The Absolute Anabelian Geometry

of Canonical Curves

Shinichi Mochizuki

Received: September 27, 2002 Revised: January 13, 2003

Abstract. In this paper, we continue our study of the issue of the extent to which a hyperbolic curve over a finite extension of the field of p-adic numbers is determined by the profinite group structure of its étale fundamental group. Our main results are that: (i) the theory of correspondences of the curve — in particular, its arithmeticity — is completely determined by its fundamental group; (ii) when the curve is a canonical lifting in the sense of “p-adic Teichmüller theory”, its isomorphism class is functorially determined by its fundamental group. Here, (i) is a consequence of a “p-adic version of the Grothendieck Conjecture for algebraic curves” proven by the author, while (ii) builds on a previous result to the effect that the logarithmic special fiber of the curve is functorially determined by its fundamental group.

2000 Mathematics Subject Classification: 14H25, 14H30 Keywords and Phrases: hyperbolic curve, étale fundamental group, anabelian, correspondences, Grothendieck Conjecture, canonical lifting, padic Teichmüller theory

Contents:

§1. Serre-Tate Canonical Liftings

§2. Arithmetic Hyperbolic Curves

§3. Hyperbolically Ordinary Canonical Liftings

Introduction

Let X_{K} be a hyperbolic curve (cf. §0 below) over a field K of characteristic 0. Denote its algebraic fundamental group by Π_{X_{K}}. Thus, we have a natural surjection

Π_{X_{K}} ↠ G_{K}

of Π_{X_{K}} onto the absolute Galois group G_{K} of K. When K is a finite extension of ℚ or ℚ_{p}, and one holds G_{K} fixed, then it is known (cf. [Tama], [Mzk6]) that one may recover the curve X_{K} in a functorial fashion from Π_{X_{K}}. This sort of result may be thought of as a “relative result” (i.e., over G_{K}).

In the present paper, we continue our study — begun in [Mzk7] — of “absolute analogues” of such relative results. Since such absolute analogues are well understood in the case where K is a finite extension of ℚ (cf. the Introduction to [Mzk7]), we concentrate on the p-adic case. In the p-adic case, it is proven in [Mzk7] (cf. [Mzk7], Theorem 2.7) — by applying the work of [Tama] and the techniques of [Mzk5]— that (if X_{K} has stable reduction, then) the “logarithmic special fiber” of X_{K} — i.e., the special fiber, equipped with its natural “log structure” (cf. [Kato]), of the “stable model” of X_{K} over the ring of integers O_{K} — may be recovered solely from the abstract profinite group Π_{X_{K}}. This result prompts the question (cf. [Mzk7], Remark 2.7.3):

What other information — e.g., the isomorphism class of X_{K} itself — can be recovered from the profinite group Π_{X_{K}}?

In this present paper, we give three partial answers to this question (cf. [Mzk7], Remark 2.7.3), all of which revolve around the central theme that:

When X_{K} is, in some sense, “canonical”, there is a tendency for substantial information concerning X_{K} — e.g., its isomorphism class — to be recoverable from Π_{X_{K}}.

Perhaps this “tendency” should not be surprising, in light of the fact that in some sense, a “canonical” curve is a curve which is “rigid”, i.e., has no moduli, hence should be “determined” by its special fiber (cf. Remark 3.6.3).

Our three partial answers are the following:

(a) The property that the Jacobian of the X_{K} be a Serre-Tate canonical lifting is determined by Π_{X_{K}} (Proposition 1.1).

(b) The theory of correspondences of X_{K} — in particular, whether or not X_{K} is “arithmetic” (cf. [Mzk3]) — is determined by Π_{X_{K}} (cf. Theorem 2.4, Corollary 2.5).

(c) The property that X_{K} be a canonical lifting in the sense of the theory of [Mzk1] (cf. also [Mzk2]) is determined by Π_{X_{K}}; moreover, in this case, the isomorphism class of X_{K} is also determined by Π_{X_{K}} (cf. Theorem 3.6).

At a technical level, (a) is entirely elementary; (b) is a formal consequence of the “p-adic version of the Grothendieck Conjecture” proven in [Mzk6], Theorem A; and (c) is derived as a consequence of the theory of [Mzk1], together with [Mzk7], Theorem 2.7.

Finally, as a consequence of (c), we conclude (cf. Corollary 3.8) that the set of points arising from curves over finite extensions of ℚ_{p} whose isomorphism classes are completely determined by Π_{X_{K}} forms a Zariski dense subset of the moduli stack over ℚ_{p}. This result (cf. Remark 3.6.2) constitutes the first application of the “p-adic Teichmüller theory” of [Mzk1], [Mzk2], to prove a hitherto unknown result that can be stated without using the terminology, concepts, or results of the theory of [Mzk1], [Mzk2]. Also, it shows that — unlike (a), (b) which only yield “useful information” concerning X_{K} in “very rare cases” — (c) may be applied to a “much larger class of X_{K}” (cf. Remarks 1.1.1, 2.5.1, 3.6.1).

Acknowledgements: I would like to thank A. Tamagawa for useful comments concerning earlier versions of this manuscript.

Section 0: Notations and Conventions

We will denote by ℕ the set of natural numbers, by which we mean the set of integers n ≥ 0. A number field is defined to be a finite extension of the field of rational numbers ℚ.

Suppose that g ≥ 0 is an integer. Then a family of curves of genus g

X → S

is defined to be a smooth, proper, geometrically connected morphism X → S whose geometric fibers are curves of genus g.

Suppose that g, r ≥ 0 are integers such that 2g−2+r > 0. We shall denote the moduli stack of r-pointed stable curves of genus g (where we assume the points to be unordered) by \overline{M}_{g,r} (cf. [DM], [Knud] for an exposition of the theory of such curves; strictly speaking, [Knud] treats the finite étale covering of \overline{M}_{g,r} determined by ordering the marked points). The open substack M_{g,r} ⊆ \overline{M}_{g,r} of smooth curves will be referred to as the moduli stack of smooth r-pointed stable curves of genus g or, alternatively, as the moduli stack of hyperbolic curves of type (g, r).

A family of hyperbolic curves of type (g, r)

X → S

is defined to be a morphism which factors X ↪ Y → S as the composite of an open immersion X ↪ Y onto the complement Y \D of a relative divisor D ⊆ Y which is finite étale over S of relative degree r, and a family Y → S of curves of genus g. One checks easily that, if S is normal, then the pair (Y, D) is unique up to canonical isomorphism. (Indeed, when S is the spectrum of a field, this fact is well-known from the elementary theory of algebraic curves. Next, we consider an arbitrary connected normal S on which a prime l is invertible (which, by Zariski localization, we may assume without loss of generality). Denote by S′ → S the finite étale covering parametrizing orderings of the marked points and trivializations of the l-torsion points of the Jacobian of Y. Note that S′ → S is independent of the choice of (Y, D), since (by the normality of S) S′ may be constructed as the normalization of S in the function field of S′ (which is independent of the choice of (Y, D) since the restriction of (Y, D) to the generic point of S has already been shown to be unique). Thus, the uniqueness of (Y, D) follows by considering the classifying morphism (associated to (Y, D)) from S′ to the finite étale covering of (M_{g,r})_{ℤ[ \frac{1}{l}]} parametrizing orderings of the marked points and trivializations of the l-torsion points of the Jacobian [since this covering is well-known to be a scheme, for l sufficiently large].)

We shall refer to Y (respectively, D; D; D) as the compactification (respectively, divisor at infinity; divisor of cusps; divisor of marked points) of X. A family of hyperbolic curves X → S is defined to be a morphism X → S such that the restriction of this morphism to each connected component of S is a family of hyperbolic curves of type (g, r) for some integers (g, r) as above.

Section 1: Serre-Tate Canonical Liftings

In this §, we observe (cf. Proposition 1.1 below) that the issue of whether or not the Jacobian of a p-adic hyperbolic curve is a Serre-Tate canonical lifting is completely determined by the abstract profinite group structure of its arithmetic profinite group.

Let p be a prime number. For i = 1, 2, let K_{i} be a finite extension of ℚ_{p}, and (X_{i})_{K_{i}} a proper hyperbolic curve over K_{i} whose associated stable curve has stable reduction over O_{K_{i}}. Denote the resulting “stable model” of (X_{i})_{K_{i}} over O_{K_{i}} by (X_{i})_{O_{K_{i}}}.

Assume that we have chosen basepoints of the (X_{i})_{K_{i}} (which thus induce basepoints of the K_{i}) and suppose that we are given an isomorphism of profinite groups Π_{(X_{1})_{K_{1}}} →^{∼} Π_{(X_{2})_{K_{2}}} , which (by [Mzk7], Lemmas 1.1.4, 1.1.5) induces a commutative diagram:

Π_{(X_{1})_{K_{1}}}  →^{∼} Π_{(X_{2})_{K_{2}}}
↓  ↓
G_{K_{1}}  →^{∼}  G_{K_{2}}

Proposition 1.1. (Group-Theoreticity of Serre-Tate Canonical Liftings) The Jacobian of (X_{1})_{K_{1}} is Serre-Tate canonical if and only if the same is true of the Jacobian of (X_{2})_{K_{2}}.

Proof. Indeed, this follows from the fact that the Jacobian of (X_{i})_{K_{i}} is a Serre-Tate canonical lifting if and only if its p-adic Tate module splits (as a G_{K_{i}}-module) into a direct sum of an unramified G_{K_{i}}-module and the Cartier dual of an unramified G_{K_{i}}-module (cf. [Mess], Chapter V: proof of Theorem 3.3, Theorem 2.3.6; [Mess], Appendix: Corollary 2.3, Proposition 2.5). ◯

Remark 1.1.1. As is shown in [DO] (cf. also [OS]), for p > 2, g ≥ 4, the Serre-Tate canonical lifting of the Jacobian of a general proper curve of genus g in characteristic p is not a Jacobian. Thus, in some sense, one expects that:

There are not so many curves to which Proposition 1.1 may be applied.

From another point of view, if there exist infinitely many Jacobians of a given genus g over finite extensions of ℚ_{p} which are Serre-Tate canonical liftings, then one expects — cf. the “André-Oort Conjecture” ([Edix], Conjecture 1.3)

— that every irreducible component of the Zariski closure of the resulting set of points in the moduli stack of principally polarized abelian varieties should be a “subvariety of Hodge type”. Moreover, one expects that the intersection of such a subvariety with the Torelli locus (i.e., locus of Jacobians) in the moduli stack of principally polarized abelian varieties should typically be “rather small”. Thus, from this point of view as well, one expects that Proposition 1.1 should not be applicable to the “overwhelming majority” of curves of genus g ≥ 2.

Section 2: Arithmetic Hyperbolic Curves

In this §, we show (cf. Theorem 2.4 below) that the theory of correspondences (cf. [Mzk3]) of a p-adic hyperbolic curve is completely determined by the abstract profinite group structure of its arithmetic profinite group. We begin by reviewing and extending the theory of [Mzk3], as it will be needed in the discussion of the present §.

Let X be a normal connected algebraic stack which is generically “schemelike” (i.e., admits an open dense algebraic substack isomorphic to a scheme). Then we shall denote by

Loc(X)

the category whose objects are (necessarily generically scheme-like) algebraic stacks Y that admit a finite étale morphism to X, and whose morphisms are finite étale morphisms of stacks Y_{1} → Y_{2} (that do not necessarily lie over X!). Note that since these stacks are generically scheme-like, it makes sense to speak of the (1-)category of such objects (so long as our morphisms are finite étale), i.e., there is no need to work with 2-categories.

Given an object Y of Loc(X), let us denote by

Loc(X)_{Y}

the category whose objects are morphisms Z → Y in Loc(X), and whose morphisms, from an object Z_{1} → Y to an object Z_{2} → Y, are the morphisms Z_{1} → Z_{2} over Y in Loc(X). Thus, by considering maximal nontrivial decompositions of the terminal object of Loc(X)_{Y} into a coproduct of nonempty objects of Loc(X)_{Y}, we conclude that the set of connected components of Y may be recovered — functorially! — from the category structure of Loc(Y). Finally, let us observe that Loc(X)_{Y} may be identified with the category

Ét(Y)

of finite étale coverings of Y (and Y -morphisms).

We would also like to consider the category

\overline{Loc}(X)

whose objects are generically scheme-like algebraic stacks which arise as finite étale quotients (in the sense of stacks!) of objects in Loc(X), and whose morphisms are finite étale morphisms of algebraic stacks. Note that \overline{Loc}(X) may be constructed entirely category-theoretically from Loc(X) by considering the “category of objects of Loc(X) equipped with a (finite étale) equivalence relation”. (We leave it to the reader to write out the routine details.)

Definition 2.1.

(i) X will be called arithmetic if \overline{Loc}(X) does not admit a terminal object.

(ii) X will be called a(n) (absolute) core if X is a terminal object in Loc(X).

(ii) X will be said to admit a(n) (absolute) core if there exists a terminal object Z in \overline{Loc}(X). In this case, \overline{Loc}(X) = \overline{Loc}(Z), so we shall say that Z is a core.

Remark 2.1.1. Let k be a field. If X is a geometrically normal, geometrically connected algebraic stack of finite type over k, then we shall write

Loc_{k}(X); \overline{Loc}_{k}(X)

for the categories obtained as above, except that we assume all the morphisms to be k-morphisms. Also, we shall say that X is k-arithmetic, or arithmetic over k (respectively, a k-core, or core over k), if \overline{Loc}_{k}(X) does not admit a terminal object (respectively, X is a terminal object in Loc_{k}(X)). On the other hand, when k is fixed, and the entire discussion “takes place over k”, then we shall often omit the “k-” from this terminology.

Remark 2.1.2. Thus, when k = ℂ, a hyperbolic curve X is k-arithmetic if and only if it is arithmetic in the sense of [Mzk3], §2. (Indeed, if X is non-arithmetic in the sense of [Mzk3], §2, then a terminal object in \overline{Loc}_{k}(X)

— i.e., a “(hyperbolic) core” — is constructed in [Mzk3], §3, so X is non-k-arithmetic. Conversely, if X is arithmetic in the sense of [Mzk3], §2, then (cf. [Mzk3], Definition 2.1, Theorem 2.5) it corresponds to a fuchsian group Γ ⊆ SL_{2}(ℝ)/{±1} which has infinite index in its commensurator C_{SL_{2}(ℝ)/{±1}}(Γ)

— a fact which precludes the existence of a k-core.) Moreover, issues over an arbitrary algebraically closed k of characteristic zero may always be resolved over ℂ, by Proposition 2.3, (ii), below.

Remark 2.1.3. If we arbitrarily choose a finite étale structure morphism to X for every object of Loc(X), then one verifies easily that every morphism of Loc(X) factors as the composite of an isomorphism (not necessarily over X!) with a (finite étale) morphism over X (i.e., relative to these arbtirary choices). A similar statement holds for Loc_{k}(X).

Definition 2.2. Let X be a smooth, geometrically connected, generically scheme-like algebraic stack of finite type over a field k of characteristic zero.

(i) We shall say that X is an orbicurve if it is of dimension 1.

(ii) We shall say that X is a hyperbolic orbicurve if it is an orbicurve which admits a compactification X ↪ \overline{X} (necessarily unique!) by a proper orbicurve \overline{X} over k such that if we denote the reduced divisor \overline{X}\X by D ⊆ \overline{X}, then \overline{X} is scheme-like near D, and, moreover, the line bundle ω_{\overline{X}/k}(D) on \overline{X} has positive degree.

Proposition 2.3. (Independence of the Base Field)

(i) Let k^{sep} be a separable closure of k; X a geometrically normal, geometrically connected algebraic stack of finite type over k. Then X is a k-core (respectively, k-arithmetic) if and only if X_{k^{sep}} =^{def} X ×_{k} k^{sep} is a k^{sep}-core (respectively, k^{sep}- arithmetic). Moreover, if X_{k^{sep}} admits a finite étale morphism X_{k^{sep}} → Z_{k^{sep}} to a k^{sep}-core Z_{k^{sep}}, then Z_{k^{sep}} descends uniquely to a k-core Z of X.

(ii) Suppose that k is algebraically closed of characteristic 0, and that X is a hyperbolic orbicurve. Next, let k′ be an algebraically closed field containing k. Then the natural functors

Loc_{k}(X) → Loc_{k′}(X ⊗_{k} k′); \overline{Loc}_{k}(X) → \overline{Loc}_{k′}(X ⊗_{k} k′)

(given by tensoring over k with k′) are equivalences of categories. In particular, X is a k-core (respectively, k-arithmetic) if and only if X ⊗_{k} k′ is a k′-core (respectively, k′-arithmetic).

Proof. First, we observe that (i) is a formal consequence of the definitions. As for (ii), let us observe first that it suffices to verify the asserted equivalences of categories. These equivalences, in turn, are formal consequences of the following two assertions (cf. Remark 2.1.3):

(a) The natural functor Ét(X) → Ét(X ⊗_{k} k′) is an equivalence of categories.

(b) If Y_{1}, Y_{2} are finite étale over X, then

Isom_{k}(Y_{1}, Y_{2}) → Isom_{k′}(Y_{1} ⊗_{k} k′, Y_{2} ⊗_{k} k′)

is bijective.

The proofs of these two assertions is an exercise in elementary algebraic geometry, involving the following well-known techniques:

(1) descending the necessary diagrams of finite étale morphisms over k′ to a subfield K ⊆ k′ which is finitely generated over k;

(2) extending orbicurves over K to orbicurves over some k-variety V with function field K;

(3) specializing orbicurves over V to closed (i.e., k-valued) points v of V;

(4) base-changing orbicurves over V to formal completions V̂_{v} of V at closed points v; (5) deforming (log) étale morphisms of orbicurves over v to morphisms over the completions V̂_{v};

(6) algebrizing such deformed morphisms (when the orbicurves involved are proper).

This “elementary exercise” is carried out (for assertion (a) above) in the case when X itself is proper in [SGA1], Exposé X, Theorem 3.8. When X is an arbitrary orbicurve as in the statement of (ii), the same arguments — centering around the rigidity of (log) étale morphisms under infinitesimal deformations — may be used, by considering compactifications (\overline{X}, D) of X as in Definition 2.2, (ii), and replacing “étale” by “étale away from D”. Note that we use the assumption that k is of characteristic zero here to ensure that all ramification is tame.

Finally, assertion (b) may be deduced by similar arguments — by applying, in (5) above, the fact (cf. Definition 2.2, (ii)) that, if \overline{Y} → \overline{X} is any finite morphism of orbicurves over k, then

H^{0}(\overline{Y} , ω_{\overline{X}/k}^{∨}(−D)|\overline{Y}) = 0

(where “∨” denotes the O\overline{X}-dual) in place of the rigidity of (log) étale morphisms used to prove assertion (a). ◯

Next, for i = 1, 2, let K_{i} be a finite extension of ℚ_{p} (where p is a prime number); let (X_{i})_{K_{i}} be a hyperbolic curve over K_{i}. Assume that we have chosen basepoints of the (X_{i})_{K_{i}}, which thus induce basepoints/algebraic closures \overline{K}_{i} of the K_{i} and determine fundamental groups Π_{(X_{i})_{K_{i}}} =^{def} π_{1}((X_{i})_{K_{i}}) and Galois

groups G_{K_{i}} =^{def} Gal(\overline{K}_{i}/K_{i}). Thus, for i = 1, 2, we have an exact sequence:

1 → Δ_{X_{i}} → Π_{(X_{i})_{K_{i}}} → G_{K_{i}} → 1

(where Δ_{X_{i}} ⊆ Π_{(X_{i})_{K_{i}}} is defined so as to make the sequence exact). Here, we shall think of G_{K_{i}} as a quotient of Π_{(X_{i})_{K_{i}}} (i.e., not as an independent group to which Π_{(X_{i})_{K_{i}}} happens to surject). By [Mzk7], Lemmas 1.1.4, 1.1.5, this quotient is characteristic, i.e., it is completely determined by the structure of Π_{(X_{i})_{K_{i}}} as a profinite group.

Theorem 2.4. (Group-Theoreticity of Correspondences) Any isomorphism α : Π_{(X_{1})_{K_{1}}} →^{∼} Π_{(X_{2})_{K_{2}}} induces equivalences of categories:

Loc_{\overline{K}_{1}}((X_{1})_{\overline{K}_{1}}) →^{∼} Loc_{\overline{K}_{2}}((X_{2})_{\overline{K}_{2}}); \overline{Loc}_{\overline{K}_{1}}((X_{1})_{\overline{K}_{1}}) →^{∼} \overline{Loc}_{\overline{K}_{2}}((X_{2})_{\overline{K}_{2}}) in a fashion that is functorial in α.

Proof. Since \overline{Loc}_{\overline{K}_{i}}((X_{i})_{\overline{K}_{i}}) may be reconstructed “category-theoretically” from Loc_{\overline{K}_{i}}((X_{i})_{\overline{K}_{i}}) (cf. the discussion at the beginning of the present §), in order to prove Theorem 2.4, it thus suffices to show that the isomorphism α induces an equivalence between the categories Loc_{\overline{K}_{i}}((X_{i})_{\overline{K}_{i}}).

Clearly, the class of objects of Loc_{\overline{K}_{i}}((X_{i})_{\overline{K}_{i}}) may be reconstructed as the class of objects of the category of finite sets with continuous Δ_{X_{i}}-action. To reconstruct the morphisms, it suffices (cf. Remark 2.1.3) to show that given any two open subgroups H_{1}, J_{1} ⊆ Π_{(X_{1})_{K_{1}}} — which we may assume, without loss of generality, to surject onto G_{K_{1}} — and an isomorphism

H_{1} →^{∼} J_{1}

that arises “K_{1}-geometrically” (i.e., from a K_{1}-scheme-theoretic isomorphism between the curves corresponding to H_{1}, J_{1}), it is necessarily the case that the corresponding isomorphism

H_{2} →^{∼} J_{2}

between open subgroups H_{2}, J_{2} ⊆ Π_{(X_{2})_{K_{2}}} arises K_{2}-geometrically.

But this follows formally from the “p-adic version of the Grothendieck Conjecture” proven in [Mzk6], Theorem A: Indeed, H_{1} →^{∼} J_{1} necessarily lies over an inner automorphism γ_{1} : G_{K_{1}} →^{∼} G_{K_{1}}. In particular, H_{2} →^{∼} J_{2} lies over an isomorphism γ_{2} : G_{K_{2}} →^{∼} G_{K_{2}}, which is obtained by conjugating γ_{1} by some fixed isomorphism (not necessarily geometric!) arising from α between the characteristic quotients G_{K_{1}} →^{∼} G_{K_{2}}. Since the property of “being an inner automorphism” is manifestly intrinsic, we thus conclude that γ_{2} is also an inner automorphism. This allows us to apply [Mzk6], Theorem A, which implies that H_{2} →^{∼} J_{2} arises K_{2}-geometrically, as desired. ◯

Corollary 2.5. (Consequences for Cores and Arithmeticity) Let

α : Π_{(X_{1})_{K_{1}}} →^{∼} Π_{(X_{2})_{K_{2}}}

be an isomorphism. Then:

(i) (X_{1})_{\overline{K}_{1}} is \overline{K}_{1}- arithmetic (respectively, a \overline{K}_{1}- core) if and only if (X_{2})_{\overline{K}_{2}} is \overline{K}_{2}- arithmetic (respectively, a \overline{K}_{2}- core).

(ii) Suppose that, for i = 1, 2, we are given a finite étale morphism (X_{i})_{K_{i}} → (Z_{i})_{K_{i}} to a K_{i}-core (Z_{i})_{K_{i}}. Then the isomorphism α extends uniquely to an isomorphism Π_{(Z_{1})_{K_{1}}} →^{∼} Π_{(Z_{2})_{K_{2}}} .

Proof. Assertion (i) is a formal consequence of Theorem 2.4 and Definition 2.1 (cf. also Remark 2.1.1). In light of Proposition 2.3, (i), assertion (ii) is a formal consequence of Theorem 2.4, at least over some corresponding finite Galois extensions K′_{1}, K′_{2} of K_{1}, K_{2}. That the resulting extension Π_{(Z_{1})_{K′_{1}}} →^{∼} Π_{(Z_{2})_{K′_{2}}} of α is unique is a formal consequence of the fact that every open subgroup of Π_{(X_{i})_{K_{i}}} has trivial centralizer in Π_{(Z_{i})_{K_{i}}} (cf. [Mzk7], Lemma 1.3.1, Corollary 1.3.3). Moreover, it follows formally from this triviality of centralizers that, by choosing corresponding normal open subgroups H_{i} ⊆ Π_{(Z_{i})_{K_{i}}} such that H_{i} ⊆ Π_{X_{K′_{i}}}, we may think of Π_{(Z_{i})_{K_{i}}} (and its various open subgroups) as subgroups of Aut(H_{i}), in a fashion which is compatible with α and its various (unique) extensions. Thus, since Π_{(Z_{i})_{K_{i}}} is generated by Π_{(Z_{i})_{K′_{i}}} and Π_{(X_{i})_{K_{i}}},

we conclude that this extension Π_{(Z_{i})_{K′_{1}}} →^{∼} Π_{(Z_{i})_{K′_{2}}} over the K′_{i} descends to

some Π_{(Z_{1})_{K_{1}}} →^{∼} Π_{(Z_{2})_{K_{2}}} , as desired. ◯

Remark 2.5.1. Recall from the theory of [Mzk3] (cf. Remark 2.1.2; Proposition 2.3, (ii), of the discussion above) that (X_{i})_{\overline{K}_{i}} is arithmetic if and only if it admits a finite étale cover which is a finite étale cover of a Shimura curve, i.e., (equivalently) if there exists a Shimura curve in \overline{Loc}_{\overline{K}i}((X_{i})_{\overline{K}_{i}}). As is discussed in [Mzk3], Theorem 2.6, a theorem of Takeuchi states that for a given (g, r), there are only finitely many isomorphism classes of hyperbolic curves of type (g, r) (over a given algebraically closed field of characteristic zero) which are arithmetic. Moreover, a general hyperbolic curve of type (g, r) is not only non-arithmetic; it is, in fact, equal to its own hyperbolic core (cf. [Mzk3], Theorem 5.3). Thus, for general curves of a given type (g, r), the structure of the category \overline{Loc}((X_{i})_{\overline{K}_{i}}) is not sufficient to determine the isomorphism class of the curve. It is not clear to the author at the time of writing whether or not, in the case when (X_{i})_{K_{i}} is arithmetic, the structure of the category \overline{Loc}((X_{i})_{\overline{K}_{i}}) is sufficient to determine the isomorphism class of (X_{i})_{K_{i}}. At any rate, just as was the case with Proposition 1.1 (cf. Remark 1.1.1), Theorem 2.4 does not allow one to recover the isomorphism class of (X_{i})_{K_{i}} for “most” (X_{i})_{K_{i}} — where here we take “most” to mean that (at least for (g, r) sufficiently large) the set of points determined by the curves for which it is possible to recover the isomorphism class of (X_{i})_{K_{i}} from the profinite group (X_{i})_{K_{i}} via the method in question fails to be Zariski dense in the moduli stack of hyperbolic curves of type (g, r) (cf. Corollary 3.8 below).

Finally, to give the reader a feel for the abstract theory — and, in particular, the state of affairs discussed in Remark 2.5.1 above — we consider the case of punctured hemi-elliptic orbicurves, in which the situation is understood somewhat explicitly:

Definition 2.6. Let X be an orbicurve (cf. Definition 2.2, (i)) over a field of characteristic zero k.

(i) We shall say that X is a hemi-elliptic orbicurve if it is obtained by forming the quotient — in the sense of stacks — of an elliptic curve by the action of ±1.

(ii) We shall say that X is a punctured hemi-elliptic orbicurve if it is obtained by forming the quotient — in the sense of stacks — of a once-punctured elliptic curve (i.e., the open subscheme given by the complement of the origin in an elliptic curve) by the action of ±1.

Proposition 2.7. (Punctured Hemi-Elliptic Cores) Let k be an algebraically closed field of characteristic 0; let X be a punctured hemielliptic orbicurve over k. Then if X is non-k-arithmetic, then X is a k-core. In particular, if X admits nontrivial automorphisms (over k), then X is k-arithmetic. Finally, there exist precisely 4 isomorphism classes of karithmetic X, which are described explicitly in [Take2], Theorem 4.1, (i).

Proof. In the following discussion, we omit the “k-”. Suppose that X is non-arithmetic. Write

Y → X

for the unique double (étale) covering by a punctured elliptic curve Y, and

Y → Z

for the unique morphism to the core (i.e., the terminal object in \overline{Loc}_{k}(X) = \overline{Loc}_{k}(Y)). Thus, the induced morphism

\overline{Y} = \overline{Y}^{crs} → \overline{Z}^{crs}

on the “coarse moduli spaces” (cf. [FC], Chapter I, Theorem 4.10) associated to the canonical compactifications \overline{Y}, \overline{Z} of the orbicurves Y, Z is a finite ramified covering morphism — whose degree we denote by d — from an elliptic curve \overline{Y} to a copy of ℙ_{k}^{1} ≅ \overline{Z}^{crs}. Note that since \overline{Y} has only one “cusp” y_{∞} (i.e. point ∈ \overline{Y} \Y), and a point of \overline{Y} is a cusp if and only if its image is a cusp in \overline{Z}, it follows that \overline{Z} also has a unique cusp z_{∞}, and that y_{∞} is the unique point of \overline{Y} lying over z_{∞}. Moreover, because \overline{Y} → \overline{Z}^{crs} arises from a finite étale morphism Y → Z, it follows that the ramification index of \overline{Y} → \overline{Z}^{crs} is the same at all points of \overline{Y} lying over a given point of \overline{Z}^{crs}. Thus, applying the Riemann-Hurwitz formula yields:

0 = −2d + ∑_{i} \frac{d}{e_{i}} (e_{i} − 1)

where the e_{i} are the ramification indices over the points of \overline{Z}^{crs} at which the covering morphism ramifies. Thus, we conclude that 2 = ∑_{i} \frac{1}{e_{i}} (e_{i} − 1). Since all of the e_{i} are integers, one verifies immediately that the only possibilities for the set of e_{i}’s are the following:

(2, 2, 2, 2); (2, 3, 6); (2, 4, 4); (3, 3, 3)

Note that it follows from the fact that y_{∞} is the unique point of \overline{Y} lying over z_{∞} that d, as well as the ramification index at z_{∞}, is necessarily equal to the largest e_{i}. In the case of (2, 2, 2, 2), we thus conclude that X = Z, so X is a core, as asserted. In the other three cases, we conclude that Y is a finite étale covering of the orbicurve determined by a “triangle group” (cf. [Take1]) of one of the following types:

(2, 3, ∞); (2, 4, ∞); (3, 3, ∞)

By [Take1], Theorem 3, (ii), such a triangle group is arithmetic, so X itself is arithmetic, thus contradicting our hypotheses. This completes the proof of the first assertion of Proposition 2.7.

The second (respectively, third) assertion of Proposition 2.7 is a formal consequence of the first assertion of Proposition 2.7 (respectively, [Take2], Theorem 4.1, (i)). ◯

Section 3: Hyperbolically Ordinary Canonical Liftings

In this §, we would like to work over a finite, unramified extension K of ℚ_{p}, where p is a prime number ≥ 5. We denote the ring of integers (respectively, residue field) of K by A (respectively, k). Since A ≅ W (k) (the ring of Witt vectors with coefficients in k), we have a natural Frobenius morphism

Φ_{A} : A → A

which lifts the Frobenius morphism Φ_{k} : k → k on k. In the following discussion, the result of base-changing over A (respectively, A; A; ℤ_{p}) with k (respectively, K; with A, via Φ_{A}; ℤ/p^{n}ℤ, for an integer n ≥ 1) will be denoted by a subscript k (respectively, subscript K; superscript F; subscript ℤ/p^{n}ℤ).

Let

(X → S =^{def} Spec(A), D ⊆ X)

be a smooth, pointed curve of type (g, r) (for which D is the divisor of marked points), where 2g−2+r > 0 — cf. §0. In the following discussion, we would like to consider the extent to which (X, D) is a canonical lifting of (X_{k}, D_{k}), in the sense of [Mzk1], Chapter III, §3; Chapter IV. We refer also to the Introduction of [Mzk2] for a survey of “p-adic Teichmüller theory” (including the theory of [Mzk1]).

Lemma 3.1. (Canonicality Modulo p^{2}) Suppose that

Y_{K} → X_{K}

is a finite ramified morphism of smooth, proper, geometrically connected curves over K which is unramified away from D_{K}. Denote the reduced induced subscheme associated to the inverse image in Y_{K} of D_{K} by E_{K} ⊆ Y_{K}. Suppose further that the reduction

Y_{k} → X_{k}

modulo p of the normalization Y → X of X in Y_{K} has the following form:

(i) Y_{k} is reduced;

(ii) Y_{k} is smooth over k, except for a total of precisely \frac{1}{2}(p−1)(2g−2+r) (≥ 2) nodes. Moreover, the “order” of the deformation of each node determined by Y is equal to 1 (equivalently: Y is regular at the nodes), and the special fiber E_{k} ⊆ Y_{k} of the closure E ⊆ Y of E_{K} in Y is a reduced divisor (equivalently: a divisor which is étale over k) at which Y_{k} is smooth.

(iii) Y_{k} has precisely two irreducible components C_{V}, C_{F}. Here, the morphism C_{V} → X_{k} (respectively, C_{F} → X_{k}) is an isomorphism (respectively, k-isomorphic to the relative Frobenius morphism Φ_{X_{k}/k} : X_{k}^{F} → X_{k} of X_{k}).

(iv) (X_{k}, D_{k}) admits a nilpotent ordinary indigenous bundle (cf. [Mzk1], Chapter II, Definitions 2.4, 3.1) whose supersingular divisor (cf. [Mzk1], Chapter II, Proposition 2.6, (3)) is equal to the image of the nodes of Y_{k} in X_{k}.

Then (X, D) is isomorphic modulo p^{2} to the canonical lifting (cf. [Mzk1], Chapter III, §3; Chapter IV) determined by the nilpotent indigenous bundle of (iv).

Remark 3.1.1. In the context of Lemma 3.1, we shall refer to the points of X_{k} which are the image of nodes of Y_{k} as supersingular points and to points which are not supersingular as ordinary. Moreover, the open subscheme of ordinary points will be denoted by

X_{k}^{ord} ⊆ X_{k}

and the corresponding p-adic formal open subscheme of X̂ (the p-adic completion of X) by X̂^{ord}. Also, we shall consider X (respectively, Y) to be equipped with the log structure (cf. [Kato] for an introduction to the theory of log structures) determined by the monoid of regular functions invertible on X_{K}\D_{K} (respectively, Y_{K}\E_{K}) and denote the resulting log scheme by X^{log} (respectively, Y^{log}). Thus, the morphism of schemes Y → X extends uniquely to a morphism of log schemes Y^{log} → X^{log}.

Proof. First, let us observe that

(Y^{log})_{k}^{ord} ≅ {(X^{log})_{k}^{ord}}^{F} ⋃ (X^{log})_{k}^{ord}

where the isomorphism is the unique isomorphism lying over X_{k}^{log}. Since (X^{log})^{ord}, (Y^{log})^{ord} are log smooth over A, it follows that the inclusion {(X^{log})_{k}^{ord}}^{F} ↪ (Y^{log})_{k}^{ord} lifts to a (not necessarily unique!) inclusion

{(X^{log})_{ℤ/p^{2}ℤ}^{ord}}^{F} ↪ (Y^{log})_{ℤ/p^{2}ℤ}^{ord}

whose composite

Ψ^{log} : {(X^{log})_{ℤ/p^{2}ℤ}^{ord}}^{F} → (X^{log})_{ℤ/p^{2}ℤ}^{ord}

with the natural morphism (Y^{log})_{ℤ/p^{2}ℤ}^{ord} → (X^{log})_{ℤ/p^{2}ℤ}^{ord} is nevertheless independent of the choice of lifting of the inclusion. Indeed, this is formal consequence of the fact that Ψ^{log} is a lifting of the Frobenius morphism on (X^{log})_{k}^{ord} (cf., e.g., the discussion of [Mzk1], Chapter II, the discussion preceding Proposition 1.2, as well as Remark 3.1.2 below).

Of course, Ψ^{log} might not be regular at the supersingular points, but we may estimate the order of the poles of Ψ^{log} at the supersingular points as follows: Since Y is assumed to be regular, it follows that the completion of Y_{ℤ/p^{2}ℤ} at a supersingular point ν is given by the formal spectrum Spf of a complete local ring isomorphic to:

R_{Y} =^{def} (A/p^{2} · A)[[s, t]]/(st − p)

(where s, t are indeterminates). Thus, modulo p, this completion is a node, with the property that precisely one branch — i.e., irreducible component — of this node lies on C_{F} (respectively, C_{V}). (Indeed, this follows from the fact that both C_{F} and C_{V} are smooth over k.) Suppose that the irreducible component lying on C_{F} is defined locally (modulo p) by the equation t = 0. Thus, the ring of regular functions on the ordinary locus of C_{F} restricted to this formal neighborhood of a supersingular point is given by k[[s]][s^{−1}]. The connected component of (Y^{log})_{ℤ/p^{2}ℤ}^{ord} determined by C_{F} may be thought of as the open subscheme “s ≠ 0”. Here, we recall that the parameter s in this discussion is uniquely determined up to multiplication by an element of R_{Y}^{×} — cf. [Mzk4], §3.7.

Next, let us write:

R′_{X} =^{def} Im(R_{Y}) ⊆ R_{Y} [s^{−1}]

for the image of R_{Y} in R_{Y} [s^{−1}]. Then since

t = s^{−1} · p ∈ R_{Y} [s^{−1}] = (A/p^{2} · A)[[s]][s^{−1}]

it follows that R′_{X} = (A/p^{2} · A)[[s]][s^{−1} · p]. In particular, if we think of

R_{X} =^{def} (A/p^{2} · A)[[s]] ⊆ R′_{X}

— so R_{X}[s^{−1}] = R′_{X}[s^{−1}] = R_{Y} [s^{−1}] — as a local smooth lifting of C_{F} at ν, we thus conclude that arbitrary regular functions on Spf(R_{Y}) restrict to meromorphic functions on Spf(R_{X}) with poles of order ≤ 1 at ν. Thus, since Ψ^{log} arises from an everywhere regular morphism Y^{log} → X^{log}, we conclude that:

Ψ^{log} has poles of order ≤ 1 at the supersingular points.

But then the conclusion of Lemma 3.1 follows formally from [Mzk1], Chapter II, Proposition 2.6, (4); Chapter IV, Propositions 4.8, 4.10, Corollary 4.9. ◯

Remark 3.1.2. The fact — cf. the end of the first paragraph of the proof of Lemma 3.1 — that the order of a pole of a Frobenius lifting is independent of the choice of smooth lifting of the domain of the Frobenius lifting may be understood more explicitly in terms of the coordinates used in the latter portion of the proof of Lemma 3.1 as follows: Any “coordinate transformation”

s ↦ s + p · g(s)

(where g(s) ∈ k[[s]][s^{−1}]) fixes — since we are working modulo p^{2} — functions of the form s^{p} + p · f(s) (where f(s) ∈ k[[s]][s^{−1}]). This shows that the order of the pole of f(s) does not depend on the choice of parameter s.

In the situation of Lemma 3.1, let us denote the natural morphism of fundamental groups (induced by (X̂^{log})^{ord} → X^{log}) by

Π_{(_{X^{log}})^{ord}} =^{def} π_{1}((X̂^{log})_{K}^{ord}) → Π_{X^{log}} =^{def} π_{1}(X_{K}^{log}) = π_{1}(X_{K}\D_{K})

(for some fixed choice of basepoints). Here, we observe that (by the main theorem of [Vala]) (X̂^{log})^{ord} is excellent, so the normalization of (X̂^{log})^{ord} in a finite étale covering of (X̂^{log})_{K}^{ord} is finite over (X̂^{log})^{ord}. Thus, (X̂^{log})_{K}^{ord} has a “well-behaved theory” of finite étale coverings which is compatible with étale localization on (X̂^{log})^{ord}. Also, before proceeding, we observe that Π_{X^{log}} fits into an exact sequence:

1 → Δ_{X^{log}} → Π_{X^{log}} → G_{K} → 1

(where Δ_{X^{log}} is defined so as to make the sequence exact).

Lemma 3.2. (The Ordinary Locus Modulo p^{2}) Let V_{𝔽_{p}} be a 2dimensional 𝔽_{p}-vector space equipped with a continuous action of Π_{X^{log}} up to ±1 — i.e., a representation

Π_{X^{log}} → GL_{2}^{±}(V_{𝔽_{p}}) =^{def} GL_{2}(V_{𝔽_{p}})/{±1}

— such that:

(i) The determinant of V_{𝔽_{p}} is isomorphic (as a Π_{X^{log}}-module) to 𝔽_{p}(1).

(ii) There exists a finite log étale Galois covering (i.e., we assume tame ramification over D)

X_{±}^{log} → X^{log}

such that the action of Π_{X^{log}} up to ±1 on V_{𝔽_{p}} lifts to a (usual) action (i.e., without sign ambiguities) of Π_{X_{±}^{log}} ⊆ Π_{X^{log}} on V_{𝔽_{p}}. This action is uniquely determined up to tensor product with a character of Π_{X_{±}^{log}} of order 2.

(iii) The finite étale covering Y_{K}^{log} → X_{K}^{log} determined by the finite Π_{X^{log}}-set of 1-dimensional 𝔽_{p}-subspaces of V_{𝔽_{p}} satisfies the hypotheses of Lemma 3.1.

(iv) Write

(Z_{±}^{log})_{K} → (X_{±}^{log})_{K}

for the finite log étale covering (of degree p^{2} − 1) corresponding to the nonzero portion of V_{𝔽_{p}}. Write

(Y_{±}^{log})_{K} → (X_{±}^{log})_{K}

for the finite log étale covering of smooth curves which is the composite (i.e., normalization of the fiber product over X_{K}^{log}) of the coverings (Z_{±}^{log})_{K}, Y_{K}^{log} of X_{K}^{log}. (Thus, (Z_{±}^{log})_{K} maps naturally to (Y_{±}^{log})_{K}, hence also to Y_{K}^{log}.) Let us refer to an irreducible component of the special fiber of a stable reduction of (Y_{±}^{log})_{K} (respectively, (Z_{±}^{log})_{K}) over some finite extension of K that maps finitely to the irreducible component “C_{F} ” (cf. Lemma 3.1) as being “of C_{F} -type” and “associated to (Y_{±}^{log})_{K} (respectively, (Z_{±}^{log})_{K})”. Then any irreducible component of C_{F} -type associated to (Z_{±}^{log})_{K} is étale and free of nodes over the ordinary locus of any irreducible component of C_{F} -type associated to (Y_{±}^{log})_{K}.

Then (after possibly tensoring V_{𝔽_{p}} with a character of Π_{X_{±}^{log}} of order 2) the

étale local system E_{𝔽_{p}}^{ord} on (X̂_{±}^{log})_{K}^{ord} determined by the Π_{(_{X_{±}^{log}})^{ord}}-module V_{𝔽_{p}}

arises from a (logarithmic) finite flat group scheme on (X̂_{±}^{log})^{ord}. Moreover, the Π_{(_{X_{±}^{log}})^{ord}}-module V_{𝔽_{p}} fits into an exact sequence:

0 → (V_{𝔽_{p}}^{etl})^{∨}(1) → V_{𝔽_{p}} → V_{𝔽_{p}}^{etl} → 0

where V_{𝔽_{p}}^{etl} is a 1-dimensional 𝔽_{p}-space “up to ±1” whose Π_{(_{X_{±}^{log}})^{ord}}-action

arises from a finite étale local system on (X_{±}^{ord})_{k} ⊆ (X_{±})_{k}, and the “∨” denotes the 𝔽_{p}-linear dual.

Proof. As was seen in the proof of Lemma 3.1, we have an isomorphism

(Y^{log})_{k}^{ord} ≅ {(X^{log})_{k}^{ord}}^{F} ⋃ (X^{log})_{k}^{ord}

which thus determines a decomposition of (Ŷ^{log})^{ord} into two connected components. Moreover, the second connected component on the right-hand side corresponds to a rank one quotient V_{𝔽_{p}} ↠ Q_{𝔽_{p}} which is stabilized by the action of Π_{(_{X^{log}})^{ord}}, while the first connected component on the right-hand side

parametrizes splittings of this quotient V_{𝔽_{p}} ↠ Q_{𝔽_{p}}. Here, we observe that Q_{𝔽_{p}}^{⊗2} admits a natural Π_{(_{X^{log}})^{ord}}-action (i.e., without sign ambiguities).

Now any choice of isomorphism between {(X̂^{log})^{ord}}^{F} and the first connected component of (Ŷ^{log})^{ord} determines a lifting of Frobenius

Φ^{log} : {(X̂^{log})^{ord}}^{F} → (X̂^{log})^{ord}

which is ordinary (by the conclusion of Lemma 3.1 — cf. [Mzk1], Chapter IV, Proposition 4.10). Thus, by the general theory of ordinary Frobenius liftings, Φ^{log} determines, in particular, a (logarithmic) finite flat group scheme annihilated by p which is an extension of the trivial finite flat group scheme “𝔽_{p}” by the finite flat group scheme determined by the Cartier dual of some étale local system of one-dimensional 𝔽_{p}-vector spaces on X_{k}^{ord} (cf. [Mzk1], Chapter III, Definition 1.6); denote the Π_{(_{X^{log}})^{ord}}-module corresponding to this étale local system by Ω_{𝔽_{p}}. Moreover, it is a formal consequence of this general theory that Φ_{K}^{log} is precisely the covering of (X̂^{log})_{K}^{ord} determined by considering splittings of this extension. Thus, since the Galois closure of this covering has Galois group given by the semi-direct product of a cyclic group of order p with a cyclic group of order p − 1, we conclude (by the elementary group theory of such a semi-direct product) that we have an isomorphism of Π_{(_{X^{log}})^{ord}}-modules: (Q_{𝔽_{p}})^{⊗−2}(1) ≅ Ω_{𝔽_{p}}(1), i.e.,

(Q_{𝔽_{p}})^{⊗−2} ≅ Ω_{𝔽_{p}}

We thus conclude that the local system E_{𝔽_{p}}^{Φ^{log}} on (X̂^{log})_{K}^{ord} determined by this (logarithmic) finite flat group scheme arising from the general theory satisfies:

E_{𝔽_{p}}^{Φ^{log}}|_{(_{X_{±}^{log}})_{K}^{ord}} ≅ E_{𝔽_{p}}^{ord} ⊗_{𝔽_{p}} Q_{𝔽_{p}}^{∨}

Next, let us write χ_{Q} for the character (valued in 𝔽_{p}^{×}) of Π_{(_{X_{±}^{log}})^{ord}} corre-

sponding to Q_{𝔽_{p}}. Now it follows formally from condition (iv) of the statement of Lemma 3.2 that the finite étale covering of (X̂_{±}^{log})_{K}^{ord} determined by Ker(χ_{Q})

W_{Q} → (X̂_{±}^{log})_{K}^{ord}

is dominated by the composite of some finite étale covering of X̂_{±}^{ord} and a “constant covering” (i.e., a covering arising from a finite extension L of K). Thus, the only ramification that may occur in the covering W_{Q} arises from ramification of the “constant covering”, i.e., the finite extension L/K. Moreover, (since (Q_{𝔽_{p}})^{⊗−2} ≅ Ω_{𝔽_{p}}) the covering determined by Ker(χ_{Q}^{2}) is unramified, so, in fact, we may take L to be the extension K(p^{\frac{1}{2}} ). This implies that we may write

χ_{Q} = χ′_{Q} · χ′′_{Q}

where the covering determined by the kernel of χ′_{Q} (respectively, χ′′_{Q}) is finite étale over X̂_{±}^{ord} (respectively, the covering arising from base-change from K to L). On the other hand, since χ′′_{Q} extends naturally to Π_{X_{±}^{log}}, we may assume (without loss of generality — cf. condition (ii) of the statement of Lemma 3.2) that χ′′_{Q} is trivial, hence that Q_{𝔽_{p}} arises from an étale local system on (X_{±}^{ord})_{k}.

But this — together with the isomorphism E_{𝔽_{p}}^{Φ^{log}}|_{(_{X_{±}^{log}})_{K}^{ord}} ≅ E_{𝔽_{p}}^{ord} ⊗_{𝔽_{p}} Q_{𝔽_{p}}^{∨} — implies the conclusion of Lemma 3.2. ◯

Lemma 3.3. (Global Logarithmic Finite Flat Group Scheme) Let V_{𝔽_{p}} be a 2-dimensional 𝔽_{p}-vector space equipped with a continuous action of Π_{X^{log}} up to ±1 which satisfies the hypotheses of Lemma 3.2. Then the étale local system E_{𝔽_{p}} on (X_{±}^{log})_{K} determined by the Π_{X_{±}^{log}}-module V_{𝔽_{p}} arises from

a (logarithmic) finite flat group scheme on X_{±}^{log}.

Proof. Write

Z → X_{±}

for the normalization of X_{±} in the finite étale covering of (X_{±}^{log})_{K} determined by the local system E_{𝔽_{p}}. Since X_{±} is regular of dimension 2, it thus follows that Z is finite and flat (by the “Auslander-Buchsbaum formula” and “Serre’s criterion for normality” — cf. [Mtmu], p. 114, p. 125) over X_{±}. Moreover, since Z_{K} is already equipped with a structure of (logarithmic) finite flat group scheme, which, by the conclusion of Lemma 3.2, extends naturally over the generic point of the special fiber of X — since it extends (by the proof of Lemma 3.2) to a regular, hence normal, (logarithmic) finite flat group scheme over the ordinary locus — it thus suffices to verify that this finite flat group scheme structure extends (uniquely) over the supersingular points of X_{±}. But this follows formally from [Mtmu], p. 124, Theorem 38 — i.e., the fact (applied to X_{±}, not Z!) that a meromorphic function on a normal scheme is regular if and only if it is regular at the primes of height 1 — and the fact that Z (hence also Z ×_{X_{±}} Z) is finite and flat over X_{±}. ◯

Lemma 3.4. (The Associated Dieudonné Crystal Modulo p) Let V_{𝔽_{p}} be a 2-dimensional 𝔽_{p}-vector space equipped with a continuous action of Π_{X^{log}} up to ±1 which satisfies the hypotheses of Lemma 3.2. Suppose, further, that the Π_{X^{log}}-module (up to ±1) V_{𝔽_{p}} satisfies the following condition:

(†_{M}) The G_{K}-module

M =^{def} H^{1}(Δ_{X^{log}}, Ad(V_{𝔽_{p}}))

(where Ad(V_{𝔽_{p}}) ⊆ End(V_{𝔽_{p}}) is the subspace of endomorphisms whose trace = 0) fits into an exact sequence:

0 → 𝔾^{−1}(M) → M → 𝔾^{2}(M) → 0

where 𝔾^{2}(M) (respectively, 𝔾^{−1}(M)) is isomorphic to the result of tensoring an unramified G_{K}-module whose dimension over 𝔽_{p} is equal to 3g − 3 + r with the Tate twist 𝔽_{p}(2) (respectively, 𝔽_{p}(−1)).

Then the “MF^{∇}-object (up to ±1)” (cf. [Falt], §2) determined by the Π_{X^{log}}- module (up to ±1) V_{𝔽_{p}} (cf. Lemma 3.3) arises from the (unique) nilpotent ordinary indigenous bundle of Lemma 3.1, (iv).

Proof. First, we recall from the theory of [Falt], §2, that the conclusion of Lemma 3.3 implies that V_{𝔽_{p}} arises from an “MF^{∇}-object (up to ±1)” on X^{log}, as in the theory of [Falt], §2. Here, the reader uncomfortable with “MF^{∇}- objects up to ±1” may instead work with a usual MF^{∇}-object over X_{±}^{log} equipped with an “action of Gal(X_{±}^{log}/X^{log}) up to ±1”. Write F_{k} for the vector bundle on (X_{±})_{k} underlying the MF^{∇}-object on X_{±}^{log} determined by E_{𝔽_{p}}.

Thus, F_{k} is a vector bundle of rank 2, whose Hodge filtration is given by a subbundle F^{1}(F_{k}) ⊆ F_{k} of rank 1. Moreover, the Kodaira-Spencer morphism of this subbundle, as well as the “Hasse invariant”

Φ_{X_{±}}^{∗}F^{1}(F_{k})^{∨} ↪ F_{k} ↠ F_{k}/F^{1}(F_{k}) ≅ F^{1}(F_{k})^{∨}

(where Φ_{X_{±}} is the Frobenius morphism on X_{±}; and the injection is the morphism that arises from the Frobenius action on the MF^{∇}-object in question), is generically nonzero. Indeed, these facts all follow immediately from our analysis of E_{𝔽_{p}} over the ordinary locus in the proof of Lemma 3.2.

Next, let us observe that this Hasse invariant has at least one zero. Indeed, if it were nonzero everywhere, it would follow formally from the general theory of MF^{∇}-objects (cf. [Falt], §2) that the Π_{X^{log}}-module (up to ±1) V_{𝔽_{p}} admits a Π_{X^{log}}-invariant subspace of 𝔽_{p}-dimension 1 — cf. the situation over the ordinary locus in the proof of Lemma 3.2, over which the Hasse invariant is, in fact, nonzero. On the other hand, this implies that Y_{K} → X_{K} admits a section, hence that Y is not connected. But this contradicts the fact (cf. the proof of Lemma 3.1) that the two irreducible components C_{V}, C_{F} of Y_{k} (cf. Lemma 3.1, (iii)) necessarily meet at the nodes of Y_{k}. (Here, we recall from Lemma 3.1, (ii), that there exists at least one node on Y_{k}.)

In particular, it follows from the fact that the Hasse invariant is generically nonzero, but not nonzero everywhere that the degree of the line bundle F^{1}(F) on (X_{±})_{k} is positive. Note that since F^{1}(F)^{⊗2} descends naturally to a line bundle L on X_{k}, we thus obtain that

1 ≤ deg(L) ≤ 2g − 2 + r

(where the second inequality follows from the fact that the Kodaira-Spencer morphism in nonzero).

Now, we conclude — from the p-adic Hodge theory of [Falt], §2; [Falt], §5, Theorem 5.3 — that the condition (†_{M}) implies various consequences concerning the Hodge filtration on the first de Rham cohomology module of Ad of the MF^{∇}-object determined by V_{𝔽_{p}}, which may be summarized by the inequality:

h^{1}(X_{k}, L^{−1}) = h^{0}(X_{k}, L ⊗_{O_{X}} ω_{X}) ≥ 3g − 3 + r

(where “h^{i}” denotes the dimension over k of “H^{i}”) — cf. [Mzk1], IV, Theorem 1.3 (and its proof), which, in essence, addresses the ℤ_{p} analogue of the present 𝔽_{p}-vector space situation. (Note that here we make essential use of the hypothesis p ≥ 5.) Thus, (cf. loc. cit.) we conclude that (since deg(L) > 0) the line bundle L ⊗_{O_{X}} ω_{X} on X_{k} is nonspecial, hence (by the above inequality) that:

deg(L) = deg(L ⊗_{O_{X}} ω_{X}) − deg(ω_{X})
= h^{0}(X_{k}, L ⊗_{O_{X}} ω_{X}) + (g − 1) − 2(g − 1)
≥ 3g − 3 + r − (g − 1) = 2g − 2 + r

Combining this with the inequalities of the preceding paragraph, we thus obtain that deg(L) = 2g − 2 + r, so the MF^{∇}-object in question is an indigenous bundle, which is necessarily equal to the indigenous bundle of Lemma 3.1, (iv), since the supersingular locus of the former is contained in the supersingular locus of the latter (cf. [Mzk1], Chapter II, Proposition 2.6, (4)). ◯

Lemma 3.5. (Canonical Deformations Modulo Higher Powers of p) Let V_{𝔽_{p}} be a 2-dimensional 𝔽_{p}-vector space equipped with a continuous action of Π_{X^{log}} up to ±1 which satisfies the hypotheses of Lemmas 3.2, 3.4. Suppose that, for some n ≥ 1:

(i) (X, D) is isomorphic modulo p^{n} to a canonical lifting (as in [Mzk1], Chapter III, §3; Chapter IV).

(ii) V_{𝔽_{p}} is the reduction modulo p of a rank 2 free ℤ/p^{n}ℤ-module V_{ℤ/p^{n}ℤ} with continuous Π_{X^{log}}-action up to ±1.

Then (X, D) is isomorphic modulo p^{n+1} to a canonical lifting, and the Π_{X^{log}}-set ℙ(V_{ℤ/p^{n}ℤ}) (of free, rank one ℤ/p^{n}ℤ-module quotients of V_{ℤ/p^{n}ℤ}) is isomorphic to the projectivization of the canonical representation modulo p^{n} (cf. [Mzk1], Chapter IV, Theorem 1.1) associated to (X, D). Finally, if the determinant of V_{ℤ/p^{n}ℤ} is isomorphic to (ℤ/p^{n}ℤ)(1), then the Π_{X^{log}}-module (up to ±1) V_{ℤ/p^{n}ℤ} is isomorphic to the canonical representation modulo p^{n}.

Proof. First, we observe that the case n = 1 is a formal consequence of Lemmas 3.1, 3.4. The case of general n is then, in essence, a formal consequence of the theory of [Mzk1], Chapter V, §1 — cf. especially, Theorem 1.7, and the discussion following it. We review the details as follows:

The space of deformations (of the projectivization) of the Π_{X^{log}}-module “up to ±1” V_{𝔽_{p}} may be thought of as a formal scheme

R

which is noncanonically isomorphic to Spf(ℤ_{p}[[t_{1}, …, t_{2(3g−3+r)}]]) (where the t_{i}’s are indeterminates) — cf. the discussion preceding [Mzk1], Chapter V, Lemma 1.5. Write R^{PD} for the p-adic completion of the PD-envelope of R at the 𝔽_{p}-valued point defined by “V_{𝔽_{p}}”. Note that R and R^{PD} are equipped with a natural G_{K}-action. Then according to the theory of loc. cit., there is a G_{K}-equivariant closed immersion of formal schemes (cf. Remark 3.5.1 below)

κ^{PD} : Spf(D̂^{Gal}) ↪ R^{PD}

where D̂^{Gal} is noncanonically isomorphic to the p-adic completion of the PDenvelope at the closed point of a power series ring ℤ_{p}[[t_{1}, …, t_{3g−3+r}]] equipped with a natural G_{K}-action.

Now it follows from the theory of loc. cit., the induction hypothesis on n, and the assumptions (i), (ii) of Lemma 3.5, that V_{ℤ/p^{n−1}ℤ} =^{def} V_{ℤ/p^{n}ℤ} ⊗ℤ/p^{n−1}ℤ corresponds (at least projectively) to the canonical representation modulo p^{n−1}, hence determines a G_{K}-invariant rational point

σ_{n−1} ∈ R^{PD}(ℤ/p^{n−1}ℤ)

that lies in the image Im(κ^{PD}) of κ^{PD}. Thus, the point

σ_{n} ∈ R^{PD}(ℤ/p^{n}ℤ)

determined by V_{ℤ/p^{n}ℤ} may be regarded as a G_{K}-invariant deformation of σ_{n−1}. Note that the set of deformations of σ_{n−1} naturally forms a torsor T over the 𝔽_{p}-vector space M of Lemma 3.4. Since σ_{n−1} ∈ Im(κ^{PD}), this torsor is equipped with a natural G_{K}-stable subspace

T′′ ⊆ T

(consisting of the deformations that lie ∈ Im(κ^{PD})) which is (by the theory of loc. cit.) a torsor over 𝔾^{−1}(M). In particular, this subspace determines a G_{K}-invariant trivialization τ′ of the 𝔾^{2}(M)-torsor

(T ↠) T′

given by the “change of structure group M ↠ 𝔾^{2}(M)”.

Now let us observe that by (†_{M}) and the fact that p ≥ 5 — so the square of the cyclotomic character G_{K} → 𝔽_{p}^{×} is nontrivial — 𝔾^{2}(M) has no nontrivial Galois invariants, i.e.:

𝔾^{2}(M)^{G_{K}} = 0

Thus, we conclude that τ′ is the unique G_{K}-invariant point of T′, hence that the image in T′ of the point τ_{n} ∈ T determined by σ_{n} is necessarily equal to τ′, i.e., τ_{n} ∈ T′′ — or, in other words, σ_{n} ∈ Im(κ^{PD}).

On the other hand, if we interpret the Galois-theoretic fact that σ_{n−1} lifts to a G_{K}-invariant σ_{n} ∈ Im(κ^{PD}) in terms of the original finite étale coverings — combinatorial information concerning which the Galois theory is intended to encode — then we obtain the following conclusion: The A-valued point

α ∈ N (A)

of the ℤ_{p}-smooth p-adic formal scheme N =^{def} N_{g,r}^{ord} (cf. [Mzk1], Chapter III, §2) determined by (X, D) and the nilpotent ordinary indigenous bundle modulo p of Lemma 3.4 not only lies — by assumption (i) of the statement of Lemma 3.5 — in the image of N (A) under the (n − 1)-st iterate Φ_{N}^{n−1} of the canonical Frobenius lifting

Φ_{N} : N → N

on (cf. [Mzk1], Chapter III, Theorem 2.8) but also in the image of N (A) under the n-th iterate Φ_{N}^{n} of Φ_{N}. (Indeed, the restriction of the covering Φ_{N}^{n−1} (respectively, Φ_{N}^{n}) to α admits a section, determined by the G_{K}-invariant rational point σ_{n−1} (respectively, σ_{n}).) Thus, we conclude that X^{log} is canonical modulo p^{n+1}. Finally, since

𝔾^{−1}(M)^{G_{K}} = 0

we conclude that σ_{n} is the unique G_{K}-invariant lifting of σ_{n−1} to ℤ/p^{n}ℤ, hence that V_{ℤ/p^{n}ℤ} corresponds (projectively) to the canonical representation modulo p^{n}, as desired. ◯

Remark 3.5.1. In some sense, it is natural to think of Im(κ^{PD}) (cf. the proof of Lemma 3.5) as the “crystalline locus” in the space of “all” representations R^{PD}.

Theorem 3.6. (Group-Theoreticity of Canonical Liftings) Let p ≥ 5 be a prime number. For i = 1, 2, let K_{i} be an unramified finite extension of ℚ_{p}, and (X_{i}, D_{i}) a smooth pointed curve of type (g_{i}, r_{i}) over O_{K_{i}}, where 2g_{i} − 2 + r_{i} > 0. Assume that we have chosen basepoints of the (X_{i})_{K_{i}}\(D_{i})_{K_{i}} (which thus induce basepoints of the K_{i}); denote the resulting fundamental group by Π_{X_{i}^{log}}. Suppose that we have been given an isomorphism of profinite groups:

Π_{X_{1}^{log}} →^{∼} Π_{X_{2}^{log}}

Then:

(i) (X_{1}, D_{1}) is a canonical lifting (in the sense of [Mzk1], Chapter III, §3; Chapter IV) if and only if (X_{2}, D_{2}) is so.

(ii) If at least one of the (X_{i}, D_{i}) is a canonical lifting, then the isomorphism on logarithmic special fibers of [Mzk7], Theorem 2.7, lifts (uniquely) to an

isomorphism (X_{1}, D_{1}) →^{∼} (X_{2}, D_{2}) over O_{K_{1}} ≅ W (k_{1}) →^{∼} W (k_{2}) ≅ O_{K_{2}}.

Proof. Let us verify (i). Since, by [Mzk7], Theorem 2.7 (and [Mzk7], Proposition 1.2.1, (vi)), the logarithmic special fibers of all stable reductions of all finite étale coverings of (X_{i})_{K_{i}}\(D_{i})_{K_{i}} are “group-theoretic”, it follows that the conditions (i), (ii), (iii), (iv) of Lemma 3.1, as well as the conditions (i), (ii), (iii), (iv) of Lemma 3.2, are all group-theoretic conditions which, moreover, (by the theory of [Mzk1], Chapter III, §3; Chapter IV) are satisfied by canonical liftings. (Here, relative to the assertion that canonical liftings satisfy Lemma 3.1, (ii), and Lemma 3.2, (iv), we remind the reader that:

(1) Since p ≥ 5, the special fiber of the curve Y of Lemma 3.1 has ≥ 2 nodes (cf. Lemma 3.1, (ii)), which implies that the curve Y is stable (i.e., even without the marked points).

(2) The smooth locus of any model of a curve over a discrete valuation ring necessarily maps to the stable model (whenever it exists) of the curve — cf., e.g., [JO]— and, moreover, whenever this map is quasi-finite, necessarily embeds as an open subscheme of the smooth locus of the stable model.)

Moreover, (since the cyclotomic character and inertia subgroup are grouptheoretic — cf. [Mzk7], Proposition 1.2.1, (ii), (vi) — it follows that) condition (†_{M}) of Lemma 3.4 is a group-theoretic condition which (by the theory of loc. cit.) is satisfied by canonical liftings. Thus, successive application of Lemma 3.5 for n = 1, 2, … implies assertion (i) of the statement of Theorem 3.6.

Next, let us observe that when one (hence both) of the (X_{i}, D_{i}) is a canonical lifting, it follows from Lemma 3.1 that the isomorphism of special fibers

(X_{1}, D_{1})_{k_{1}} →^{∼} (X_{2}, D_{2})_{k_{2}}

(lying over some isomorphism k_{1} →^{∼} k_{2}) determined by [Mzk7], Theorem 2.7 is compatible with the nilpotent ordinary indigenous bundles on either side that give rise to the canonical liftings. On the other hand, by the theory of loc. cit., the lifting of (X_{i}, D_{i})_{k_{i}} over O_{K_{i}} ≅ W (k_{i}) is determined uniquely by the fact that this lifting is “canonical” (i.e., when it is indeed the case that it is canonical!). This completes the proof of assertion (ii) of the statement of Theorem 3.6. ◯

Remark 3.6.1. Thus, (cf. Remarks 1.1.1, 2.2.1) unlike the situation with Proposition 1.1, Theorem 2.2:

Theorem 3.6 provides, for each hyperbolic (g, r), “lots of examples” — in particular, an example lifting a general curve of type (g, r) in characteristic p ≥ 5 — of (X_{K}, D_{K}) which are group-theoretically determined solely by the profinite group π_{1}(X_{K}\D_{K}).

In fact, the exact same arguments of the present §3 show that the analogue of Theorem 3.6 also holds for “Lubin-Tate canonical curves (for various tones ϖ)” — i.e., the curves defined by considering canonical points (cf. [Mzk2], Chapter VIII, §1.1) associated to the canonical Frobenius lifting of [Mzk2], Chapter VIII, Theorem 3.1, in the case of a “VF-pattern Π of pure tone ϖ”.

One feature of the Lubin-Tate case that differs (cf. [Mzk7], Proposition 1.2.1, (vi)) from the “classical ordinary” case of [Mzk1] is that it is not immediately clear that the “Lubin-Tate character”

χ : G_{K} → W (𝔽_{q})^{×}

(where 𝔽_{q} is a finite extension of 𝔽_{p}) is group-theoretic. Note, however, that at least for the “portion of χ modulo p”

G_{K} → 𝔽_{q}^{×}

group-theoreticity — at least after passing to a finite unramified extension of the original base field K (which does not present any problems, from the point of view of proving the Lubin-Tate analogue of Theorem 3.6) — is a consequence of the fact that the field structure on (the union of 0 and) the torsion in G_{K}^{ab} of order prime to p is group-theoretic (cf. [Mzk7], Lemma 2.6, Theorem 2.7). This much is sufficient for the Lubin-Tate analogues of Lemmas 3.1, 3.2, 3.3, 3.4, 3.5 (except for the last sentence of the statement of Lemma 3.5, which is, at any rate, not necessary to prove Theorem 3.6), and hence of Theorem 3.6. Finally, once one has proved that the curve in question is (Lubin-Tate) canonical and recovered its canonical representation V_{ℤ_{p}}, at least projectively, then it follows, by considering the W (𝔽_{q})-module analogue of the exact sequence of 𝔽_{p}-vector spaces in the conclusion of Lemma 3.2, over the ordinary locus, that (even if one only knows V_{ℤ_{p}} “projectively”) one may recover the Lubin-Tate character by forming

Hom(third nonzero term of the exact sequence,

first nonzero term of the exact sequence)

— at least up to un “étale twist”, i.e., up to multiplication by some unramified character G_{K} → W (𝔽_{q})^{×}. (We leave the remaining routine technical details of the Lubin-Tate case to the enthusiastic reader.) At the time of writing, it is not clear to the author whether or not it is possible to eliminate this “étale twist”. Since this étale twist corresponds to the well-known dependence of the Lubin-Tate group on the choice of uniformizer, this indeterminacy with respect to an étale twist may be thought of as being related to the fact that (at the present time) the author is unable to prove the group-theoreticity of the natural uniformizer “p” — cf. [Mzk7], Remark 2.7.2.

At any rate, the theory of the present §3 constitutes the case of “tone 0”. Moreover, one checks easily — by considering the “FL-bundle” (as in [Mzk1], Chapter II, Proposition 1.2) determined by the lifting modulo p^{2} — that canonical curves of distinct tones are never isomorphic. Thus, (at least when r = 0) the Lubin-Tate canonical curves give rise to (strictly) more examples of (proper) hyperbolic X_{K} which are group-theoretically determined solely by the profinite group π_{1}(X_{K}). This prompts the following interesting question: Is this “list” complete? — i.e.:

Do there exist any other curves (X_{K}, D_{K}) that are group-theoretically determined solely by the profinite group π_{1}(X_{K}\D_{K})?

At the time of writing, the author does not even have a conjectural answer to this question.

Remark 3.6.2. One interesting aspect of the theory of the present §3 is that, to the knowledge of the author:

It constitutes the first application of the “p-adic Teichmüller theory” of [Mzk1], [Mzk2], to prove a hitherto unknown result (cf. Corollary 3.8 below) that lies outside — i.e., can be stated without using the terminology, concepts, or results of — the theory of [Mzk1], [Mzk2].

Moreover, not only does this constitute the first application of “p-adic Teichmüller theory” to prove a new result (cf. the proof of the irreducibility of the moduli stack via p-adic Teichmüller theory in [Mzk2], Chapter III, §2.5 — an application, albeit to an old result), it is interesting relative to the original philosophical motivation for this theory, involving the analogy to uniformization theory/Teichmüller theory over the complex numbers, which was to construct a p-adic theory that would allow one to prove a “p-adic version of the Grothendieck Conjecture” as in [Mzk6] — cf. [Mzk2], Introduction, §0.10, for more on these ideas.

Remark 3.6.3. One interesting point of view is the following:

For a hyperbolic curve X_{K} over a finite extension K of ℚ_{p}, consideration of the profinite group Π_{X_{K}} should be thought of as the arithmetic analogue of considering a hyperbolic curve (which is given a priori) over 𝔽_{p}[[t]][t^{−1}] — where t is an indeterminate which, perhaps, should be thought of as the symbol “p” — “in the absolute”, i.e., “stripped of its structure morphism to a specific copy of 𝔽_{p}[[t]][t^{−1}]”, or, alternatively, “when we allow the indeterminate t to vary freely (in 𝔽_{p}[[t]])”.

Thus, from this point of view, it is natural to expect that the hyperbolic curves X_{K} most likely to be recoverable from the absolute datum Π_{X_{K}} are those which are “defined over some (fictitious) absolute field of constants inside K”

— hence which have moduli that are invariant with respect to changes of variable t ↦ t + t^{2} + . . .. In particular, since one expects canonical curves to be arithmetic analogues of “curves defined over the constant field” (cf. [Mzk2], Introduction, §2.3), it is perhaps not surprising that they should satisfy the property of Theorem 3.6.

Moreover, this point of view suggests that:

Perhaps it is natural to regard Theorem 3.6 as the proper analogue for hyperbolic canonical curves of the fact that Serre-Tate canonical liftings (of abelian varieties) are defined over number fields (cf. [Mzk2], Introduction, §2.1, Open Question (7)).

Indeed, one way of showing that Serre-Tate canonical liftings — or, indeed, arbitrary abelian varieties with lots of endomorphisms — are defined over number fields is by thinking of the algebraic extension \overline{ℚ}_{p} of ℚ_{p} over which such abelian varieties are a priori defined “in the absolute”, i.e., as a transcendental extension of ℚ and considering what happens when one transports such abelian varieties via arbitrary field automorphisms of \overline{ℚ}_{p}. Such field automorphisms are reminiscent of the “changes of variable” appearing in the approach to thinking about“recovering X_{K} from the absolute datum Π_{X_{K}} ” described above.

Remark 3.6.4. The “rigidity” of canonical curves — in the sense that they are determined by the existence of the unique G_{K}-invariant lifting “V_{ℤ_{p}}” of the representation V_{𝔽_{p}} (cf. Lemma 3.5 and its proof) — is reminiscent, at least at a technical level, of the theory of deformations of representations applied in Wiles’ famous proof of the “modularity conjecture” (cf. [Wiles]). It would be interesting if this analogy could be pursued in more detail in the future.

Remark 3.6.5. Just as the Serre-Tate canonical coordinates are used in [Mzk5], §9, to prove a weak p-adic Grothendieck Conjecture-type result for hyperbolic curves over p-adic fields whose Jacobians have ordinary reduction, the techniques of the present § may be applied — by using the canonical coordinates of [Mzk1]— to prove a similar (but in some sense even weaker) padic Grothendieck Conjecture-type result for hyperbolic curves over absolutely unramified p-adic fields which are isomorphic to canonical curves (as in [Mzk1]) modulo p^{2}. Thus, the true significance of the theory of the present § lies in its wide applicability in the canonical lifting case (cf. Corollary 3.8 below), a feature which differs substantially from the theory in the case of ordinary Jacobians (cf. Remark 1.1.1).

Definition 3.7. Let Y_{L} be a hyperbolic curve over a finite extension L of ℚ_{p}. Then we shall say that Y_{L} is absolute if for every other hyperbolic curve Y′_{L′} over a finite extension L′ of ℚ_{p},

π_{1}(Y_{L}) ≅ π_{1}(Y′_{L′})

(as profinite groups) implies that Y′_{L′} is isomorphic as a ℚ_{p}-scheme to Y_{L}. Also, we shall refer to points in moduli stacks of hyperbolic curves over ℚ_{p} that are defined by absolute hyperbolic curves as absolute.

Corollary 3.8. (Application of p-adic Teichmüller Theory)

(i) A general pointed smooth curve (X_{k}, D_{k}) of type (g, r), where 2g−2+r > 0, over a finite field k of characteristic p ≥ 5 may be lifted to a pointed smooth curve (X_{K}, D_{K}) over the quotient field K of the ring of Witt vectors A = W (k) such that the hyperbolic curve X_{K}\D_{K} is absolute.

(ii) In particular, for each (g, r), p ≥ 5, there exists a Zariski dense — hence (at least when 3g − 3 + r ≥ 1) infinite — set of absolute points, valued in absolutely unramified finite extensions of ℚ_{p}, of the moduli stack of hyperbolic curves of type (g, r) over ℚ_{p}.

Proof. Assertion (i) follows formally from Theorem 3.6; [Mzk7], Lemmas 1.1.4, 1.1.5; and [Mzk7], Proposition 1.2.1, (v). Assertion (ii) follows formally from assertion (i) and the following elementary argument: The scheme-theoretic closure of the points of assertion (i) in the (compactified) moduli stack over ℤ_{p} forms a ℤ_{p}- flat proper algebraic stack Z with the property that Z ⊗𝔽_{p} is equal to the entire moduli stack over 𝔽_{p}, hence is smooth of dimension 3g−3+r. But this implies — by ℤ_{p}-flatness — that Z ⊗ ℚ_{p} is also of dimension 3g − 3 + r, hence equal to the entire moduli stack over ℚ_{p}, as desired. ◯

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Shinichi Mochizuki Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502, Japan motizuki@kurims.kyotou.ac.jp