dR}^{+}[\frac{1}{t}] is Fontaine’s p-adic period field.

C((z − ζ)) = C[[z − ζ]][\frac{1}{t}] is its analogue.

It is filtered by putting F il^{n}B_{dR} := t^{n}B_{dR}^{+}. It is filtered by putting

F il^{n}C((z − ζ)) := t^{n}C[[z − ζ]] for n ∈ ℤ.

B_{dR} is a successive extension of the Galois modules C(n) for n ∈ ℤ as follows (see 1.5)

0 → t^{n+1}B_{dR}^{+} → t^{n}B_{dR}^{+} → C(n) → 0 .

C((z − ζ)) is a successive extension of the Galois modules C(n) ≅ C for n ∈ ℤ as follows (see 1.5)

0 → t^{n+1}C[[z − ζ]] → t^{n}C[[z − ζ]] → C → 0 .

This implies (B_{dR})^{G_{K}} = K. This implies the unfavorable fact that C((z − ζ))^{G_{L}} = L^{p}((z − ζ)) ⊋ L which spoils the use of G_{L}-representations; see Section 5.

B̃_{dR}^{+} naturally contains B̃^{[1,1]} and if p^{−n} ∈ [s̄, r̄] for some n ∈ ℤ then this induces an inclusion φ^{−n} : B̃^{[s̄,r̄]} ↪ B_{dR}^{+}.

C[[z − ζ]] naturally contains C⟨\frac{z}{ζ} , \frac{ζ}{z} ⟩ and if q^{n} ∈ [r, s] for some n ∈ ℤ this induces an inclusion σ^{−n} : C⟨\frac{z}{ζ^{r}} , \frac{ζ^{s}}{z} ⟩ ↪ C[[z − ζ]].

By [11] the following sequence is exact

It corresponds to the exact sequence

0 → ℚ_{p} → (B̃_{max})^{φ=1} → B_{dR}/B_{dR}^{+} → 0. 0 → 𝔽_{q}((z)) → O_{C}[[z, \frac{ζ}{z} ⟩[\frac{1}{z}]^{σ=1} → \frac{C((z − ζ))}{C[[z − ζ]]} → 0.

Note that one can construct relative versions of B_{dR}^{+} and B_{dR} over affinoid K-algebras; see [9, 44] There are obvious R-rational versions R[[z − ζ]] and R[[z − ζ]][\frac{1}{z−ζ}] of these rings over an arbitrary affinoid L-algebra R.

3 P -DIVISIBLE GROUPS, LOCAL SHTUKA AND CRYSTALS 12

3 p-Divisible Groups, Local Shtuka and Crystals

3.1 Barsotti-Tate Groups

Let S be a scheme and h ≥ 0 be an integer. A Barsotti-Tate group or p-divisible group of height h over S is an inductive system

G = (G_{1} ⟶^{i_{1}} G_{2} ⟶^{i_{2}} G_{3} ⟶^{i_{3}} …)

where for each n ≥ 1

• G_{n} is a finite commutative group scheme over S of order p^{nh},

• the sequence of group schemes over S is exact

0 → G_{n} ⟶^{i_{n}} G_{n+1} ⟶^{p^{n}} G_{n+1} .

Obviously multiplication with p on G_{n} induces on the Lie algebra of G_{n} multiplication by the scalar p, compare 1.2.

Let S be an 𝔽_{q}[z]-scheme and denote the image of z in O_{S} by ζ. Let d, h ≥ 0 be integers. A divisible local Anderson module of height h and dimension d over S is an inductive system of finite 𝔽_{q}[[z]]- module schemes over S

G = (G_{1} ⟶^{i_{1}} G_{2} ⟶^{i_{2}} G_{3} ⟶^{i_{3}} …)

where for each n ≥ 1

• the 𝔽_{q}-module scheme G_{n} can be embedded into an 𝔽_{q}-vector group scheme over S,

• the order of G_{n} is q^{hn},

• the following sequence of 𝔽_{q}[[z]]-module schemes over S is exact

0 → G_{n} ⟶^{i_{n}} G_{n+1} ⟶^{z^{n}} G_{n+1} ,

• (z − ζ)^{d} = 0 on Lie G_{n},

• d = max_{s∈S, n≥1}{ dim_{κ(s)}(Lie G_{n} ⊗_{O_{S}} κ(s)) }.

So z does not need to act on Lie G_{n} as the scalar ζ, compare 1.2

3.2 Local Shtuka

Let k ⊃ 𝔽_{p} be a perfect field. A Dieudonné crystal over k is a pair (M, F_{M}) where M is a finite free W (k)-module and F_{M} : M → M is a φlinear endomorphism with pM ⊂ F_{M}(M).

Equivalently we can set φ^{∗}M := M ⊗_{W (k),φ}W (k) and linearize F_{M} to a homomorphism of W (k)- modules F_{M} : φ^{∗}M → M which satisfies p = 0 on coker F_{M}.

Let S be a formal scheme over 𝔽_{q}[[ζ]]. A local shtuka of rank n over S is a pair (M, F_{M}) consisting of a sheaf of O_{S}[[z]]-modules on S and an isomorphism F_{M} : σ^{∗}M[\frac{1}{z−ζ}] ⟶^{∼} M[\frac{1}{z−ζ}], where σ^{∗}M := M ⊗_{O_{S}[[z]],σ}O_{S}[[z]], such that the following conditions hold:

• locally for the Zariski topology on S, M is a free O_{S}[[z]]-module of rank n,

• there exists an integer e such that F_{M}(σ^{∗}M) ⊂ (z − ζ)^{−e}M and the quotient (z−ζ)^{−e}M/F_{M}(σ^{∗}M) is locally free and coherent as an O_{S}-module.

A local shtuka is called effective if F_{M} is actually a morphism σ^{∗}M → M.

Then the functor which assigns to a Barsotti-Tate group its contravariant Dieudonné module is an anti-equivalence between the category of Barsotti-Tate groups over k and the category of Dieudonné crystals over k, see [14, p. 71].

Then the category of divisible local Anderson modules over S is anti-equivalent to the category of effective local shtuka over S; see [26].

For a Barsotti-Tate group over k the properties of being étale or connected and the description of isogenies reflect in its Dieudonné module.

For a divisible local Anderson module the properties of being étale or having connected fibers, and the description of isogenies can be read off from its associated local shtuka.

Barsotti-Tate groups arise from abelian varieties as their subgroups of p-power torsion. They are of most interest when p equals the characteristic of the ground field.

Also divisible local Anderson modules arise from global objects like Drinfeld-modules [15] and abelian t-modules [1] as their subgroups of zpower torsion. Similarly local shtuka arise from global objects like shtuka [35], t-motives [1], or abelian sheaves [23] by completing with respect to z.

The parallel between Barsotti-Tate groups or more generally F -crystals and local shtuka is close. It ranges from the classification over algebraically closed fields (see 3.6), the behavior of Newton and Hodge polygons, like the Grothendieck-Katz Specialization Theorem, to their deformation theory; see Katz [31], Grothendieck [20], Messing [38], Hartl [26] and [23, §§6–8].

3.3 Tate Modules

Let K be a complete discretely valued field extension of ℚ_{p} with perfect residue field k.

Let L be a field extension of 𝔽_{q}((ζ)) which is complete with respect to an absolute value | . | : L → ℝ_{≥0} extending the absolute value on 𝔽_{q}((ζ)).

The Tate module of a Barsotti-Tate group G = lim_{⟶} (G_{n}, i_{n}) over O_{K} is the ℤ_{p}[G_{K}]-module

T_{p}G := lim_{⟵_{n}} (G_{n}(K^{alg}), p) .

The Tate module of a divisible local Anderson module G = lim_{⟶} (G_{n}, i_{n}) over O_{L} is the 𝔽_{q}[[z]][G_{L}]- module

T_{z}G := lim_{⟵_{n}} (G_{n}(L^{sep}), z) .

The Tate module T_{z}(M, F_{M}) of a local shtuka (M, F_{M}) over O_{L} is the 𝔽_{q}[[z]]-dual of (M, F_{M})^{F =1}(L^{sep}) which by definition equals

{ m ∈ M ⊗_{O_{L}[[z]]} L^{sep}[[z]] : F_{M}(σ^{∗}m) = m } .

There is a description of T_{p}G in terms of the Dieudonné crystal associated with G, see Fontaine [18, §V.1].

If M(G) is the local shtuka over 𝔽_{q}[[ζ]] associated with G then the 𝔽_{q}[[z]][G_{L}]-modules T_{z}G and T_{z}M(G) are canonically isomorphic; see [26].

3.4 Lubin-Tate Formal Groups

Let G be the Lubin-Tate formal group over ℤ_{p} on which p acts as x ↦ (1 + x)^{p} − 1, see [36].

Let G be the formal additive group over 𝔽_{q}[[ζ]] on which we let z act by x ↦ ζx + x^{q}. Then G is a Then G is a Barsotti-Tate group of height 1 over ℤ_{p}. It is isomorphic over ℤ_{p} to the Lubin-Tate formal group on which p acts as x ↦ px + x^{p}. divisible local Anderson module of height 1 and dimension 1. If we identify z with ζ then G is the Lubin-Tate formal group over 𝔽_{q}[[ζ]] on which ζ acts as x ↦ ζx + x^{q}. The local shtuka associated with G is M(G) = (𝔽_{q}[[ζ]][[z]], (z − ζ) · σ).

The Tate module of G is generated over ℤ_{p} by

(ε^{(n)})_{n∈ℕ} ∈ T_{p}G = lim_{⟵_{n}} (G_{n}(K^{alg}), p) ,

where ε^{(n)} was defined in 1.3. Thus G_{ℚ_{p}} acts on T_{p}G through the cyclotomic character from 1.3.

For the Tate module we obtain

T_{z}G = T_{z}M(G) = 𝔽_{q}[[z]] · t_{+} ,

where t_{+}, as defined in 1.3, is viewed as the map

M(G)^{F =1}(𝔽_{q}((ζ))^{sep}) → 𝔽_{q}[[z]] , (t_{+})^{−1} ↦ 1 .

Thus G_{𝔽_{q}((ζ))} acts on T_{z}G through the cyclotomic character from 1.3.

3.5 Crystals and Isocrystals

Let k be a perfect field containing 𝔽_{p}. An F - (iso)crystal over k is a pair (D, F_{D}) where

• D is a finite free module over W (k) (respectively over W (k)[\frac{1}{p}] ),

• F_{D} : φ^{∗}D → D is a homomorphism of W (k)-modules with p-torsion cokernel (respectively which is an isomorphism).

Every Dieudonné crystal over k is an F -crystal.

Let ℓ ⊃ 𝔽_{q} be a field which is an 𝔽_{q}[[z]]-algebra in which the image ζ of z is zero. A z-(iso)crystal over ℓ is a pair (D, F_{D}) where

• D is a finite free module over ℓ[[z]] (respectively over ℓ((z)) ),

• put σ^{∗}D := D⊗_{ℓ[[z]],σ} ℓ[[z]], then F_{D} : σ^{∗}D → D is a homomorphism of ℓ[[z]]-modules with z-torsion cokernel (respectively which is an isomorphism).

So a z-crystal is nothing but an effective local shtuka over ℓ.

3.6 The Dieudonné-Manin Classification

For integers r, d with r > 0 and (r, d) = 1 consider the F -isocrystal D_{d,r} = (W (k)[\frac{1}{p}]^{⊕r}, F_{D}) over k with

F_{D} = ( 0 p^{−d} 1 1 0 )· φ .

For integers r, d with r > 0 and (r, d) = 1 consider the z-isocrystal D_{d,r} = (ℓ((z))^{⊕r}, F_{D}) over ℓ with

F_{D} = ( 0 z^{−d} 1 1 0 )· σ .

If k is algebraically closed every F -isocrystal over k is isomorphic to a direct sum ⨁_{i} D_{d_{i},r_{i}} for uniquely determined pairs of integers (r_{i}, d_{i}) up to permutation; see Manin [37].

If ℓ is algebraically closed every z-isocrystal over ℓ is isomorphic to a direct sum ⨁_{i} D_{d_{i},r_{i}} for uniquely determined pairs of integers (r_{i}, d_{i}) up to permutation; see Laumon [34, §B.1].

3.7 Filtered Isocrystals

Let K be a complete discretely valued extension of ℚ_{p} with perfect residue field k and set K_{0} := W (k)[\frac{1}{p}].

Let L be as in 3.1. Let O_{L} be its valuation ring and let ℓ be its residue field. Assume that there is a fixed section ℓ ↪ O_{L} of the residue map O_{L} → ℓ. This induces in particular a homomorphism ℓ((z)) → L[[z − ζ]] into the power series ring over L sending z to z = ζ + (z − ζ).

A filtered isocrystal \underline{D} = (D, F_{D}, F il^{•}D_{K}) over K consists of

• (D, F_{D}) an F -isocrystal over k and

• F il^{•}D_{K} a decreasing separated exhaustive filtration of D_{K} := D ⊗_{K_{0}} K by Ksubspaces.

One does not require any compatibility between F_{D} and F il^{•}.

A filtered isocrystal \underline{D} = (D, F_{D}, 𝔮_{D}) over L consists of

• (D, F_{D}) a z-isocrystal over ℓ and

• 𝔮_{D} an L[[z − ζ]]-lattice inside the L((z − ζ))- vector space σ^{∗}D ⊗_{ℓ((z))} L((z − ζ)).

𝔮_{D} is called a Hodge-Pink structure on (D, F_{D}). One also sets 𝔭_{D} := σ^{∗}D ⊗_{ℓ((z))} L[[z − ζ]]. For the significance of L[[z − ζ]] and its analogue in mixed characteristic see 2.9.

The filtration F il^{•}D_{K} defines a G_{K}-stable B_{dR}^{+}- lattice 𝔮_{D} (see 2.9) in 𝔭_{D} ⊗_{B_{dR}^{+}} B_{dR} where 𝔭_{D} := D_{K} ⊗_{K} B_{dR}, by setting

𝔮_{D} := F il^{0}(D_{K} ⊗_{K} B_{dR}) .

Conversely any such lattice determines a filtration by

F il^{i}D_{K} = ((𝔭_{D} ∩ t^{i}𝔮_{D})/(t 𝔭_{D} ∩ t^{i}𝔮_{D}))^{G_{K}} .

This defines a 1-1-correspondence between filtrations and G_{K}-stable lattices.

Every Hodge-Pink structure determines in particular a Hodge-Pink filtration F il^{•} on the L-vector space D_{L} := 𝔭_{D}/(z − ζ)𝔭_{D} by letting F il^{i}D_{L} be the subspace

(𝔭_{D} ∩ (z − ζ)^{i}𝔮_{D}) / ((z − ζ)𝔭_{D} ∩ (z − ζ)^{i}𝔮_{D})

of D_{L}. Note that z acts on D_{L} as the scalar ζ. However, as was observed by Pink [39], the fact that z and ζ both play part of the role of p makes it necessary to consider Hodge-Pink structures instead of only Hodge-Pink filtrations to get a reasonable category. See [24, Remark 2.2.3] for a detailed discussion of this phenomenon.

The Hodge-Tate weights of \underline{D} are the integers h for which F il^{−h}D_{K} ≠ F il^{−h+1}D_{K} or equivalently, the elementary divisors of the B_{dR}^{+}-lattice 𝔮_{D} relative to 𝔭_{D}. Note that the definition differs by a minus sign from the corresponding definition in equal characteristic.

The Hodge-Pink weights of (D, F_{D}, 𝔮_{D}) are the elementary divisors of 𝔭_{D} relative to 𝔮_{D}. More precisely if e is a large enough integer such that (z − ζ)^{e}𝔭_{D} ⊂ 𝔮_{D} and

𝔮_{D}/(z − ζ)^{e}𝔭_{D} ≅ ⨁_{i=1}^{n} L[[z − ζ]]/(z − ζ)^{w_{i}+e}

then the Hodge-Pink weights are the integers w_{1}, …, w_{n}.

3.8 Weak Admissibility

Let \underline{D} = (D, F_{D}, F il^{•}D_{K}) be a filtered isocrystal of rank n over K and define Let \underline{D} = (D, F_{D}, 𝔮_{D}) be a filtered isocrystal of rank n over L and define t_{N}(\underline{D}) := ord_{p}(det F_{D}) the Newton slope and

t_{N}(\underline{D}) := ord_{z}(det F_{D}) the Newton slope and

t_{H}(\underline{D}) := ∑_{i∈ℤ}i · dim_{K} gr_{F il^{•}}^{i}D_{K} =

= max{ i ∈ ℤ : F il^{i}(∧^{n}D_{K}) = ∧^{n}D_{K} } the Hodge slope of \underline{D}, which also equals the negative of the sum of the Hodge-Tate weights counted with multiplicity. t_{H}(\underline{D}) := ∑_{i∈ℤ}i · dim_{L} gr_{F il^{•}}^{i}D_{L} the Hodge slope

of \underline{D}, which also equals the integer e such that ∧^{n}𝔮_{D} = (z − ζ)^{−e} ∧^{n} 𝔭_{D} or the sum of the Hodge-Pink weights counted with multiplicity.

\underline{D} is called weakly admissible if t_{H}(\underline{D}) = t_{N}(\underline{D}) and t_{H}(\underline{D}′) ≤ t_{N}(\underline{D}′) for any subobject

\underline{D} is called weakly admissible if t_{H}(\underline{D}) = t_{N}(\underline{D})
and t_{H}(\underline{D}′) ≤ t_{N}(\underline{D}′) for any subobject

\underline{D}′ = (D′, F_{D}|_{D′}, F il^{•}D′_{K}) ⊂ \underline{D}, where D′ ⊂ D is an F_{D}-stable K_{0}-subspace and F il^{•}D′_{K} is the induced filtration on D′_{K} = D′ ⊗_{K_{0}} K.

\underline{D}′ = (D′, F_{D}|_{D′}, 𝔮_{D′}) ⊂ \underline{D}, where D′ ⊂ D is an F_{D}-stable ℓ((z))-subspace and the lattice 𝔮_{D′} equals 𝔮_{D} ∩ σ^{∗}D′ ⊗_{ℓ((z))} L((z − ζ)).

4 The Slope Filtration Theorem for Frobenius Modules

4.1 Frobenius Modules

Let K be as in Section 2 and set R = B_{K,rig}^{†}, or R = B̃_{rig}^{†}, or R = B_{K}^{†}, see 2.4 and 2.8.

Let L, C, and R be as in Section 2 and set R =

⋃_{r→∞} R⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} or R = ⋃_{r→∞} R⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}], see 2.4 and

2.8. Note that we allow R = L = C.

A φ-module over R consists of a finite free Rmodule M and an isomorphism F_{M} : φ^{∗}M ⟶^{∼} M where φ^{∗}M := M ⊗_{R,φ} R.

A σ-module over R consists of a finite free Rmodule M and an isomorphism F_{M} : σ^{∗}M ⟶^{∼} M where σ^{∗}M := M ⊗_{R,σ} R.

4.2 Dieudonné-Manin Decompositions

For integers r, d with r > 0 and (r, d) = 1 consider the φ-module M_{d,r} = (R^{⊕r}, F_{M}) over R with

F_{M} = ( 0 p^{−d} 1 1 0 )· φ .

For integers r, d with r > 0 and (r, d) = 1 consider the σ-module M_{d,r} = (R^{⊕r}, F_{M}) over R with

F_{M} = ( 0 z^{−d} 1 1 0 )· σ .

Every φ-module over B̃_{rig}^{†} is isomorphic to a direct sum ⨁_{i} M_{d_{i},r_{i}} for uniquely determined pairs of integers (r_{i}, d_{i}) up to permutation; see Kedlaya [32, Theorem 4.5.7].

Every σ-module over ⋃_{r} C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} is isomorphic to a direct sum ⨁_{i} M_{d_{i},r_{i}} for uniquely determined pairs of integers (r_{i}, d_{i}) up to permutation; see Hartl-Pink [27, Theorem 11.1].

A φ-module M over B_{K,rig}^{†} is called (isoclinic of) slope λ if M ⊗_{B_{K,rig}^{†}} B̃_{rig}^{†} is isomorphic to M_{d,r}^{⊕e}

where d, r ∈ ℤ, r > 0, (d, r) = 1, λ = \frac{d}{r} and re = rk M.

A σ-module M over ⋃_{r} L⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} is called (iso-
clinic of) slope λ if M ⊗_{⋃_{r} L⟨ \frac{z}{ζ^{r}} , \frac{1}{z} \}} ⋃_{r} C⟨\frac{z}{ζ^{r}} , \frac{1}{z}\}
is isomorphic to M_{d,r}^{⊕e} where d, r ∈ ℤ, r > 0,
(d, r) = 1, λ = \frac{d}{r} and re = rk M.

4.3 The Slope Filtration Theorem

Let M be a φ-module over B_{K,rig}^{†}. Then there exists a unique filtration 0 = M_{0} ⊂ M_{1} ⊂ … ⊂ M_{ℓ} = M of M by saturated φ-submodules, such that the quotients M_{i}/M_{i−1} are isoclinic of some slopes λ_{i}, and λ_{1} < … < λ_{ℓ}, see Kedlaya [32, Theorem 6.4.1].

Let M be a σ-module over ⋃_{r} L⟨\frac{z}{ζ^{r}} , \frac{1}{z}\}. Then there exists a unique filtration 0 = M_{0} ⊂ M_{1} ⊂ … ⊂ M_{ℓ} = M of M by saturated σ-submodules, such that the quotients M_{i}/M_{i−1} are isoclinic of some slopes λ_{i}, and λ_{1} < … < λ_{ℓ}, see Hartl [24, Theorem 1.7.7].

The base change functor from isoclinic φmodules over B_{K}^{†} of slope λ to isoclinic φmodules over B_{K,rig}^{†} is an equivalence of categories; see [32, Theorem 6.3.3].

The base change functor from isoclinic σ-modules over ⋃_{r} L⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}] of slope λ to isoclinic σ-modules over ⋃_{r} L⟨\frac{z}{ζ^{r}} , \frac{1}{z}\} is an equivalence of categories; see [24, Theorem 1.7.5].

Due to the limitations of the Theory of the field of norms it is not yet possible to prove a relative version of this equivalence.

There exists a relative version of this equivalence over arbitrary affinoid L-algebras R in place of L. This is the main tool to prove the Theorem in 6.2.

5 Galois Representations

Let K be as in Section 2 and denote by G_{K} := Gal(K^{alg}/K) the absolute Galois group of K. Fontaine Theory is the theory of the category Rep_{ℚ_{p}} G_{K} of continuous representations of G_{K} in finite dimensional ℚ_{p}-vector spaces and various full subcategories like crystalline, or semi-stable, or de Rham representations, etc.

Let L be as in Section 2 and let G_{L} := Gal(L^{sep}/L) be the absolute Galois group of L. The theory of the category Rep_{𝔽_{q}((z))} G_{L} of continuous representations of G_{L} in finite dimensional 𝔽_{q}((z))-vector spaces is spoiled by various unpleasant facts, such as the ones mentioned in 1.4, 1.5, 2.5, 2.9, or 5.2 below.

5.1 (φ, Γ)-Modules

By the Theorem of the field of norms (see 2.1) and Katz’ [30, Proposition 4.1.1] the category Rep_{ℚ_{p}} H_{K} is equivalent to the category of slope zero φ-modules over B_{K} (see 2.3) by mapping the H_{K}-representation V to the φ-module D(V) := (B ⊗_{ℚ_{p}} V)^{H_{K}} and conversely mapping the φ-module M to the H_{K}-representation V_{p}(M) := (B ⊗_{B_{K}} M)^{φ=1}.

By the analogue of Katz’ theorem the category Rep_{𝔽_{q}((z))} G_{L} is equivalent to the category of slope zero σ-modules over L((z)) by mapping the G_{L}-representation V to the σ-module (L^{sep}((z))⊗_{𝔽_{q}((z))}V )^{G_{L}} and conversely mapping the σ-module M to the G_{L}-representation V_{z}(M) := (L^{sep}((z)) ⊗_{L((z))} M)^{σ=1}.

There is a relative version for certain affinoid Kalgebras by Andreatta and Brinon [4].

There is a relative version for arbitrary affinoid Lalgebras R by replacing G_{L} with the étale fundamental group π_{1}^{ét}(Sp R) (see [29]) and L((z)) with R[[z]][\frac{1}{z}].

As a consequence the category Rep_{ℚ_{p}} G_{K} is equivalent to the category of slope zero (φ, Γ_{K})- modules over B_{K}, that is slope zero φ-modules with Γ_{K} := G_{K}/H_{K}-action commuting with φ. Namely, V is mapped to D(V) = (B ⊗_{ℚ_{p}} V)^{H_{K}} which inherits the Γ_{K}-action from G_{K}.

Since the distinction between G_{L} and H_{L} collapses there is no Γ_{L}-action, see 2.1.

5.2 Overconvergence

The main theorem of Cherbonnier-Colmez [10] says that every representation V in Rep_{ℚ_{p}} G_{K} is overconvergent, that is that D(V) has a basis consisting of elements of D^{†}(V) := (B^{†} ⊗_{ℚ_{p}} V)^{H_{K}}, see 2.4. Equivalently every slope zero (φ, Γ_{K})-module over B_{K} has a basis on which φ acts by a matrix with coefficients in B_{K}^{†}.

Overconvergence fails in equal characteristic as the slope zero σ-module (M, F_{M}) with

M = 𝔽_{q}((ζ))((z)) , F_{M} = (∑_{i=0}^{∞} ζ^{q^{−i}}z^{i}) · σ

shows. Since in mixed characteristic overconvergence is most important for the theory of padic Galois representations we propose to view σmodules over ⋃_{r} R⟨\frac{z}{ζ^{r}} ⟩[\frac{1}{z}], see 2.4, as the appropriate analogue in equal characteristic of p-adic Galois representations.

5.3 Crystalline Galois Representations

A representation V in Rep_{ℚ_{p}} G_{K} is called crystalline if dim_{ℚ_{p}} V equals the dimension over (B̃_{rig})^{G_{K}} = K_{0}, see 2.7, of

D_{cris}(V) := (B̃_{rig} ⊗_{ℚ_{p}} V)^{G_{K}} .

The Frobenius φ on B̃_{rig} makes D_{cris}(V) into an F -isocrystal over k.

Due to the unfavorable fact that

O_{C}[[z, \frac{1}{z}\}^{G_{L}} = O_{L^{p}}[[z, \frac{1}{z}\} ⊋ ℓ((z)) ,

see 2.5, the notion of crystalline Galois representations is problematic. But see the next item.

5.4 Rigidified Local Shtuka

The Breuil-Kisin classification [33, Theorem 0.1] of crystalline G_{K}-representations states the following: Let 𝔖 := W (k)[[u]] and let π ∈ K be a uniformizer with Eisenstein polynomial E(u) ∈ 𝔖. Let φ : 𝔖 → 𝔖 extend the Frobenius lift φ on W (k) and map u ↦ u^{p}. Let Mod_{/𝔖}^{φ} be the Tannakian category of finite free 𝔖-modules 𝔐 with an isomorphism F_{𝔐} : φ^{∗}𝔐[\frac{1}{E(u)}] ⟶^{∼} 𝔐[\frac{1}{E(u)}], where φ^{∗}𝔐 := 𝔐 ⊗_{𝔖,φ} 𝔖.

A rigidified local shtuka (M, F_{M}, δ_{M}) over O_{L} consists of a local shtuka (M, F_{M}) over O_{L} and, posing (D, F_{D}) := (M, F_{M}) ⊗_{O_{L}[[z]]} ℓ((z)), an isomorphism δ_{M}

M ⊗_{O_{L}[[z]]} O_{L}[[z, \frac{1}{z}\}[\frac{1}{t_{−}}] ⟶^{∼} D ⊗_{ℓ((z))} O_{L}[[z, \frac{1}{z}\}[\frac{1}{t_{−}}]

which satisfies δ_{M} ∘F_{M} = F_{D}∘σ^{∗}δ_{M} and which reduces to the identity modulo 𝔪_{L}. If L is discretely valued the forgetful functor (M, F_{M}, δ_{M}) ↦ (M, F_{M}) is an equivalence of categories by [19, Lemma 2.1], see also [24, Lemma 2.3.1].

Then the category of crystalline representations is tensor equivalent to a full subcategory of the isogeny category of Mod_{/𝔖}^{φ}. The essential image can be characterized in terms of a connection.

From a (rigidified) local shtuka (M, F_{M}) over O_{L} one obtains a Galois representation

V_{z}(M) := (L^{sep}((z)) ⊗_{O_{L}[[z]]} M)^{σ=1}

as in 3.3. The tensor functor (M, F_{M}) ↦ V_{z}(M) is faithful; see [24, Proposition 2.1.4].

5.5 The Mysterious Functor

If V is a crystalline G_{K}-representation, then D_{cris}(V) is an F -isocrystal over k, see 5.3. The embedding B̃_{rig} ⊂ B_{dR} (see 2.7 and 2.9) equips D_{cris}(V) ⊗_{K_{0}} K with a filtration F il^{•} by Ksubspaces, such that (D_{cris}(V), F il^{•}) is a filtered isocrystal over K, see 3.7. This is Fontaine’s functor

V ↦ (D_{cris}(V), F il^{•})

whose existence was conjectured by Grothendieck (the “mysterious functor”) in case V equals the étale cohomology of a smooth proper Kvariety with good reduction.

If (M, F_{M}, δ_{M}) is a rigidified local shtuka over O_{L} then (D, F_{D}) := (M, F_{M}) ⊗_{O_{L}[[z]]} ℓ((z)) is a z-isocrystal over ℓ and

𝔮_{D} := σ^{∗}δ_{M} ∘ F_{M}^{−1}(M ⊗_{O_{L}[[z]]} L[[z − ζ]])

is a Hodge-Pink structure on (D, F_{D}), such that (D, F_{D}, 𝔮_{D}) is a filtered isocrystal over L, see 3.7. The functor

ℍ : (M, F_{M}, δ_{M}) ↦ (D, F_{D}, 𝔮_{D})

is the analogue of the mysterious functor.

A filtered isocrystal in the essential image of Fontaine’s functor is called admissible.

A filtered isocrystal in the essential image of ℍ is called admissible.

5.6 Weakly Admissible Implies Admissible

By the Colmez-Fontaine Theorem [13] Fontaine’s Functor is an equivalence of categories between crystalline G_{K}-representations and weakly admissible filtered isocrystals over K, see 3.8. (Here K has to be discretely valued with perfect residue field.)

The functor ℍ is an equivalence between rigidified local shtuka over O_{L} and weakly admissible filtered isocrystals (see 3.8) if the completion L̃ of the compositum ℓ^{alg}L inside C does not contain an element a with 0 < |a| < 1 such that for all n ∈ ℕ the q^{n}-th roots of a also lie in L̃, see [24, Theorem 2.5.3]. This condition is for example satisfied if the value group of L is not q-divisible.

6 Period Spaces for Filtered Isocrystals

6.1 Period Spaces

On a fixed F -isocrystal (D, F_{D}) over 𝔽_{p}^{alg} the filtrations from 3.7 are parametrized by a projective partial flag variety over W (𝔽_{p}^{alg})[\frac{1}{p}].

On a fixed z-isocrystal (D, F_{D}) over 𝔽_{q}^{alg} the Hodge-Pink structures from 3.7 are parametrized by a quasi-projective partial jet bundle over a partial flag variety over 𝔽_{p}^{alg}((ζ)). The jets arise since a Hodge-Pink structure contains more information than just the Hodge-Pink filtration.

The weakly admissible filtrations (see 3.8) form a rigid analytic subspace F^{wa} of this partial flag variety by Rapoport-Zink [40, Proposition 1.36]. This is a p-adic period space. When viewed as a Berkovich space it is even a connected open Berkovich subspace of the partial flag variety; see [25, Proposition 1.3].

The weakly admissible Hodge-Pink structures (see 3.8) form a rigid analytic subspace F^{wa} of this partial jet bundle by [24, Theorem 3.2.5] called a period space for Hodge-Pink structures. When viewed as a Berkovich space it is even a connected open Berkovich subspace of the partial jet bundle.

6.2 A Conjecture of Rapoport and Zink

Rapoport and Zink [40, p. 29] conjecture the existence of an étale morphism F′ → F^{wa} of rigid analytic spaces, which is bijective on Kvalued points with K finite over W (𝔽_{p}^{alg})[\frac{1}{p}], and a p-adic representation of the étale fundamental group

π_{1}^{ét}(F′) ⟶ GL_{n}(ℚ_{p})

which induces the universal filtered isocrystal over F′.

The analogue of Rapoport and Zink’s conjecture is a theorem; see [24, Theorem 3.4.3]: There exists a unique maximal open Berkovich subspace F^{a} of the period space F^{wa} containing all its L-valued points for fields L as in 5.6, there exists an admissible formal scheme Y over 𝔽_{q}^{alg}[[ζ]] whose associated Berkovich space Y^{Berko} is an étale covering space of F^{a}, and there exists a rigidified local shtuka (see 5.4) over Y inducing the universal filtered isocrystal over F^{a}. This local shtuka gives rise to a representation π_{1}^{ét}(F^{a}) → GL_{n}(𝔽_{q}((z))) of the étale fundamental group. The proof relies on the relative descent mentioned in 4.3 which is lacking in mixed characteristic.

6.3 Rapoport-Zink Spaces

Let 𝔼 be a fixed p-divisible group over 𝔽_{q}^{alg}. The functor which assigns to every formal W (𝔽_{p}^{alg})- scheme S the set of isomorphism classes of pairs (E, ρ) where

• E is a p-divisible group over S,

• ρ : 𝔼S̄ → ES̄ is a quasi-isogeny over S̄, the closed subscheme of S defined by the ideal p O_{S},

is representable by an adic formal scheme X locally formally of finite type; see [40, Theorem 2.16]. X is called a Rapoport-Zink space.

Let 𝕄 be a fixed local shtuka over 𝔽_{q}^{alg}. The functor which assigns to every formal 𝔽_{q}^{alg}[[ζ]]-scheme S the set of isomorphism classes of pairs (M, ρ) where

• M is a local shtuka over S,

• ρ : M ⊗_{O_{S}[[z]]} OS̄[[z]][\frac{1}{z}] ⟶^{∼} 𝕄_{𝔽_{q}^{alg}[[z]]}OS̄[[z]][\frac{1}{z}]

is an isomorphism where S̄ is the closed subscheme of S defined by the ideal ζ O_{S},

is representable by an adic formal scheme X locally formally of finite type; see [26]. X is called a Rapoport-Zink space.

6.4 Period Morphisms

The Hodge-Tate filtration on the Dieudonné crystal associated with the universal p-divisible group over the formal scheme X from 6.3 defines an étale period morphism X^{rig} → F^{wa} by [40, Proposition 5.15]. When viewed as a morphism of Berkovich spaces it has open image and identifies the Berkovich space associated with X^{rig} with an étale covering space of this image.

Applying the mysterious functor from 5.5 to the universal local shtuka over the formal scheme X from 6.3 defines an étale period morphism from X^{Berko} to the space F^{a} from 6.2. It identifies X^{Berko} with the étale covering space Y^{Berko} of F^{a} from 6.2, see [26].

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Urs Hartl University of Freiburg Institute of Mathematics Eckerstr. 1 D – 79104 Freiburg Germany E-mail: urs.hartl@math.uni-freiburg.de