Étale fundamental groups of non-Archimedean analytic spaces

A.J. de Jong*

Dedicated to Frans Oort

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Étale coverings of Berkovich spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. The case of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4. Berkovich spaces and local systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5. Étale coverings of rigid analytic spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6. Filtered F -crystals, p-divisible groups and local systems . . . . . . . . . . . . . . . . . . . . . . . . 18 7. Some examples of étale covering spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1 Introduction

How to define the étale fundamental group of a non-Archimedean analytic space X? As we know from the work of Grothendieck on the algebraic fundamental group for schemes, an étale fundamental group should be part of a dualism whose other constituent is the category of étale covering spaces. The definition of étale covering spaces was given by Berkovich, see [B2, Remark 6.3.4 (ii)]. This category includes the finite étale coverings of X and the topological covering spaces of X, of these two classes the first is dual to the algebraic fundamental group π_{1}^{alg}(X) of X, the second gives rise to the topological fundamental group π_{1}^{top}(X). A genuine example of an étale covering map is the logarithm

log : {z ∈ ℂ_{p}; |z − 1| < 1} ⟶ ℂ_{p},

given by the usual powerseries. It is a Galois covering with group ⋃_{n} μ_{p^{n}}(ℂ_{p}), and its existence evidences the nontriviality of the étale fundamental group of the affine line over ℂ_{p}, in contrast to the triviality of both the algebraic and topological fundamental group

of 𝔸1_{ℂ_{p}}.

We may motivate the study of this étale fundamental group by analogy to the case of schemes or the case of complex analytic spaces. However, at least part of the theory in this work (especially in Sections 4, 6) was developed by the author in an attempt to understand the nature of the p-adic period maps introduced by Rapoport and Zink [RZ]. These maps are étale rigid analytic morphisms of the form

π : M^{rig} ⟶ F^{wa},

* The research of Dr. A.J. de Jong has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. and the logarithm is an example of such a morphism, see [RZ, Proposition 5.40]. In this formula, M denotes a formal scheme over Spf(ℤ_{p}) which represents a moduli problem for p-divisible groups quasi-isogenous to a fixed p-divisible group X_{0}. The space F^{wa} denotes the admissible open subset of weakly admissible filtrations in a flag variety of filtrations of the F -isocrystal associated to X_{0}. Both the moduli problem and the flag variety involve polarizations and endomorphisms. We remark that in all cases π may be considered as an étale morphism of Berkovich spaces. Let U be the paracompact separated rigid variety whose associated analytic space (see [B2]) is the image of π, considered as a morphism of Berkovich spaces. The morphism j : U → F^{wa} is injective and identifies local rings, and if Fontaine’s conjecture - weakly admissible ⇔ admissible - holds then j is actually a surjection (unless M^{rig} is empty, see [RZ, Proposition 5.20]). However, even granting this conjecture, in general it cannot be expected that j is an isomorphism, see [RZ, 5.41].

By definition the morphism π factors as j ∘ π′ for some

π′ : M^{rig} ⟶ U.

Using the results of this article it follows that π′ is an étale covering map. More precisely, it follows that there is a natural local system of ℚ_{p}-vectorspaces V on U, such that M^{rig} is identified with a union of connected components in the space of lattices in V. The arguments for this - see Section 7 - are given only in a special case, namely the case where M is the Lubin-Tate moduli space, treated by Hopkins and Gross, [HG]. In this case, we have that F^{wa} = F = ℙ^{h−1}, and, by the results of [HG], we know that U ≅ ℙ^{h−1}. In particular, from the fact that π is not an isomorphism we deduce that π_{1}(ℙh−1_{ℂ_{p}}) ≠ {1}.

Moreover, the local system V corresponds to a continuous homomorphism

π_{1}(U) → G(ℚ_{p}),

where G is a certain algebraic group associated to the moduli problem. It is natural to ask what the image of this homomorphism is, or what the image of the geometric fundamental group π_{1}(U_{ℂ_{p}}) is. In the Lubin-Tate case we have G = GL_{h}, where h is the height of the associated formal group and we can answer these questions. We prove in the final section that the map

π_{1}(ℙh−1_{ℂ_{p}}) → SL_{h}(ℚ_{p})

is a surjection.

The general theory of the étale fundamental group is exposed using the k-analytic spaces developed by Berkovich. In the proof of the duality theorem it is used in an essential way that such a space is locally path connected; I do not know how to formulate the corresponding fact for rigid spaces other than referring to these k-analytic spaces. However, in Section 5 we translate the results back to rigid analytic varieties. Indeed, when we speak in Section 6 about p-divisible groups and filtered F -crystals over analytic spaces, it seems more natural to use the terminology of rigid analytic varieties. The result of this section is that the category of p-divisible groups up to isogeny over a rigid analytic space X is equivalent to a full subcategory of the category of filtered F -crystals, and in case X is reduced, equivalent to a full subcategory of the category of local systems of ℚ_{p}-vector spaces. For example, the local system V above is given by a p-divisible group up to isogeny over U.

The author thanks Prof. M. Rapoport and Prof. T. Zink for many discussions concerning the p-adic period maps and other subjects, for example the idea of having a local sytem of ℚ_{p}-vector spaces on U is already contained in [RZ]. Further, he would like to thank Prof. D. Gross and Prof. J.-K. Yu for many discussions concerning the structure of the p-adic period map in the Lubin-Tate case, let us mention here that it was proved first by Prof. Yu that the period map is an étale covering map in this case. He also thanks Prof. M. Hopkins, Prof. W. Messing, and Prof. M. van der Put. Thanks are due to the referee, who corrected some mistakes, and gave a simpler proof of Proposition 7.5 which originally was proved only for special fields k. Finally, it is his pleasure to thank Prof. F. Oort, to whom this article is dedicated, for his constant help and advice during the last years.

2 Étale covering spaces of Berkovich spaces

Let k be a field complete with respect to a non-Archimedean valuation | |; we do not assume that | | is nontrivial. In this section we work with k-analytic spaces as defined in [B2, page 17]. Let X be such a space. We remark that the underlying topological space |X| is locally connected, see [B1, Corollary 2.2.8].

2.1 Definition. (See [B2, Remark 6.3.4 (ii)].) Let f : Y → X be a morphism of k-An. We say that Y is an étale covering space of X or that f is an étale covering map if for all x ∈ X there exists an open neighbourhood U ⊂ X of x such that f^{−1}(U) is a disjoint union of spaces V_{i}, each mapping finite étale to U. We say that Y is a topological covering space of X if we can choose U and the V_{i} as above such that all the maps V_{i} → U are isomorphisms.

We note that finite étale morphisms Y → X are étale covering spaces; these are the coverings used in algebraic geometry to define the algebraic fundamental group. A morphism of étale covering spaces Y_{1}, Y_{2} of X is a morphism Y_{1} → Y_{2} compatible with the structural maps Y_{i} → X. From the definition it follows that a fibre product Y_{1} ×_{Y} Y_{2} of étale covering spaces is an étale covering space. If g : Y_{1} → Y_{2} is a morphism of such covering spaces then g is an étale covering map. However, it is in general not true that a composition of étale covering maps is an étale covering map. Finite disjoint unions of étale covering spaces are étale covering spaces; the empty space is an étale covering space of X. Infinite disjoint unions of étale covering spaces do not give étale covering spaces in general.

2.2 Lemma. Let f : Y → X be an étale covering map.

(i) The morphism f is étale and separated. (ii) For any morphism Z → X of An_{k} (see [B2, page 25]) the fibre product Y ×_{X} Z is an étale covering space of Z. (iii) If X is paracompact, then so is Y. In this case, if Y is connected, the fibres of f are at most countable. (iv) The image of f is a union of connected components of X.

Proof. Statements (i), (ii) and (iv) follow immediately from the definitions. To prove (iii) we note that since |X| is Hausdorff and |Y | → |X| separated, we get that |Y | is Hausdorff. By definition there exists an open covering X = ⋃_{i} U_{i} such that f^{−1}(U_{i}) = ∐ V_{ij}, with V_{ij} → U_{i} finite étale. Consider a closed locally finite refinement X = ⋃_{s} Z_{s} of the covering X = ⋃_{i} U_{i} ([En, Theorem 5.1.11]). For each s we choose an i such that Z_{s} ⊂ U_{i} and put W_{sj} = f^{−1}(Z_{s}) ∩ V_{ij}. These are closed subsets of Y. The map W_{sj} → Z_{s} has finite fibres, Z_{s} is paracompact, hence W_{sj} is paracompact [En, Theorem 5.1.35].

Take y ∈ Y and let x = f(y). By definition we may choose an open neigbourhood U of x such that the set {s′|Z_{s′} ∩ U ≠ ∅} is finite, say it consists of s_{1}, …, s_{r}. We replace U by U \Z_{s_{i}} if x ∉ Z_{s_{i}}. Hence we may assume x ∈ Z_{s_{i}} for all i. Shrinking U we may therefore assume that U ⊂ U_{i}, where U_{i} is the chosen subset such that Z_{s_{i}} ⊂ U_{i}. Finally, we may choose U so small that for some connected open neighbourhood V of y the map V → U is finite étale. It is clear that for each i = 1, …, r there is a unique j_{i} such that V ⊂ V_{ij_{i}}. Thus among the W_{sj} only the sets W_{s_{i}j_{i}} meet V. We get that the covering Y = ⋃ W_{sj} is locally finite and Y is paracompact [En, Theorem 5.1.34]. If Y is connected, then Y is Lindelöf [En, Theorem 5.1.27]. Since the fibre f^{−1}(x) ⊂ Y is a discrete closed subspace of Y, it is Lindelöf and hence countable. □

2.3 Lemma. Let F be a sheaf on the étale site X_{ét} of X. Suppose that {g_{i} : U_{i} → X} is a covering for the étale topology of X. The sheaf F is representable by an étale covering space of X if and only if F_{U_{i}} is representable by an étale covering space of U_{i} for each i.

Proof. The only if statement follows from Lemma 2.2 and [B2, Corollary 4.1.4]. Assume that F_{U_{i}} is representable by an étale covering space for each i. Take x ∈ X. By assumption there exists an open neighbourhood U ⊂ X, a finite étale morphism g : V → U and a point y ∈ V mapping to x such that F_{V} is representable by an étale covering space Y_{V} over V. By shrinking U and V we may even assume that V is Galois over U, say with group G. The action of G on V lifts to an action on Y_{V}, since this action lifts to F_{V}. Using that Y_{V} is an étale covering map, we may shrink V further to get the situation where Y_{V} is a disjoint union of spaces finite étale over V. There is an étale covering space Y_{U} → U such that there is a G-equivariant isomorphism Y_{U} ×_{U} V ≅ Y_{V}. To prove this, write Y_{V} = ∐ V_{j} such that V_{j} → V is finite étale and V_{j} ⊂ Y_{V} is G-stable. This is possible as G is finite. The spaces V_{j} and V may be viewed as (finite étale) schemes over (any affinoid subdomain of) U, see [B2, Section 2.6]. In the category of schemes we know the result; this gives the spaces V_{j}/G over U and the disjoint union of these gives Y_{U}. Obviously, Y_{U} represents F_{U} and is an étale covering space of U. We leave it to the reader to glue the spaces Y_{U} to an étale covering space Y representing F (use [B2, Proposition 1.3.3]). □

2.4 Lemma. Consider an étale covering space Y of X. Suppose R ⊂ Y ×_{X} Y is a union of connected components, which is an equivalence relation on Y over X. The quotient sheaf Y/R on X_{ét} is representable by an étale covering space of X.

Proof. By the previous lemma the question is local on X. By definition of étale covering spaces this reduces the question to the case where Y → X is finite étale. In case X is affinoid, the spaces Y and R are affinoid also and the result is known by algebraic geometry [SGA1, Exposé V ]. As usual we leave the gluing process to the reader. □

2.5 Example. Suppose that g : Y_{1} → Y_{2} is a morphism of étale covering spaces of X. The fibre product R = Y_{1} ×_{Y_{2}} Y_{1} is a union of connected components of Y_{1} ×_{X} Y_{1}, since the map R → Y_{1} ×_{X} Y_{1} is an injective étale covering map. Thus R is an equivalence relation on Y_{1} over X. Of course the quotient Y_{1}/R is represented by Im(g) ⊂ Y_{2}, a union of connected components of Y_{2} (Lemma 2.2). The result of Lemma 2.4 is that any equivalence relation as in the lemma is realized in this fashion. □

2.6 Lemma. The category of topological covering spaces of X is equivalent to the category of covering spaces of |X|.

Proof. Suppose T → |X| is a covering space. We define a sheaf F = F(T ) on X_{ét} as follows: The sections of F over g : U → X étale are morphisms |U| → T of topological spaces over |X|. Using Lemma 2.3 we see immediately that F is representable by a topological covering space T → X with T ≅ |T |. This defines a quasi-inverse to the functor T ↦ |T |. □

To define the fundamental group of X we need some notations. We use the symbole \underline{Cov}_{X} (resp. \underline{Cov}_{X}^{top}, \underline{Cov}_{X}^{alg}) to denote the category of étale covering spaces of X (resp. topological covering spaces of X, finite étale covering spaces of X). A geometric point x̄ of X is a morphism x̄ : M(K) → X of An_{k} where K is an algebraically closed complete valued extension of k. As in [SGA1] we consider the fibre functor

Fx̄ : \underline{Cov}_{X} ⟶ \underline{Sets}

defined by the formula

Fx̄(Y) = { ȳ: M(K) → Y | f(ȳ) = x̄ }.

Note that Fx̄ is compatible with fibre products and disjoint unions. We denote F top ; x̄, resp. F alg ; x̄ the restriction of Fx̄ to the subcategory \underline{Cov}_{X}^{top}, resp. \underline{Cov}_{X}^{alg}.

The fundamental group of X with base point x̄ is a topological group. The underlying abstract group is

π_{1}(X, x̄) = Aut(Fx̄),

i.e., it is the group of isomorphisms of functors Fx̄ → Fx̄. We define a topology on π_{1}(X, x̄) as follows. For any pair (Y, ȳ), Y ∈ \underline{Cov}_{X} and ȳ ∈ Fx̄(Y) we put

H(Y, ȳ) ⊂ π_{1}(X, x̄)

equal to the stabilizer of ȳ for the natural action of π_{1}(X, x̄) on Fx̄(Y). The family F of subgroups so obtained satisfies i) H, H′ ∈ F ⇒ H ∩ H′ ∈ F and ii) H ∈ F, γ ∈ π_{1}(X, x̄) ⇒ γHγ^{−1} ∈ F. The first property follows from the existence of fibre products in \underline{Cov}_{X}; the second since γH(Y, ȳ)γ^{−1} = H(Y, γ · ȳ). Consequently, there is a unique topology on π_{1}(X, x̄) making it into a topological group such that the familly of H ∈ F forms a cofinal system of open neighbourhoods of the identity element (see [Bou, General topology III §2]).

The same definition, but using the functor F top ; x̄ (resp. F alg ; x̄), gives a topological group π_{1}^{top}(X, x̄) (resp. π_{1}^{alg}(X, x̄)). These are called the topological (resp. algebraic) fundamental group of X. By Lemma 2.6 the topological group π_{1}^{top}(X, x̄) depends only on the pair (|X|, x), where x ∈ X denotes the unique point in the image of x̄. There are restriction maps Aut(Fx̄) → Aut(F top ; x̄) and Aut(Fx̄) → Aut(F alg ; x̄); these give rise to continuous homomorphisms π_{1}(X, x̄) → π_{1}^{top}(X, x̄) and π_{1}(X, x̄) → π_{1}^{alg}(X, x̄). Finally, there is an obvious functoriality of these fundamental groups with respect to morphisms of pairs (Y, ȳ) → (X, x̄).

2.7 Lemma. The topological group π_{1}(X, x̄) is Hausdorff and prodiscrete. More precisely, the map

π_{1}(X, x̄) ⟶ lim_{\overleftarrow{H∈F}} π_{1}(X, x̄)/H

is an isomorphism of topological spaces. Similar for the topological resp. algebraic fundamental group of X.

Proof. Suppose that (γ_{H})_{H∈F} is an element of the inverse limit. For each pair (Y, ȳ), Y ∈ \underline{Cov}_{X}, ȳ ∈ Fx̄(Y) we can define γ · ȳ as γ̃_{H} · ȳ where H = H(Y, ȳ) and γ̃_{H} ∈ π_{1}(X, x̄) is a lift of the element γ_{H} ∈ π_{1}(X, x̄)/H. It is clear that this is well defined. We leave it to the reader to see that this defines an element γ ∈ Aut(Fx̄) = π_{1}(X, x̄). This is the inverse to the natural map of the lemma; the statement on topologies is clear. □

Let us introduce some notation. If G is a topological group we write G-\underline{Sets} to denote the category of pairs (V, ρ), where V is a set and ρ : G × V → V is an action of G on V which is continuous for the discrete topology on V. This means that the stabilizer of any element of V is open in G. Let us write finite-G-\underline{Sets} for the full subcategory consisting of pairs (V, ρ) with V finite.

It follows immediately from the definitions that Fx̄ may be considered as a functor

Fx̄ : \underline{Cov}_{X} ⟶ π_{1}(X, x̄)-\underline{Sets}.

The same holds for F top ; x̄; the functor F alg ; x̄ is seen as a functor

F alg ; x̄ : \underline{Cov}_{X}^{alg} ⟶ finite-π_{1}^{alg}(X, x̄)-\underline{Sets}.

2.8 Example. Suppose that X = {x} consists of one point and that x̄ : M(K) → X is given by the embedding H(x) → K. We write H(x)^{sep} for the separable closure of H(x) in K. In this case, étale covering spaces Y → X with Y connected correspond to finite separable field extensions H(x) ⊂ L. The set Fx̄(Y) is equal to the set of H(x)-embeddings L → K.

From this discussion it follows that \underline{Cov}_{X} = Ind(\underline{Cov}_{X}^{alg}). By Grothendieck’s theory of the fundamental group [SGA1, Exposé V] we get the equalities π_{1}(X, x̄) = π_{1}^{alg}(X, x̄) = Gal(H(x)^{sep}/H(x)) and the duality statements that \underline{Cov}_{X}^{alg} ≅ finite-π_{1}^{alg}(X, x̄)-\underline{Sets} and \underline{Cov}_{X} ≅ π_{1}(X, x̄)-\underline{Sets}. Finally, we note the trivial fact that π_{1}^{top}(X, x̄) = \{1\}. □

2.9 Theorem. Suppose X is connected. For any two geometric points x̄, x̄′ there exists an isomorphism of functors Fx̄ ≅ F_{}_{x̄′}.

This will be proved later on; first we deduce the duality theorem from it.

2.10 Theorem. Here x̄ is a geometric point of the connected k-analytic space X.

(i) The functor Fx̄

\underline{Cov}_{X} ⟶ π_{1}(X, x̄)-\underline{Sets}

is fully faithful; any π_{1}(X, x̄)-set which consits of a single orbit is in the essential image of Fx̄. Thus the category π_{1}(X, x̄)-\underline{Sets} is naturally equivalent to the category of disjoint unions of étale covering spaces of X. An analogous result holds for the functor F top ; x̄. The functor F alg ; x̄

\underline{Cov}_{X}^{alg} ⟶ finite-π_{1}^{alg}(X, x̄)-\underline{Sets}

is an equivalence. (ii) The topology on π_{1}^{top}(X, x̄) is defined by a fundamental system of open normal subgroups. In the category \underline{Cov}_{X}^{top} any connected object is covered by a Galois covering of X. (iii) The topological group π_{1}^{alg}(X, x̄) is profinite. Any continuous map of π_{1}(X, x̄) to a pro-finite group factors through π_{1}^{alg}(X, x̄). (iv) The maps π_{1}(X, x̄) → π_{1}^{top}(X, x̄) and π_{1}(X, x̄) → π_{1}^{alg}(X, x̄) have dense image.

Proof. For any Y ∈ An_{k} and geometric points ȳ, ȳ′ of Y we call a path connecting ȳ to ȳ′ an isomorphism α : Fȳ → F_{}_{ȳ′}. This is functorial: If f : Y → Z is a morphism in An_{k} we get a path f(α) on Z connecting f ∘ ȳ to f ∘ ȳ′. Use Lemma 2.2. If f ∘ ȳ = f ∘ ȳ′ then we get a loop f(α) ∈ π_{1}(Z, z̄) with z̄ = f ∘ȳ. If we are in the special situation that f : Y → X is an étale covering map and ȳ, ȳ′ ∈ Fx̄(Y) then γ = f(α) ∈ π_{1}(X, x̄) will be an element such that γ · ȳ = ȳ′. Therefore, Theorem 2.9 implies that the orbits of π_{1}(X, x̄) on Fx̄(Y) correspond bijectively to the connected components of Y.

This implies that the functor Fx̄ is fully faithful: morphisms Y_{1} → Y_{2} in \underline{Cov}_{X} correspond 1-1 with connected components Γ of Y_{1} ×_{X} Y_{2} such that Fx̄(Γ) → Fx̄(Y_{1}) is bijective. Consider the π_{1}(X, x̄)-set π_{1}(X, x̄)/H, where H ⊂ π_{1}(X, x̄) is an open subgroup. By definition this means that H(Y, ȳ) ⊂ H for some Y ∈ \underline{Cov}_{X} and ȳ ∈ Fx̄(Y). We have to show that H = H(Y′, ȳ′) for some pair (Y′, ȳ′). The connected components of Y ×_{X} Y correspond 1-1 to orbits in Fx̄(Y ×_{X} Y) = Fx̄(Y) × Fx̄(Y). Let R ⊂ Y ×_{X} Y correspond to the union of the orbits of the points (ȳ, hȳ) where h ∈ H. It is easily seen that R is an equivalence relation on Y over X. The result follows by taking Y′ = Y/R (see Lemma 2.4).

The same arguments work for the case of topological covering spaces. Regarding finite étale coverings and the algebraic fundamental group of X we may refer to [SGA1, Exposé V]. Further, statement (iii) is also clear.

To see that any connected topological covering space Y → X is dominated by a Galois covering is a topological question (see Lemma 2.6). For any open covering U : |X| = ⋃ U_{i} there exists a universal covering space X̃_{U} → |X| dominating any covering split over the members of U. The construction of X̃_{U} works for arbitrary locally connected spaces X. Let U : X = ⋃ U_{i} be an open covering with connected U_{i}. Let N(U) be the nerve of the covering U. The category of topological covering spaces of X split over U is equivalent to the category of coverings of N(U). The result follows as there is a universal covering Ñ of the simplicial set N(U).

The final statement is a formal consequence of the above and the fact that the functors \underline{Cov}_{X}^{top} → \underline{Cov}_{X} and \underline{Cov}_{X}^{alg} → \underline{Cov}_{X} are fully faithful. □

2.11 Remarks. (i) The map π_{1}(X, x̄) → π_{1}^{alg}(X, x̄) is not surjective in general. To see this take X to be two copies of the projective line (over k algebraically closed) glued together in 0 and ∞. In this case π_{1}^{alg}(X, x̄) = ℤ̂ and π_{1}^{top}(X, x̄) = ℤ (since π_{1}^{top}(|ℙ^{1}|, x̄) = {1}, see [B1, 4.2.2]). The map π_{1}(X, x̄) → π_{1}^{alg}(X, x̄) = ℤ̂ factors through ℤ ⊂ ℤ̂ in this case. We are able to show that it is surjective in certain cases, see Proposition 7.5. (ii) We do not know whether π_{1}^{top}(X, x̄) is in general a discrete topological group. This would follow if the topological space |X| were semi-locally simply connected (in the sense of covering spaces). For the case of curves, see 3.9. (iii) We do not know whether the map π_{1}(X, x̄) → π_{1}^{top}(X, x̄) is surjective in general. This is true if π_{1}^{top}(X, x̄) is a discrete topological group: if in this case (Y, ȳ) is the universal topological covering space, γ ∈ π_{1}^{top}(X, x̄) and α is a path connecting ȳ to γȳ then γ̃ = f(α) ∈ π_{1}(X, x̄) maps to γ.

Proof of Theorem 2.9. It suffices to prove the theorem for connected affinoid spaces X. Thus we assume X is affinoid; using the functoriality of paths the reader may even suppose X is strictly affinoid.

For any finite open covering U : X = ⋃_{i=1}^{n} U_{i} of X we denote \underline{Cov}_{X,U} ⊂ \underline{Cov}_{X} the full subcategory of étale covering maps f : Y → X that are split over the members of U: the spaces f^{−1}(U_{i}) are disjoint unions of spaces lying finite étale over U_{i}. By our definitions we have

\underline{Cov}_{X} = lim_{\overrightarrow{U}} \underline{Cov}_{X,U}.

Let us denote by F_{x̄,U} (resp. F_{x̄′,U}) the fibre functor Fx̄ (resp. F_{}_{x̄′}) restricted to \underline{Cov}_{X,U}. It is clear that

Isom(Fx̄, F_{}_{x̄′}) = lim_{\overleftarrow{U}} Isom(F_{x̄,U}, F_{x̄′,U}).

By [B1, 3.2.1] the space |X| is arcwise connected. Let us choose a closed subset ℓ ⊂ |X| and an isomorphism [0, 1] → ℓ such that 0 corresponds to x and 1 corresponds to x′ (see [En, 6.3.11 & 6.3.12]). Let us consider coverings U : X = ⋃_{i=1}^{n} U_{i} such that there exists an m, 1 ≤ m ≤ n with

\begin{aligned}U_{1} ∩ ℓ & = [0, t_{1}), \\ U_{i} ∩ ℓ & = (s_{i}, t_{i}), \\ U_{m} ∩ ℓ & = (s_{m}, 1] \\ U_{i} ∩ ℓ & = ∅, \end{aligned}  \begin{matrix}i = 2, …, m − 1 \\ and \\ i ≥ m + 1 \end{matrix}

where

0 < s_{2} < t_{1} < … < s_{m} < t_{m−1} < 1.

We note that these coverings are cofinal in the system of all finite open coverings of X. For any such U we choose

x_{i} ∈ U_{i} ∩ U_{i+1} ∩ ℓ, i = 1, …, m − 1

and a geometric point x̄_{i} lying over x_{i}. For convenience we put x_{0} = x, x̄_{0} = x̄, x_{m} = x′ and x̄_{m} = x̄′.

By [SGA1, Exposé V] we can find isomorphisms

α_{i} : F_{x̄_{i},U_{i+1}}^{alg} ⟶ F_{x̄_{i+1},U_{i+1}}^{alg}

as functors on \underline{Cov}_{U_{i+1}}^{alg}. We remark that this will also induce an isomorphism

α_{i} : F_{x̄_{i},U} ⟶ F_{x̄_{i+1},U},

since each F_{x̄_{i},U} factors in two ways:

\xymatrix{ & Ind(\underline{Cov}_{U_{i}}^{alg}) \ar[dr]^{F_{x̄_{i},U_{i}}^{alg}} \\ _{\underline{Cov}_{X,U}} \ar[ur] \ar[dr] &  & \underline{Sets}. \\  & _{Ind(\underline{Cov}{alg \atop U_{i+1}})} \ar@{-->}[ur]_{F_{x̄_{i},U_{i+1}}^{alg}}}

We put

K_{U} ⊂ Isom(F_{x̄,U}, F_{x̄′,U})

equal to all possible compositions α_{m−1} ∘ … ∘ α_{0} of choices α_{i} as described above.

Note that the set Isom(F_{x̄,U}, F_{x̄′,U}) has a natural Hausdorff topology which makes it into a principal homogeneous space under the topological group Aut(F_{x̄,U}). (Topologies defined as before.) The essential remark is that the subspaces K_{U} are (nonempty) compact. This follows from the fact that what we just defined was a continuous map

Isom(F_{x̄_{0},U_{1}}^{alg}, F_{x̄_{1},U_{1}}^{alg}) × … × Isom(F_{x̄_{m−1},U_{m}}^{alg}, F_{x̄_{m},U_{m}}^{alg}) ⟶ Isom(F_{x̄,U}, F_{x̄′,U})

of a compact topological space onto K_{U}. (Each of the spaces occuring on the left is homogeneous under a profinite group.) Finally, if U′ refines U the K_{U′} maps into K_{U} under the natural map

Isom(F_{x̄,U′}, F_{x̄′,U′}) ⟶ Isom(F_{x̄,U}, F_{x̄′,U}).

Modulo the usual arguments, this boils down to the fact that the subspace K_{U} is independent of the choice of the points x_{i} in the intersections U_{i} ∩ U_{i+1} ∩ ℓ. To see this, for another set of choises x′_{i}, choose paths

γ_{i} : F_{x̄_{i},U_{i}∩U_{i+1}}^{alg} ⟶ F_{x̄′_{i},U_{i}∩U_{i+1}}^{alg}.

(This is possible, x_{i} and x′_{i} are connected by a segment of ℓ in U_{i} ∩ U_{i+1}, hence lie in the same connected component.) We get α_{m−1}∘…∘α_{0} = α′_{m−1}∘…∘α′_{0} with α′_{i} = γ_{i+1}∘α_{i}∘γ_{i}^{−1} (use that γ_{i} can be viewed both as a path in U_{i} and in U_{i+1}).

The theorem follows:

∅ ≠ lim_{\overleftarrow{U}} K_{U} ⊂ lim_{\overleftarrow{U}} Isom(F_{x̄,U}, F_{x̄′,U})

since the limit of nonempty compact spaces is nonempty. □

Let us prove some general properties of these fundamental groups. Given a connected k-analytic space and a geometric point x̄, the structure map X → M(k) gives a homomorphism of fundamental groups.

2.12 Proposition. If X is geometrically connected then the map

π_{1}(X, x̄) ⟶ π_{1}(M(k), x̄) ≅ Gal(k^{sep}/k)

is surjective; for general X the image of this map is open.

Proof. The fact that X is geometrically connected means that Gal(k^{sep}/k) is a quotient of π_{1}^{alg}(X, x̄). Hence Theorem 2.10 implies that the map has dense image in this case. Thus it suffices to prove the image is open in the general case. This assertion does not depend on the choice of the base point x of X. Hence, by [B2, Lemma 7.3.3], we may choose x such that the image of π_{1}(x, x̄) → Gal(k^{sep}/k) has finite index. This map factors through π_{1}(X, x̄), thus the result is clear. □

We write ℂ for a completion of a separable closure of k; note that ℂ is algebraically closed. Further, we choose a geometric point x̄ of the ℂ-analytic space X_{ℂ}. Note that x̄ can also be considered as a geometric point of X using the morphism X_{ℂ} → X of An_{k}.

2.13 Proposition. If X is geometrically connected the sequence

π_{1}^{alg}(X_{ℂ}, x̄) ⟶ π_{1}^{alg}(X, x̄) ⟶ Gal(k^{sep}/k) ⟶ 1

is exact.

Proof. It is clear that the composition is trivial and that the right arrow is surjective. Let Y → X be finite étale and suppose there is a connected component Y′ ⊂ Y_{ℂ} mapping isomorphically to X_{ℂ}. We have to construct a map X_{k′} → Y for some finite separable extension k ⊂ k′. See [SGA 1, page 139]. Since by the lemma below the connected component Y′ already exists over such a k′ we are done. □

2.14 Lemma. Let X ∈ k-An.

(i) The map φ : |X_{ℂ}| → |X| is open and compact. (ii) Let X be connected. The space X_{ℂ} has finitely many connected components; there exists a finite separable field extension k ⊂ k′ such that the connected components of X_{k′} correspond to those of X_{ℂ}.

Proof. Compactness of the map X_{ℂ} → X is in [B2, page 30]. If x ∈ X_{ℂ} and x ∈ U ⊂ X_{ℂ} is an open neighbourhood, then by the argument of [B2, page 103] there exists a finite separable extenision k ⊂ k′ ⊂ ℂ and an open U′ ⊂ X_{k′} such that x ∈ U ′_{ℂ} ⊂ U. Thus openess follows as the finite étale morphism X_{k′} → X is open.

To prove (ii) we take x ∈ X as in [B2, Lemma 7.3.3]. Since φ is open any connected component of X_{ℂ} has a nontrivial intersection with the set φ^{−1}(x). This set is equal to M(H(x) ˆ⊗ℂ), which is finite by our choice of x. Thus we see that the action of Gal(k^{sep}/k) on the set of connected components of X_{ℂ} is continuous. We leave it to the reader to see that these components actually exist over the field k′ whose Galois group acts trivially on the set of components. □

2.15 Remark. The author doesn’t know whether the sequence π_{1}(X_{ℂ}, x̄) → π_{1}(X, x̄) → Gal(k^{sep}/k) is exact. However, an argument similar to the above shows that the closure of the image of the first arrow is the kernel of the second arrow.

3 The case of curves

In this section we consider a k-analytic space X, which is pure of dimension 1 and separated. Let us call such a space a curve. We put together some results on curves proved in the liturature. An important result, proved by Berkovich, is that |X| is locally simply connected.

3.1 Lemma. An affinoid curve X has a finite boundary ∂X. In general the boundary ∂X of a curve X is a discrete subset of |X|.

Proof. By [B1, 2.5.12] we may assume X strictly affinoid. Since X allows a finite morphism to a closed disc E = E(0, 1), it suffices to do the case X = E, use [B1, 2.5.8 & 2.5.13]. The boundary ∂E of E consists of the single point corresponding to the supremum norm; this can be seen by embedding E into the projective line and using [B1, 2.5.8 & 2.5.13]. The second assertion follows since the boundary of X is closed and its intersection with any affinoid is at most finite. □

3.2 Proposition. [FM] If |k^{∗}| ≠ {1}, then any irreducible compact curve is either affinoid or projective.

Proof. This result in case X is a strict k-analytic space is in [FM]. In the general case, take a finite affinoid G-covering X = ⋃ U_{i}. We can choose a field extension k ⊂ K_{r} such that all the affinoid spaces U_{i} ˆ⊗K_{r} and (U_{i} ∩ U_{j}) ⊗ K_{r} are strictly affinoid. The strictly K_{r}-analytic space X ˆ⊗K_{r} is still irreducible. Suppose it is affinoid. In this case it follows that the banach k-algebra A = Γ(X, O_{X}) is such that A ˆ⊗K_{r} is affinoid. Consequently, we get that A is affinoid and that X ≅ M(A).

If X ˆ⊗K_{r} is not affinoid then it is projective. It follows that X is closed, see [B2, 1.5.5]. By Lemma 3.5 we see that X is a strict k-analytic space. Hence we may apply [FM]. □

3.3 Remark. If |k^{∗}| = 1 the first part of the proof given above still works. To deal with the case X ˆ⊗K_{r} projective, we reduce X and normalize. (We may even reduce to the case k algebraically closed.) Take L = Ω_{X}^{1} in case the genus g of X ˆ⊗K_{r} is at least 2 or put L = (Ω_{X}^{1})^{−1} for g = 0. The graded ring R = ⨁ Γ(X, L^{⊗n}) is such that Proj(R⊗K_{r}) ≅ X ˆ⊗K_{r}, since we know that X ˆ⊗K_{r} is projective and L ˆ⊗K is ample. We deduce that X ≅ Proj(R).

For the case g = 1, the difficulty remains to find a point x ∈ X whose residue field is a finite extension of k.

3.4 Corollary. Any curve is a good k-analytic space.

Proof. Any point has a compact analytic domain as neighbourhood. This domain has finitely many components, each of which may be assumed to be affinoid by the proposition or the remark in case |k^{∗}| = 1. In this case the domain is affinoid, since its normalisation is affinoid. □

3.5 Lemma. Suppose Y is a closed k-analytic space. If the valuation of k is nontrivial then Y has a unique structure of a strictly k-analytic space.

Proof. A closed k-analytic space is good, see [B2, Section 1.5]. Unicity by [B2, 1.2.17]. Since Y → M(k) is good and closed, we may see Y as an analytic space in the sense of [B1] (use [B2, Section 1.5]) and use [B1, Proposition 3.1.2 (ii)] to conclude. □

3.6 Remark. This implies, if √|k^{∗}| ≠ {1}, that a curve X has a big open part U = X \∂X which is a strict k-analytic space. This is true as ∂U = ∅. In particular, any smooth curve, in the sense of [B2], is strict.

3.7 Proposition. [B1, LP] Any curve is paracompact.

Proof. This may be deduced from the results of [B1, Section 4]; but here is another proof. It suffices to prove the result when the valuation on k is nontrivial: consider the surjective compact map |X ˆ⊗K| → |X| and use [En, 5.1.33]. We may even assume that |k^{∗}| = ℝ_{+}^{∗} by taking K to be a complete valued field over k with |K^{∗}| = ℝ_{+}^{∗}. Clearly, this implies that X is strictly k-analytic, so that we may apply [LP]. □

3.8 Proposition. [B1] The space |X| is locally simply connected in the sense of coverings.

Proof. This means that any point x ∈ X has a neighbourhood U such that π_{1}^{top}(U, x) = {1}. By Lemma 3.4 we may assume X affinoid. The result follows from [B1, 4.3.3] if the valuation of k is nontrivial, from [B1, 4.3.5] if the valuation is trivial. □

3.9 Corollary. (i) The topological group π_{1}^{top}(X, x̄) is a discrete group; the homomorphism π_{1}(X, x̄) → π_{1}^{top}(X, x̄) is surjective. (ii) The kernel of π_{1}(X, x̄) → π_{1}^{top}(X, x̄) is topologically generated by profinite subgroups.

Proof. For the first statement, see Remark 2.11. For the second, we consider for all geometric points x̄′ and paths α : Fx̄ → F_{}_{x̄′} the map φ_{x̄′,α} : π_{1}(x′, x̄′) → π_{1}(X, x̄) given by γ ↦ α^{−1} ∘ γ ∘ α. The closure N of the subgroup generated by the images of the maps φ_{x̄′,α} is contained in the kernel of π_{1}(X, x̄) → π_{1}^{top}(X, x̄). To prove these subgroups are equal it suffices to show that any étale covering map f : Y → X where N acts trivially on Fx̄(Y) is a topological covering space. It is easily seen that N acts trivially if and only if for all y′ ∈ Y the field extension H(f(y′)) ⊂ H(y′) is trivial.

Take a point x′ ∈ X. There exists an open neighbourhood U of x′ such that f^{−1}(U) = ∐ V_{i}, each V_{i} → U finite étale. By the lemma below the maps V_{i} → U are topological covering maps. By the proposition above we could have chosen U to be simply connected. In this case we see that f splits over U, i.e., we see that f is a topological covering map. □

3.10 Lemma. If f : Z → Y is a finite étale morphism and all the field extensions H(f(z)) ⊂ H(z) are trivial, then f is a topological covering map.

Proof. This is easy, compare [B2, Section 6.3]. □

3.11 Remark. Another proof of the result of the proposition can be deduced from Proposition 5.3 below. As in [B1] this relies on the stable reduction theorem for algebraic curves.

4 Berkovich spaces and local systems

Let k be a non-Archimedean valued field and let X be an analytic space over k. We fix any prime number ℓ.

The most convenient way to define a local system of ℤ_{ℓ}-lattices is the following: It is an abelian sheaf F on Ét(X), which is ℓ-divisible, ℓ-power torsion and such that F[ℓ] is locally constant with finite fibres. This means that F = ⋃ F_{n}, with F_{n} = F[ℓ^{n}], each F_{n} is a finite locally free sheaf of ℤ/ℓ^{n}ℤ-modules on Ét(X). It is clear what morphisms of such local systems are. Let us denote ℤ_{ℓ}-\underline{Loc}_{X} the ℤ_{ℓ}-linear category so obtained. Note that this category has a tensor product and internal Hom satisfying the usual axioms. For any morphism Y → X in An_{k} there are pullback functors ℤ_{ℓ}-\underline{Loc}_{X} → ℤ_{ℓ}-\underline{Loc}_{Y}.

Suppose x̄ is a geometric point of X. We define the stalk Fx̄ of F at x̄ to be

Fx̄ = lim_{\overleftarrow{n}} F_{n,x̄},

where the transition maps are given by multiplication by ℓ. This is a finite free ℤ_{ℓ}-module. The rank of this module is a locally constant function on |X|. Further, it is clear that the ℤ_{ℓ}-module Fx̄ has a continuous π_{1}^{alg}(X, x̄)-action. In fact if X is connected this gives an equivalence of ℤ_{ℓ}-\underline{Loc}_{X} with the category of such π_{1}^{alg}(X, x̄)-modules. This equivalence is compatible with tensor product and internal Hom. By Theorem 2.10 we may also replace π_{1}^{alg}(X, x̄) by the topological group π_{1}(X, x̄).

Suppose that we have a site C and a category p : S → C fibred in ℤ_{ℓ}-linear categories over C. There is an obvious category S ⊗ ℚ_{ℓ} fibred in ℚ_{ℓ}-linear categories over C. The objects of the categories are the same, for objects A, B in the fibre category S_{x} = p^{−1}(x), x ∈ C we have

Hom_{(S⊗ℚ_{ℓ})_{x}}(A, B) = Hom_{S_{x}}(A, B) ⊗_{ℤ_{ℓ}} ℚ_{ℓ}.

Even if S is a stack, the category S ⊗ ℚ_{ℓ} is not a stack in general. The stack associated to S ⊗ ℚ_{ℓ} is denoted S_{ℚ}. Note that if S has a tensor product and internal Hom then so does S_{ℚ}.

We note that ℤ_{ℓ}-\underline{Loc} is a stack over the category of k-analytic spaces with the étale topology. In this case we write ℚ_{ℓ}-\underline{Loc} for the associated stack in ℚ_{ℓ}-linear categories.

4.1 Definition. A local system of ℚ_{ℓ}-vectorspaces V over X is an object of ℚ_{ℓ}-\underline{Loc}_{X}, where ℚ_{ℓ}-\underline{Loc} denotes the stack associated to the fibred category ℤ_{ℓ}-\underline{Loc} ⊗ ℚ_{ℓ}.

By going through the definitions, we see that such a local system V is given by the following data

V = ({U_{i} → X}, F_{i}, ϕ_{ij})

where – {U_{i} → X} is an étale covering of X,

– for each i there is given an object F_{i} ∈ ℤ_{ℓ}-\underline{Loc}_{U_{i}},

– for each pair i, j there is given an isomorphism ϕ_{ij} : F_{i}|_{U_{i}×_{X}U_{j}} → F_{j}|_{U_{i}×_{X}U_{j}} in the

fibre category of ℤ_{ℓ}-\underline{Loc} ⊗ ℚ_{ℓ} over U_{i} ×_{X} U_{j}. These data are subject to the cocycle condition pr_{ij}^{∗}(ϕ_{ij}) ∘ pr_{jk}^{∗}(ϕ_{jk}) = pr_{ik}^{∗}(ϕ_{ik}) on the triple product U_{i} ×_{X} U_{j} ×_{X} U_{k}. A refinement of the covering {U_{i} → X} gives by definition an isomorphic object. Thus morphisms V → V′ need only be defined for systems given over the same covering {U_{i} → X}. In this case (after possibly refining the covering {U_{i} → X} further) such a morphism is defined by a collection of morphisms ϕ_{i} : F_{i} → F′_{i} from ℤ_{ℓ}-\underline{Loc} ⊗ ℚ_{ℓ} over U_{i} satisfying ϕ′_{ij} ∘ pr_{i}^{∗}(ϕ_{i}) = pr_{j}^{∗}(ϕ_{j}) ∘ ϕ_{ij}.

This presentation of objects also indicates how to define tensor products and internal Hom. It is clear that ℚ_{ℓ}-\underline{Loc}_{X} is a ℚ_{ℓ}-linear abelian tensor category with internal Hom. If X is connected then ℚ_{ℓ}-\underline{Loc}_{X} is also rigid. The functor ℤ_{ℓ}-\underline{Loc}_{X} → ℚ_{ℓ}-\underline{Loc}_{X} is denoted F ↦ F ⊗ ℚ_{ℓ}; it is compatible with tensor product and internal Hom. Remark that a morphism ϕ : F_{1} ⊗ℚ_{ℓ} → F_{2} ⊗ℚ_{ℓ} induces a morphism on stalks ϕx̄ : F_{1,x̄} ⊗ℚ_{ℓ} → F_{2,x̄} ⊗ℚ_{ℓ} (just choose any geometric point lying over x̄ on one of the U_{i} and consider the action of ϕ_{i} on the stalks at this point). By definition a quasi-isogeny of F_{1} to F_{2} is an isomorphism F_{1} ⊗ ℚ_{ℓ} → F_{2} ⊗ ℚ_{ℓ}.

Let G be a topological group. We write \underline{Rep}_{ℚ_{ℓ}}(G) for the category of continuous representations of G in finite dimensional ℚ_{ℓ}-vectorspaces. This is a ℚ_{ℓ}-linear tensor category. There is an obvious faithfull exact functor ω : \underline{Rep}_{ℚ_{ℓ}}(G) → \underline{Vect}_{ℚ_{ℓ}}.

4.2 Theorem. For any geometric point x̄ of X there is a natural ℚ_{ℓ}-linear tensor functor

ωx̄ : ℚ_{ℓ}-\underline{Loc}_{X} ⟶ \underline{Rep}_{ℚ_{ℓ}}(π_{1}(X, x̄)).

If X is connected then ωx̄ is an equivalence.

Proof. We consider the sheaf G of lattices on V ∈ ℚ_{ℓ}-\underline{Loc}_{X}. It is defined as follows: a section of G over U → X étale is given by a pair (F, φ) up to isomorphism, where (i) F is an object of ℤ_{ℓ}-\underline{Loc}_{U} and (ii) φ is an isomorphism F ⊗ ℚ_{ℓ} → V|_{U}. We leave it to the reader to see that G is indeed a sheaf (use that ℤ_{ℓ}-\underline{Loc} is a stack). If V is given by the triple V = ({U_{i} → X}, F_{i}, ϕ_{ij}) then it follows immediately that G|_{U_{i}} is representable by a disjoint union of spaces finite étale over U_{i}. Hence Lemma 2.3 implies that G is representable by an étale covering space Y over X.

Consider the universal object (F, φ) over Y. Since V lives over X we get a quasiisogeny

ϕ : pr_{1}^{∗}(F) ⊗ ℚ_{ℓ} ⟶ pr_{2}^{∗}(F) ⊗ ℚ_{ℓ}

over Y ×_{X} Y satisying the cocycle relation over Y ×_{X} Y ×_{X} Y. The morphisms F_{n} → Y are finite étale, hence F_{n} may be considered as an étale covering space of X also. Consider the set

W = lim_{\overleftarrow{n}} Fx̄(F_{n}).

It has a prodiscrete topology and a continuous π_{1}(X, x̄)-action. There is a map W → Fx̄(Y) compatible with the π_{1}(X, x̄)-action. Further, there is a natural group structure on W over Fx̄(Y), given by the fact that the fibre of W → Fx̄(Y) at ȳ is the stalk Fȳ of F at ȳ. Let us write w_{1} ∼ w_{2} for w_{i} ∈ W if and only if ϕ_{y_{1}y_{2}}(w_{1}) = w_{2}, where y_{i} ∈ Fx̄(Y) is the image of w_{i} and ϕ_{y_{1}y_{2}} : F_{y_{1}} ⊗ ℚ_{ℓ} → F_{y_{2}} ⊗ ℚ_{ℓ} is the action of ϕ on the stalks at the point (y_{1}, y_{2}) of Y ×_{X} Y. This is an equivalence relation as follows immediately from the cocycle condition. We note that ∼ is compatible with the π_{1}(X, x̄)-action and hence that the topological space

ωx̄(V) = W/ ∼

is a continuous π_{1}(X, x̄)-set.

Let us show that for any pair w_{1}, w_{2} ∈ W there are equivalent w′_{1}, w′_{2} such that w′_{1} and w′_{2} are in the same fibre Fȳ. To do this, note that for some n >> 0 we have ℓ^{n}ϕ_{y_{1}y_{2}}(w_{1}) ∈ F_{y_{2}} (a priori this lies in F_{y_{2}} ⊗ ℚ_{ℓ}). We take ȳ ∈ Fx̄(Y) corresponding to the lattice ℓ^{−n}F_{y_{2}}. Clearly, w′_{1} = ϕ_{y_{1}y}(w_{1}) and w′_{2} = ϕ_{y_{2}y}(w_{2}) work. This means that we can define an addition on ωx̄(V), putting [w_{1}] + [w_{2}] = [w′_{1} + w′_{2}] and a scalar multiplication, putting λ · [w_{1}] = [λw_{1}], for λ ∈ ℤ_{ℓ}. We leave it to the reader to check this is well defined and that the ℤ_{ℓ}-module so defined is in fact ℓ-divisible, hence a ℚ_{ℓ}-vectorspace. Finally, the action of π_{1}(X, x̄) preserves this vectorspace structure and is continuous. This defines our functor ωx̄. We leave it to the reader to check it is compatible with tensor products and internal Hom.

Remark that for any ȳ ∈ Fx̄(Y) there is a natural isomorphism Fȳ ⊗ ℚ_{ℓ} ≅ ωȳ(V). Using this and internal Hom one may check that ωx̄ is fully faithfull, if X is connected.

Let us show that ωx̄ is essentially surjective. Suppose that

ρ : π_{1}(X, x̄) ⟶ GL_{n}(ℚ_{ℓ})

is a continuous homomorphism. This gives a continous action on the (discrete) set of lattices GL_{n}(ℚ_{ℓ})/GL_{n}(ℤ_{ℓ}). An orbit O of this action corresponds to an étale covering map f : Y → X with Fx̄(Y) = O, see Theorem 2.10. Take ȳ ∈ Fx̄(Y) corresponding to the lattice Λ(ȳ) ∈ ℚ_{ℓ}^{n}. The action of π_{1}(Y, ȳ) on ℚ_{ℓ}^{n} stabilizes Λ(ȳ), hence we get a local system of ℤ_{ℓ}-lattices F(ȳ) over Y associated to this action. First let us show that F(ȳ) is independent of the choice of ȳ. For another point ȳ′ lying over x̄ we choose a path α : Fȳ → F_{}_{ȳ′}, this determines an element γ = f(α) in π_{1}(X, x̄) such that γȳ = ȳ′, i.e., ρ(γ)Λ(ȳ) = Λ(ȳ′). Thus we get F(ȳ)ȳ ≅ F(ȳ)_{}_{ȳ′} via α and Λ(ȳ) ≅ Λ(ȳ′) via ρ(γ). Together these define an isomorphism F(ȳ)_{}_{ȳ′} ≅ F(ȳ′)_{}_{ȳ′} compatible with π_{1}(Y, ȳ′)-actions. This gives the desired isomorphism F(ȳ) ≅ F(ȳ′).

This means that F = F(ȳ) is a local system of ℤ_{ℓ}-lattices such that at each point ȳ′ the fibre F_{}_{ȳ′} ⊗ ℚ_{ℓ} is identified with ℚ_{ℓ}^{n} with π_{1}(Y, ȳ′) action given by the composite of ρ and π_{1}(Y, ȳ′) → π_{1}(X, x̄). The existence of a quasi-isogeny over Y ×_{X} Y satisfying the cocycle condition follows formally from this. Thus we get V ∈ ℚ_{ℓ}-\underline{Loc}_{X} as desired. □

4.3 Remark. Suppose that x̄′ is a second geometric point of the connected analytic space X. The equivalence of categories \underline{Rep}_{ℚ_{ℓ}}(π_{1}(X, x̄)) ≅ \underline{Rep}_{ℚ_{ℓ}}(π_{1}(X, x̄′)) gotten from applying the theorem twice is isomorphic to the equivalence one gets from the isomorphism π_{1}(X, x̄) ≅ π_{1}(X, x̄′) one gets by choosing a path from x̄ to x̄′.

4.4 Corollary. A local system of ℚ_{ℓ}-vectorspaces V can always be given as V = ({U_{i} → X}, F_{i}, ϕ_{ij}), where the maps U_{i} → X are open immersions.

Proof. This means that for any point x of X we can find a neighbourhood x ∈ U ⊂ X and a lattice of V over U. Let Y denote the space of lattices in V as in the proof of the theorem above. We have to prove that there exists a point y ∈ Y with H(x) = H(y). This means exactly that there exists a lattice Λ ⊂ ωx̄(V) stable for the action of π_{1}(x, x̄). Such a lattice exists since π_{1}(x, x̄) is a compact group. □

5 Étale coverings of rigid analytic spaces

Here we just remark how to transpose the definitions and results above to the category of rigid analytic spaces. In this section the field k has a nontrivial valuation. For simplicity we only work with quasi-separated rigid spaces. We use the terminology developed in [JP].

Let us define when a morphism of rigid analytic spaces f : Y → X is an étale covering space. If X is affinoid, we demand that Y be paracompact and that the associated morphism of Berkovich spaces (see [B2, 1.6.1]) is an étale covering space. This can be spelled out as follows: for any analytic point a of our affinoid X there exists a wide affinoid neighbourhood U of a such that Y ×_{X} U is a disjoint union of affinoids finite étale over U. If X is arbitrary, we demand that any base change Y ×_{X} X′ → X′ with X′ affinoid be an étale covering space. Similar definition for topological covering spaces.

We note that if the space X comes from a Hausdorff strictly k-analytic space X_{B} as in [B2, Section] then the categories of étale covering spaces of X and X_{B} coincide. (Use that if Y_{B} → X_{B} is an étale covering space of X_{B} in k-An then Y_{B} has a unique structure of a strictly k-analytic space such that Y_{B} → X_{B} in an arrow of st-k-An.)

For any geometric point (called étale point in [JP]) x̄ of the rigid space X there is a fibre functor and fundamental groups π_{1}(X, x̄), π_{1}^{top}(X, x̄) and π_{1}^{alg}(X, x̄). The results of Theorems 2.9 and 2.10 hold in this situation also.

Instead of the definition of étale covering map given above we could have considered the following (seemingly more natural) condition on the morphism f : Y → X: there exists an admissible affinoid covering X = ⋃ U_{i} such that each f^{−1}(U_{i}) is a disjoint union of affinoids finite étale over U_{i}. Let us call these maps almost étale covering maps for the moment. The author has not been able to show that any almost étale covering space is an étale covering space (nor does he beleive strongly that this should be the case). Going through the usual arguments (compare [SGA4, Exposé IV Exercice 2.7.5]), one sees that the fundamental group associated to this category is a strict pro-system of topological groups, but it is not clear that this system is ”realized” by a topological group.

Let us define ℤ_{ℓ}-\underline{Loc}_{X} as in the case of Berkovich spaces, but using the étale topology on X as defined in [JP] (or [SS]). Let us define ℚ_{ℓ}-\underline{Loc} as the stack associated to ℤ_{ℓ}-\underline{Loc}⊗ℚ_{ℓ} using this same topology. In particular, objects of ℚ_{ℓ}-\underline{Loc}_{X} are given by triples V = (\{U_{i} → X}, F_{i}, ϕ_{ij}) exactly as before, but here {U_{i} → X} is a étale covering of X in the rigid étale topology. In particular any admissible affinoid covering of X will do. In view of the above it is interesting to note that this does give the right concept.

5.1 Proposition. If the rigid analytic variety X comes from a paracompact strictly kanalytic space X_{B}, then there is a natural equivalence of categories ℚ_{ℓ}-\underline{Loc}_{X_{B}} → ℚ_{ℓ}-\underline{Loc}_{X}.

Proof. It is clear that there is a natural functor ℚ_{ℓ}-\underline{Loc}_{X_{B}} → ℚ_{ℓ}-\underline{Loc}_{X} which is fully faithfull, by [B2, 1.6.1]. To show that an object V ∈ ℚ_{ℓ}-\underline{Loc}_{X} is in the essential image of this functor we have to show that for any x ∈ X_{B} there is an open neighbourhood U ⊂ X_{B} such that V has a lattice over the admissible open subset U_{0} of X = (X_{B})_{0}.

We define the sheaf of lattices in V exactly as in the proof of 4.2. The same proof works to show that this is representable by an almost étale covering map f : Y → X. The same argument as in the proof of Corollary 4.4 gives an analytic point y of Y such that H(x) = H(y). By definition of almost étale covering spaces we can find finitely many affinoids U_{1}, …, U_{n} ⊂ X_{B} with x ∈ U_{i} such that ⋃ U_{i} is a neighbourhood of x and such that f^{−1}(U_{i,0}) is a disjoint union of affinoids finite étale over U_{i,0}. By shrinking U_{i} we may assume that the connected component V_{i,0} of f^{−1}(U_{i,0}) containing y maps isomorphically onto U_{i,0} (compare [JP]). At this point it is clear that ⋃ V_{i,0} ⊂ Y maps isomorphically to ⋃ U_{i,0} (use that f is separated). Thus we may take U to be any open neighbourhood of x contained in ⋃ U_{i}. □

5.2 Corollary. If X is a connected rigid analytic space then ℚ_{ℓ}-\underline{Loc}_{X} is equivalent to the category \underline{Rep}_{ℚ_{ℓ}}(π_{1}(X, x̄)).

Proof. In the case that X is paracompact, this is clear from the above. In the general case the result follows by gluing. □

Let us consider a connected rigid analytic curve X, i.e., X is a separated purely one dimensional rigid analytic space. By [LP] this space has a paracompact formal model 𝔛, i.e., a separated admissible formal scheme 𝔛 over Spf(k^{∘}) which has a locally finite covering by affinoids. If the base field k is algebraically closed and X is smooth we may even assume that 𝔛 is semi-stable, see [LP]. This means that 𝔛 locally looks like an open subset of the completion of a semi-stable curve C → Spec(k^{∘}). In particular, the special fibre 𝔛_{s} is a semi-stable curve over Spec(k^{∘}/k^{∘∘}), which may have infinitely many components. Let Γ denote the graph associated to this semi-stable curve 𝔛_{s}. Here is a result which is contained in [B1, Section 4.3]; we give a ‘classical’ proof to indicate how to prove results like Proposition 3.9 using the methods of formal and rigid geometry.

5.3 Proposition. In the situation described above there is a natural isomorphism π_{1}^{top}(X, x̄) ≅ π_{1}(Γ, sp(x)).

Proof. The category of topological covering spaces of X is equal to the direct limit of the categories of topological covering spaces of 𝔛′_{s} where 𝔛′ runs through the semi-stable models of X. This follows as any admissible affinoid covering of X may be refined by a locally finite one and subsequently refined by a pure covering coming from a semi-stable blowing up 𝔛′ of 𝔛 (this may involve blowing in an ideal with infinite support). If a topological covering space Y is split by a pure covering coming from the model 𝔛′ then Y has a formal model 𝔜′ mapping to 𝔛′ such that 𝔜′_{s} → 𝔛′_{s} is a topological covering map. Of course these correspond bijectively to coverings of the graph.

However, the blowing up 𝔛′ → 𝔛 does not change the homotopy type of the graphs associated to the special fibre. Indeed, the blowing up 𝔛′ → 𝔛 can be factored into a sequence of simple blowing ups of points. Thus, the graph Γ′ can be obtained from Γ by subsequently adding a vertex and connecting this by one edge to Γ (blowing up of a smooth point of 𝔛_{s}) or replacing one edge by two edges joined in one point (blowing up of one of the double points of 𝔛_{s}). □

6 Filtered F-crystals, p-divisible groups and local systems

In this section we work with a complete discrete valuation ring O of mixed characteristic (0, p). We assume the residue field k = O/πO is perfect. The quotient field of O is denoted K. We introduce three categories:

– Sch_{k} is the category of schemes of finite type over k,

– FS_{O} is the category of formal schemes of finite type over Spf(O),

– Rig_{K} denotes the category of quasi-separated rigid analytic spaces over K. Each of these we consider as a site with the rigid topology on Rig_{K} and the Zariski topology on both FS_{O} and Sch_{k}.

We are going to discuss a construction which associates to a stack over Sch_{k} (resp. FS_{O}) a stack over FS_{O} (resp. Rig_{K}). For example suppose p : S → Sch_{k} is a stack. We define p^{for} : S^{for} → FS_{O} by putting

(p^{for})^{−1}(𝔛) = p^{−1}(𝔛_{red}).

The reader checks immediately that S^{for} is a stack; actually S^{for} is the pushforward (see [Gi]) of S under the morphism of sites Sch_{k} → FS_{O} given by the functor 𝔛 ↦ 𝔛_{red}. (Another possibility is to consider the functor 𝔛 ↦ Spec(O_{𝔛}/pO_{𝔛}).)

The second construction. Take a stack p : S → FS_{O}. We want to construct a stack p^{rig} : S^{rig} → Rig_{K}. Suppose X ∈ Rig_{K} is quasi-compact. In this case we define

(p^{rig})^{−1}(X) = lim_{\overrightarrow{𝔛}} p^{−1}(𝔛).

The limit is taken over the category of models of X, see [BL]. It suffices to consider Oflat formal schemes. It follows from the result [BL, 4.1] that this defines a stack over QCRig_{K}, the full subcategory of Rig_{K} of quasi-compact rigid spaces. Any stack over QCRig_{K} extends canonically to a stack over Rig_{K}, since we consider only quasi-separated rigid varieties. Thus we get

p^{rig} : S^{rig} ⟶ Rig_{K}

associated to S. We remark that both constructions are functorial with respect to morphisms of stacks.

Notation: if S → Sch_{k} is a stack, we write S^{rig} in stead of (S^{for})^{rig}.

6.1 Examples. (i) Take S = Isoc to be the stack of convergent isocrystals over Sch_{k}. This means that S_{T} is the category of convergent isocrystals over T. See [B]. Per convention the associated stack over Rig_{K} is denoted Isoc^{rig}. The same can be done with F -isoc, the stack of convergent F -isocrystals over Sch_{k}. (ii) Take S = BT to be the stack over FS_{O} whose category of sections over 𝔛 ∈ FS_{O} is the category of p-divisible groups over 𝔛 (see [dJ]). There results the stack of p-divisible goups BT^{rig} over Rig_{K}. □

Note that there is an obvious morphism of stacks BT^{rig} → ℤ_{p}-\underline{Loc}. This associates to a p-divisible group its Tate-module. Notation G ↦ T_{p}(G). Recall that BT_{ℚ} is the ℚ_{p}-linear stack associated to the ℤ_{p}-linear stack BT. See Section 4 for notations. The functor T_{p} transforms quasi-isogenies into quasi-isogenies, hence it extends to a functor T_{p} : (BT_{ℚ})^{rig} → ℚ_{p}-\underline{Loc}.

6.2 Proposition. Let X be an object of Rig_{K}.

(i) There is a natural equivalence (BT_{ℚ})^{rig} = (BT^{rig})_{ℚ}. The common value is denoted

BTrig_{ℚ}.

(ii) The functor T_{p} : BT_{X}^{rig} → ℤ_{p}-\underline{Loc}_{X} is faithfull. It is fully faithfull if X is reduced. The same for T_{p} : BTrig_{ℚ} → ℚ_{p}-\underline{Loc}. (iii) Both BT^{rig} and BTrig_{ℚ} are stacks for the étale topology of Rig_{K}, see [SS, JP]. (iv) Any object F ∈ ℤ_{p}-\underline{Loc}_{X} such that F ⊗ ℚ_{p} is in the essential image of BT_{ℚ,X}^{rig} → ℚ_{p}-\underline{Loc}_{X} is isomorphic to T_{p}(G) for some G ∈ BT_{X}^{rig}.

Proof. By definition of the procedure S ↦ S^{rig} it suffices to work in the categories FS_{O} and QCRig_{K}. In the rest of the proof all rigid spaces will be quasi-compact and quasiseparated, all coverings will be finite.

To prove (i) we compare objects. An object of (BT^{rig})_{ℚ,X} is given by a triple (X = ⋃ U_{i}, G_{i}, ϕ_{ij}) where X = ⋃ U_{i} is a finite covering, G_{i} ∈ BT_{U_{i}}^{rig} and ϕ_{ij} is an arrow of BT^{rig} ⊗ ℚ_{p} over U_{i} ∩ U_{j} (compare with Section 4). This means that G_{i} is given by a p-divisible group over a formal model 𝔘_{i} of U_{i} and ϕ_{ij} is given over a formal model 𝔘_{ij} of U_{i} ∩ U_{j} dominating both 𝔘_{i} and 𝔘_{j}. The cocycle condition is fullfilled on a formal model 𝔘_{ijk} of U_{ijk}. By the lemma below we may assume that the formal schemes 𝔘_{i}, 𝔘_{ij} and 𝔘_{ijk} are formal open subschemes of the model 𝔛 of X. Thus we get an object of (BT_{ℚ})_{X}^{rig}. We leave it to the reader to treat morphisms.

Let us prove the second statement. The functor on BT_{X}^{rig} given by restriction to the reduction of X is faithfull, hence it suffices to do the case of reduced X.

Let X = Sp(A) be a reduced affinoid and let A^{∘} ⊂ A denote the subring of power bounded elements. Let G, H be p-divisible groups over the formal model 𝔛 = Spf(A^{∘}). Thus A = A^{∘}[1/p]. We consider the map

Hom(G, H) ⟶ Hom(G_{A}, H_{A}).

Working through the definitions we see that in order to prove (ii) we have to show that this map is a bijection. Let A′ be the normalization of the ring A; it is a Tate algebra. We note that (A′)^{∘} is normal; so that by Tate’s theorem [T] we have the result for X′ = Sp(A′). The result for A follows from the fact that A^{∘} = A ∩ (A′)^{∘}.

We come to the third statement. We only indicate how to deal with étale descent for objects. An étale descent datum over X for an object of BT^{rig} is typically given by a triple (U, G, ϕ), where U → X is a surjective étale morphism of QCRig_{K}, G is a p-divisible group given over a model 𝔘 of U and ϕ is an isomorphism between the two pullbacks of G over a formal model 𝔘_{2} of U ×_{X} U. The cocycle condition is verified over a formal model 𝔘_{3} of U ×_{X} U ×_{X} U. By the lemma below we may assume the model 𝔘 to be a flat quasi-finite covering of a model 𝔛 of X. Again by the lemma, we may assume that 𝔘_{2} is actually equal to 𝔘 ×_{𝔛} 𝔘 and similar for 𝔘_{3}. This means that we get descent data on G with respect to the flat quasi-finite covering 𝔘 → 𝔛, which is effective.

The case of BTrig_{ℚ} is a little more tricky. Again the descent datum is given by a triple (U, G, ϕ) as above. In this case G is an object of BT_{ℚ,U}^{rig}, hence itself given by a triple (U = ⋃ U_{i}, G_{i}, ψ_{ij}). The morphism ϕ is given over a covering U ×_{X} U = ⋃ U_{ijα}, where U_{i} ×_{X} U_{j} = ⋃ U_{ijα} is a finite covering. Over U_{ijα} we have ϕ_{ijα} = p^{−n}χ_{ijα} where χ_{ijα} is an isogeny of pr_{1}^{∗}(G_{i}) into pr_{2}^{∗}(G_{j}). The integer n may be chosen large enough so as to work independently of i, j and α. Thus it is clear that χ_{ijα} = χ_{ijβ} on U_{ijα} ∩ U_{ijβ}. This means we get in fact a quasi-isogeny ϕ_{ij} of pr_{1}^{∗}(G_{i}) into pr_{2}^{∗}(G_{j}) over U_{i} ×_{X} U_{j}. Redefining U = ∐ U_{i}, G = ∐ G_{i} over ∐ U_{i} and ϕ = ∐ ϕ_{ij}, we see that we get (U, G, ϕ) where G ∈ BT_{U}^{rig} and ϕ is a quasi-isogeny.

Next, we note that (U, T_{p}(G), T_{p}(ϕ)) gives rise to a local system of ℚ_{p}-vectorspaces V over X, by definition of ℚ_{p}-\underline{Loc}_{X}. By Corollary 4.4, there exists a covering X = ⋃ X_{i} such that V has a lattice F_{i} over each X_{i}. Over U_{i} = U ×_{X} X_{i} we have a quasi-isogeny ρ_{i} : T_{p}(G)_{U_{j}} → F_{i}, which we may suppose to be an isogeny by replacing F_{i} by p^{−m}F_{i} for some m sufficiently large.

Suppose G_{U_{i}} is given over the formal model 𝔘_{i} of U_{i}. The space parametrising finite locally free closed subgroup schemes of G[p^{n}] is represented by a formal scheme ℨ projective over 𝔘_{i}. The subgroup Ker(ρ_{i}) ⊂ T_{p}(G)[p^{n}]_{U_{i}} (here n is taken large enough) determines a morphism 𝔘_{i}^{rig} → ℨ^{rig}. This comes from a morphism 𝔘′_{i} → ℨ for some admissible blow up 𝔘′_{i} → 𝔘_{i}. Hence, the subgroup Ker(ρ_{i}) ⊂ T_{p}(G)[p^{n}]_{U_{i}} extends to a finite flat subgroup scheme N_{i} ⊂ G[p^{n}]_{𝔘′_{i}}. We conclude that there exists a morphism G_{U_{i}} → G′_{i} in BT_{U_{i}}^{rig} such that F_{i} = T_{p}(G′_{i}).

The quasi-isogeny ϕ determines a quasi-isogeny ϕ_{i} of pr_{1}^{∗}(G_{i}) into pr_{2}^{∗}(G_{i}) over U_{i} ×_{X} U_{i}. Since F_{i} = T_{p}(G′_{i}) we get that T_{p}(ϕ_{i}) is an isomorphism. We leave it to the reader to show that this implies ϕ_{i} is actually an isomorphism in BT^{rig}. The cocycle condition is verified on U_{i} ×_{X} U_{i} ×_{X} U_{i}. The descent datum (U_{i} → X_{i}, G′_{i}, ϕ_{i}) so obtained is effective by what has gone before. There results a p-divisible group G_{i} over X_{i}.

The quasi-isogeny

G_{i}|_{U_{i}∩U_{j}} ⟶ G|_{U_{i}∩U_{j}} ⟶ G_{j}|_{U_{i}∩U_{j}}

is compatible with ϕ_{i} and ϕ_{j}, hence descents to a quasi-isogeny ϕ_{ij} over X_{i} ∩ X_{j}. This gives the desired object (X = ⋃ X_{i}, G_{i}, ϕ_{ij}) of BTrig_{ℚ} over X.

The proof of the last statement is the same as the proof above that F_{i} = T_{p}(G_{i}) for some G_{i}. □

6.3 Lemma. [BL] Let f : U → X be an étale morphism of QCRig_{K}. There exists a flat (quasi-finite) formal model 𝔣 : 𝔘 → 𝔛. Any other model 𝔘′ of U is dominated by a model of the form 𝔘 ×_{𝔛} 𝔛′ for some admissible blowing up 𝔛′ → 𝔛.

Proof. The reference [BL] provides the flat formal model. To prove the last statement, by [BL], we may assume 𝔘′ → 𝔘 is an admissible blowing up. By [BL], we may find a admissible blowing up 𝔛′ → 𝔛 such that the strict transform 𝔘′′ of 𝔘′ is flat over 𝔛′. The morphism 𝔘′′ → 𝔘 ×_{𝔛} 𝔛′ is flat and a blowing up, hence an isomorphism. □

Next, we relate Isoc^{rig} to flat vectorbundles. The stack FVB of flat vectorbundles over Rig_{K} has as category of sections FVB_{X} over X the category of pairs (E, ∇) where E is a finite locally free sheaf of O_{X}-modules and ∇ : E → E ⊗ O_{X} is an integrable connection. This is a K-linear tensor category with tensor product and internal Hom.

Suppose 𝔛 ∈ FS_{O} is affine, say 𝔛 = Spf(A), with maximal ideal of definition I ⊂ A.

The associated affinoid rigid space is X = 𝔛^{rig}. We can choose a surjection

W {x_{1}, …, x_{n}} ⟶ A,

since W = W (k) ⊂ O is finite. Let B be the completion of W {x_{1}, …, x_{n}} in the preimage of the ideal I. (Note: this topological W -algebra is not of topologically finite type over W in general.) The formal scheme 𝔜 = Spf(B) is formally smooth over Spf(W). The rigid space U = 𝔜^{rig} over K_{0} (see [B, 2.4; dJ, Section 7]) is a tube for 𝔛_{red} = Spec(A/I). The surjection B → A defines a closed immersion

j : X = 𝔛^{rig} ⟶ U = 𝔜^{rig}.

Suppose that E is a convergent isocrystal on 𝔛_{red}. The realization of E on U is a finite locally free O_{U}-module E_{U} with integrable connection. The pullback E_{X} = j^{∗}(E_{U}) is a flat vectorbundle over X, i.e., an object of FVB_{X}. The construction Isoc(𝔛_{red}) → FVB_{X} just described does not depend on the choice of 𝔛 → 𝔜. Any two of these can be compared to a third, hence we need only compare in the case we have the situation 𝔛 → 𝔜 → 𝔜′. The result follows since E_{U′}|_{U} ≅ E_{U} by definition of converent isocrystals.

6.4 Proposition. The construction above induces a morphism of stacks

Isoc^{rig} ⟶ FVB.

This functor is denoted E ↦ E_{X} on E over X.

Suppose that in the affine situation 𝔛 = Spf(A) described above, we are given a pdivisible group G over 𝔛. There are two ways to get a flat vectorbundle over X. The first is to take 𝔹(G_{red})_{X}, i.e., to take the flat vectorbundle associated to the convergent F -isocrystal of G_{red} = G × 𝔛_{red}. (This is the convergent F -isocrystal associated to the Dieudonné crystal 𝔻(G_{red}) over 𝔛_{red}, see [B].) The second is to take (𝔻(G)_{𝔛})^{rig}, i.e., to take the flat vectorbundle on X associated to the value over 𝔛 of the Dieudonné crystal of G, which is a flat vector bundle 𝔻(G)_{𝔛} on 𝔛. The canonical isomorphism

𝔹(G_{red})_{X} = (𝔻(G)_{𝔛})^{rig}

is easily constructed, compare [dJ, 5.3].

Note that the value 𝔻(G)_{𝔛} over 𝔛 of the Dieudonné crystal of G is endowed with the canonical Hodge-filtration ω_{G} ⊂ 𝔻(G)_{𝔛}. This means that we may associate to G the pair (𝔹(G_{red}), ω_{G}^{rig}), consisting of a convergent F -isocrystal on X and a filtration on the associated flat vectorbundle.

6.5 Definition. A filtered F -isocrystal over X ∈ Rig_{K} is a pair (E, F^{·}) where E ∈ F -isoc_{X}^{rig} and F^{·} = F^{·}E_{X} is a decreasing filtration by locally direct summands. The stack on Rig_{K} this gives rise to is denoted FFC.

The construction above gives rise to a morphism of stacks BT^{rig} → FFC. Since this is a functor which transforms quasi-isogenies into isomorphisms we get a natural extension

(𝔹, F^{·}) : BTrig_{ℚ} ⟶ FFC.

6.6 Proposition. The functor (𝔹, F^{·}) is fully faithfull.

Proof. The question is whether the functor BT_{𝔛} → FFC_{X} is fully faithfull up to isogeny in the affine situation described above. Let us write S = 𝔛_{red} and DC_{S} for the category of Dieudonné crystals over S. We know that the functor DC_{S} → F -isoc_{S} is fully faithfull up to isogeny by [B] (see also [dJ, 5.5]). We know that the functor BT_{S} → DC_{S} is fully faithfull up to isogeny by [dJ].

Suppose given a homomorphism φ : (𝔹(G_{2,red}), ω_{G_{2}}^{rig}) → (𝔹(G_{1,red}), ω_{G_{1}}^{rig}), where G_{1}, G_{2} are p-divisible groups over 𝔛. By the above we get a morphism of p-divisible groups ψ_{red} : G_{1,red} → G_{2,red} such that 𝔹(ψ) = p^{n}φ on the isocrystals. Since φ is compatible with the filtrations, we see that some p^{m}ψ_{red} will lift (uniquely) to a homomorphism ψ : G_{1} → G_{2} over 𝔛 (the argument for this can be found in [dJ, proof of 5.7.8], see also the proof of Lemma 5.4.2 in [dJ]). □

7 Some examples of étale covering spaces

The motivating example is perhaps the logarithm map

log : D ⟶ 𝔸^{1}

from the open disc D of radius 1 around 1 over ℚ_{p} to the affine line over ℚ_{p}. This map is of course given by the usual formula log(1 + z) = z − z^{2}/2 + z^{3}/3 … Notice that it is a homomorphism of analytic groups, if D is seen as a subgroup of 𝔾_{m} and 𝔸^{1} is given the usual additive group structure.

Let E be a sufficiently small closed disc around 0 ∈ 𝔸^{1}. On E the exponential function exp is an inverse to log with image a closed disc U around 1. The other connected components of log^{−1}(E) can be described as follows: for any n ∈ ℕ there is one:

U_{n} = { u · ζ | u ∈ U, ζ a p^{n}-th primitive root of 1 }

If we replace ℚ_{p} by ℂ (the completion of the algebraic closure of ℚ_{p}), then U_{n} splits into φ(p^{n}) = (p − 1)p^{n−1} connected components. Furthermore, each of these components over ℂ maps isomorphically to E. We have shown that the inverse image log^{−1}(E) is a disjoint union of affinoids finite étale over E.

Let E_{m} = {x ∈ 𝔸^{1} | p^{m}x ∈ E}. It follows that

log^{−1}(E_{m}) = ∐_{n∈ℕ}{ x | x^{p^{m}} ∈ U_{n} }.

It is easily seen that each of the terms on the right maps finite étale onto E_{m}. Since each point of 𝔸^{1} lies in the interior of one of the E_{m} it follows that log is an étale covering map.

It is clear from the above that log_{ℂ} is a Galois covering with group ⋃_{n} μ_{p^{n}}(ℂ). We have proved the following result on the fundamental group.

7.1 Proposition. The covering log_{ℂ} gives rise to a continuous surjection of the fundamental group of 𝔸1_{ℂ} onto the discrete group ⋃_{n} μ_{p^{n}}(ℂ). The fundamental group of 𝔸^{1} surjects onto the semidirect product ⋃_{n} μ_{p^{n}}(ℚ̄_{p}) >< Gal(ℚ̄_{p}/ℚ_{p}).

The logarithm is an example of a species of rigid analytic period maps for p-divisible groups introduced by Rapoport and Zink [RZ]. In certain cases these give rise to étale covering spaces of projective homogeneous spaces. We only explain this in the case studied by Gross and Hopkin’s [HG]. We refer to the articles [RZ, HG] for proofs and references for the following statements. We use freely the terminology from Section 6.

Let X_{0} be a divisible group of dimension 1 and height h over 𝔽̄_{p}. Let W = W (𝔽̄_{p}) denote the ring of Witt vectors over 𝔽̄_{p}. Consider schemes S over W where p is locally nilpotent. We write S̄ = S × 𝔽̄_{p}. We consider pairs (X, ρ) where X is a p-divisible group over S and ρ : X_{0} × S̄ → X × S̄ is a quasi-isogeny. If we consider these pairs up to isomorphism we get a functor M which is representable by a formal scheme

M = ∐_{n∈ℤ}M^{(n)}

over Spf(W). Thus it carries a universal pair (X, ρ). The integer n corresponds to log_{p} deg(ρ) and each of the components M^{(n)} is isomorphic to Spf(W [[t_{1}, …, t_{n−1}]]).

Let K_{0} be the quotient field of W. Put V = 𝔻(X_{0})_{W} ⊗ K_{0}. This is the rational Dieudonné module of X_{0}, it is an F -isocrystal over K_{0}. The p-adic period mapping is a rigid analytic étale mapping

π : M = M^{rig} ⟶ P = ℙ(V).

Let E ∈ F -isoc_{P}^{rig} denote the constant convergent F -isocrystal over P associated to X_{0}, i.e., with value V. Let F^{·} ⊂ E_{P} = V ⊗ O_{P} be the filtration given by the fact that P is the space of lines in V. Let X^{rig} denote the object of BT_{M}^{rig} associated to X over M. The quasi-isogeny ρ determines an isomorphism of F -isocrystals 𝔹(X^{rig}) ≅ π^{∗}(E), i.e., with the constant F -isocrystal with fibre V. The morphism π is defined by the property that π^{∗}(E, F^{·}) = (𝔹, F^{·})(X^{rig}). Explicitely, if m ∈ M is a point then ρ defines an isomorphism 𝔹(X^{rig})_{m} ≅ V ⊗ κ(m). Thus the Hodge filtration ω_{X,m} ⊂ 𝔹(X^{rig})_{m} defines a point π(m) of ℙ(V) over κ(m).

7.2 Proposition. There is an object V ∈ ℚ_{p}-\underline{Loc}_{P} having the following properties.

(i) It is an object of the full subcategory BT_{ℚ,P}^{rig} ⊂ ℚ_{p}-\underline{Loc}_{P} and (𝔹, F^{·})(V) = (E, F^{·}). (ii) There is an isomorphism π^{∗}(V) ≅ X^{rig} ⊗ ℚ_{p}. (iii) The space M over P represents the sheaf of lattices in V over P, see proof of 4.2. In particular M is an étale covering space of P.

Proof. We use the following result of [HG]: there are finitely many affinoids U_{i} ⊂ M such that ⋃ π(U_{i}) = ℙ, [HG, Corollary 23.15 & Corollary 23.17]. This means that M → P is a covering for the rigid étale topology of P. Since (𝔹, F^{·}) is fully faithful by Proposition 6.6 and (E, F^{·}) lives over P, we get an étale descent datum on X^{rig} ⊗ ℚ_{p}. By Proposition 6.2 we get an object V of BTrig_{ℚ} over P with (𝔹, F^{·})(V) = (E, F^{·}). This proves (i) and (ii).

The last assertion follows from (iv) of Proposition 6.2 and the definition of M. □

According to Theorem 4.2 the local system V corresponds to a continuous homomorphism

π_{1}(P, x̄) ⟶ GL_{h}(ℚ_{p}).

Let us base change to the algebraically closed field ℂ. Note that M_{ℂ} is still the space of lattices in V_{ℂ}. Of course M_{ℂ} = ∐ M(n)_{ℂ} is a countable union of open polydiscs. The action of p on lattices by multiplying (F ↦ pF) shifts the integer n by h. This already implies that the fundamental group of P_{ℂ} is nontrivial.

7.3 Lemma. The fundamental group of P_{ℂ} does not have any abelian quotients.

Proof. We have to show that H_{ét}^{1}(P_{ℂ}, M) is zero for any abelian group M. By the usual arguments it suffices to do this when either M is a finite group or M ≅ ℤ. The case of a finite group is OK since there are no nontrivial finite étale covering spaces of P_{ℂ}. Consider the spectral sequence associated to the morphism of sites π : (P_{ℂ})_{ét} → |P_{ℂ}|. By [B2, Proposition 4.2.4], one has R^{1}π_{∗}(ℤ_{P_{ℂ}}) = 0, and therefore H_{ét}^{1}(P_{ℂ}, ℤ) = H^{1}(|P_{ℂ}|, ℤ). The latter group is zero by [B1, Theorem 6.1.5]. □

The lemma implies that the homomorphism π_{1}(P_{ℂ}, x̄) → GL_{h}(ℚ_{p}) maps into SL_{h}(ℚ_{p}). This could have been seen also by the following argument: the diagram

π_{1}(P, x̄) ⟶ GL_{h}(ℚ_{p})
↓ ↓^{det}
Gal(ℂ/K_{0}) ⟶^{χ}  ℚ_{p}^{∗}

is commutative. This is true since, by Raynaud, the Galois group acts on the determinant of the Tate module of a p-divisible group G by the character χ^{dim G}. (Here χ is the Teichmüller character.) We see from this that π_{1}(P, x̄) maps into the subgroup GL′_{h}(ℚ_{p}) ⊂ GL_{h}(ℚ_{p}) consisting of elements g having ord_{p}detg = 0.

7.4 Proposition. The continuous map π_{1}(P_{ℂ}, x̄) → SL_{h}(ℚ_{p}) constructed above is a surjection. It identifies SL_{h}(ℚ_{p}) with a topological quotient of the fundamental group of

P_{ℂ} ≅ ℙh−1_{ℂ}.

Proof. We first prove an analogous statement over the field K_{0}. Let us take a point x ∈ M and let us choose our geometric point x̄ lying over x. We may also see x̄ as a geometric point of P. We define Γ as the image of the homomorphism π_{1}(P, x̄) → GL_{h}(ℚ_{p}). We endow it with the quotient topology coming from the surjection π_{1}(P, x̄) → Γ. Hence we have a continuous injection i : Γ → GL_{h}(ℚ_{p}). We claim the set

(1)  i(Γ)\GL_{h}(ℚ_{p})/p^{ℤ} · GL_{h}(ℤ_{p})

has h elements. This follows from the remark above that the space M has h connected components modulo the action of p^{ℤ}, and the identification of M with the space of lattices in V.

Furthermore, let F be the lattice of V|_{M} over M given by Proposition 7.2. Let us take a basis of Fx̄. If we compute the monodromy of V using this basis, then we see that i^{−1}(GL_{h}(ℤ_{p})) equals the image of the composition π_{1}(M, x̄) → π_{1}(P, x̄) → GL_{h}(ℚ_{p}). In particular, we can look at the image Γ_{x} of the composition

π_{1}(x, x̄) ⟶ π_{1}(M, x̄) ⟶ GL_{h}(ℤ_{p}).

Note that since the first group is profinite, the group Γ_{x} is a closed subgroup of GL_{h}(ℤ_{p}). We claim that we can choose x ∈ M(K_{0}) such that Γ_{x} has finite index in GL_{h}(ℤ_{p}). For this it suffices that Γ_{x} is Zariski dense in GL_{h,ℚ_{p}}.

This just means that there exists a p-divisible group G over W (𝔽̄_{p}), whose special fibre is isogeneous to X_{0} and such that the Galois representation on the Tate module has the corresponding property. Such may be constructed, for example by using Fontaine’s theory: we give a filtered Dieudonné module (V, Φ, F^{1}) over K_{0} such that the ℚ_{p}-linear tensor category generated by it is isomorphic to Rep(GL_{h}). We give a sufficient criterium for (V, Φ, F^{1}) to have this property. The σ-linear map Φ has slope 1/h hence we can find a basis v_{1}, …, v_{h} of V consisting of elements such that Φ^{h}(v_{i}) = pv_{i}. (This basis is unique up to a base change give by a matrix with coefficients in ℚ_{p^{h}}.) It suffices to take F^{1} = K_{0}(v_{1} + a_{2}v_{2} + … + a_{h}v_{h}), where a_{i} ∈ K_{0} are algebraically independent over ℚ_{p}.

From all of this we conclude that i^{−1}(GL_{h}(ℤ_{p})) ⊂ Γ contains a profinite subgroup Γ_{x} which maps to a subgroup of finite index in GL_{h}(ℤ_{p}). It follows that the map i is an open and closed immersion and that the image is not discrete. We conclude that i induces an isomorphism of topological groups Γ → GL′_{h}(ℚ_{p}): use the remarks preceding the proposition, the number of elements of the set (1) and a theorem on non-discrete closed subgroups of finite covolume [Ma, page 95].

At this point we would be done if we knew that the sequence π_{1}(P_{ℂ}, x̄) → π_{1}(P, x̄) → Gal(ℂ/K_{0}) were exact, but we don’t know this (see Remark 2.15). We do know the corresponding statement for the algebraic fundamental group of M, and this is what we are going to use.

Let us define Γ_{ℂ} as the image of the map π_{1}(P_{ℂ}, x̄) → SL_{h}(ℚ_{p}), endowed with the quotient topology. There is a continuous injection i_{ℂ} of Γ_{ℂ} into SL_{h}(ℚ_{p}).

First, remark that the preceding result implies that the monodromy over M gives a surjection π_{1}(M, x̄) → GL_{h}(ℤ_{p}). This factors through π_{1}^{alg}(M, x̄), see Theorem 2.10. Let N ⊂ SL_{h}(ℤ_{p}) be the image of π_{1}^{alg}(M_{ℂ}, x̄) in the monodromy representation given by the local system of ℤ_{p}-lattices F_{ℂ} over M_{ℂ}. By Proposition 2.13 we get that N is a (closed) normal subgroup of SL_{h}(ℤ_{p}). Hence, it is either a finite subgroup acting diagonally, or it has finite index. Let M(n) ⊂ M be the connected component x̄ maps into. The first possibility would imply that all Galois representations associated to the p-divisible groups corresponding to (K_{0}-valued) points of M(n) are isomorphic as representations into PGL_{h}(ℤ_{p}). Clearly, this is absurd. Hence, we get that N has finite index. We want to show that π_{1}(M_{ℂ}, x̄) maps onto N.

Let us make a general remark about the algebraic fundamental group of the open polydisc D^{h−1} ≅ M(n)_{ℂ} over ℂ. For any closed polydisc E^{h−1} ⊂ D^{h−1} we get a continuous homomorphism π_{1}^{alg}(E^{h−1}, x̄) → π_{1}^{alg}(D^{h−1}, x̄). Furthermore, for any surjection π_{1}^{alg}(D^{h−1}, x̄) → G onto a finite group G, we can choose our closed polydisc such that the composition π_{1}^{alg}(E^{h−1}, x̄) → π_{1}^{alg}(D^{h−1}, x̄) → G is surjective still. This follows as the open polydisc is an increasing union of closed polydiscs.

Now note that there exists a finite quotient N → G having the following property: any closed subgroup N′ ⊂ N surjecting onto G is equal to N. The proof of this statement is left to the reader. Combining the above, we see that there exists a closed polydisc E ⊂ M(n)_{ℂ} whose algebraic fundamental group maps onto N. By Proposition 7.5 there exists a profinite subgroup K ⊂ π_{1}(E, x̄) mapping onto π_{1}^{alg}(E, x̄). Mapping K into π_{1}(M_{ℂ}, x̄) we get a profinite subgroup K′ ⊂ π_{1}(M_{ℂ}, x̄) which surjects onto N. Conclusion: the topological group Γ_{ℂ} contains a profinite subgroup K′′ mapping isomorphically onto a subgroup of finite index in SL_{h}(ℤ_{p}).

The rest of the argument is similar to the arguments we gave to deal with the group Γ. For example the analogue of the set (1) in this case is also finite since the connected components of M_{ℂ} agree with the connected components of M. □

In the above proof we used some results on the fundamental groups of the space X = E(0, 1)^{n}, the closed n-dimensional unit disc. We work over the complete nontrivially valued field k. The maximal point x_{max} of X is the point x_{max} ∈ X = M(k⟨T_{1}, …, T_{n}⟩) corresponding to the supremum norm | | on the algebra k⟨T_{1}, …, T_{n}⟩. We consider the homomorphism of profinite groups

α : π_{1}(x_{max}, x̄) ⟶ π_{1}^{alg}(X, x̄),

where x̄ is a geometric point lying over x_{max}. We remark that composing α with the natural map to Gal(k^{sep}/k) gives a surjection; this follows as the field extension k ⊂ H(x) is separable and k is algebraically closed in H(x). The following proposition holds more generally for any strictly affinoid X over k with nonsingular reduction over k^{∘}/k^{∘∘}.

7.5 Proposition. The map α is surjective. It follows that the canonical map π_{1}(X, x̄) → π_{1}^{alg}(X, x̄) is surjective.

Proof. We have to show that any finite étale covering f : Y → X of degree d ≥ 2 such that Y_{x_{max}} = {p_{1}, …, p_{d}} splits. This implies that f is a local isomorphism at each point p_{i}. Consider first the case n = 1. We can find an affinoid domain U ⊂ X, which is a neighbourhood of x_{max} such that f^{−1}(U) = ∐_{i=1}^{d} V_{i} each V_{i} mapping isomorphically to U. Put V = X \ U. We note that V ⊂ ℙ^{1} is open and that ℙ^{1} \ V is a closed analytic domain of ℙ^{1}. The reason for this is that U contains x_{max} and ∂(X/ℙ^{1}) = {x_{max}}. We glue Y with d copies of the space ℙ^{1} \ V via the isomorphisms V_{i} → U, see [B2, Proposition 1.3.3 b)]. The space Y′ so obtained maps to ℙ^{1}. The morphism f′ : Y′ → ℙ^{1} is finite étale, since it is true over the members of the admissible G-covering ℙ^{1} = X ∪ (ℙ^{1} \ V). Hence, the finite étale covering Y′ → ℙ^{1} comes from a finite étale covering T → Spec(k). We get the same for Y and by the remark before the proposition we see that Y is split.

In the case n ≥ 2, consider the projection pr : X → X_{1} to the first n − 1 coordinates. By the first case, f is split over the fibre X_{x_{1}} of pr at the maximal point x_{1} of X_{1}. It follows, by induction, that f is split over the fibres of the projection to the nth coordinate, and we are done. □

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Dr. A.J. de Jong Universiteit Utrecht, Mathematisch Instituut, Postbus 80.010, 3508 TA Utrecht, The Netherlands. E-mail: dejong@math.ruu.nl