topics in absolute anabelian geometry ii

7/27/2019 Topics in Absolute Anabelian Geometry II

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TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II:

DECOMPOSITION GROUPS AND ENDOMORPHISMS

Shinichi Mochizuki

April 2012

Abstract. The present paper, which forms the second part of a three-part seriesin which we study absolute anabelian geometry from an algorithmic point of

view, focuses on the study of the closely related notions of decomposition groupsand endomorphisms in this anabelian context. We begin by studying an abstractcombinatorial analogueof the algebro-geometric notion of a stable polycurve [i.e., asuccessive extension of families of stable curves] and showing that the geometryof log divisors on stable polycurves may be extended, in a purely group-theoreticfashion, to this abstract combinatorial analogue; this leads to various anabelianresults concerning configuration spaces. We then turn to the study of the absolutepro- anabelian geometryof hyperbolic curves over mixed-characteristic local fields,for a set of primes of cardinality 2 that contains the residue characteristic of thebase field. In particular, we prove a certain pro-p resolution of nonsingularitiestype result, which implies a conditional anabelian result to the effect that thecondition, on an isomorphism of arithmetic fundamental groups, of preservation ofdecomposition groups of most closed points implies that the isomorphism arises

from an isomorphism of schemes i.e., in a word, point-theoreticity impliesgeometricity; a non-conditional version of this result is then obtained for pro-curves obtained by removing from a proper curve some set of closed points whichis p-adically dense in a Galois-compatible fashion. Finally, we study, from analgorithmic point of view, the theory of Belyi and elliptic cuspidalizations, i.e.,group-theoretic reconstruction algorithms for the arithmetic fundamental group ofan open subschemeof a hyperbolic curve that arise from consideration of certainendomorphismsdetermined by Belyi maps and endomorphisms of elliptic curves.

Contents:0. Notations and Conventions1. A Combinatorial Analogue of Stable Polycurves2. Geometric Uniformly Toral Neighborhoods3. Elliptic and Belyi Cuspidalizations

Introduction

In the present paper, which forms the second part of a three-part series, we

continue our discussion of various topics in absolute anabelian geometry from

2000 Mathematical Subject Classification. Primary 14H30; Secondary 14H25.

Typeset by AMS-TEX

1


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2 SHINICHI MOCHIZUKI

agroup-theoretic algorithmicpoint of view, as discussed in the Introductionto [Mzk15]. The topics presented in the present paper center around the followingtwo themes:

(A) [the subgroups of arithmetic fundamental groups constituted by] de-composition groupsof subvarieties of a given variety [such as closedpoints, divisors] as a crucial tool that leads to absolute anabelian results;

(B) hidden endomorphisms which may be thought of as hiddensymmetries of hyperbolic curves that give rise to various absoluteanabelian results.

In fact, decomposition groups and endomorphisms are, in a certain sense, re-lated notions that is to say, the monoid of endomorphisms of a variety maybe thought of as a sort of decomposition group of the generic point!

With regard to the theme (B), we recall that the endomorphisms of an abelianvariety play a fundamental role in the theory of abelian varieties [e.g., ellipticcurves!]. Unlike abelian varieties, hyperbolic curves [say, in characteristic zero]do not have sufficient endomorphisms in the literal, scheme-theoretic sense toform the basis for an interesting theory. This difference between abelian varietiesand hyperbolic curves may be thought of, at a certain level, as reflecting the dif-ference between linear Euclidean geometriesand non-linear hyperbolic geometries.From this point of view, it is natural to search for hidden endomorphisms thatare, in some way, related to the intrinsic non-linear hyperbolic geometryof a hy-perbolic curve. Examples [that appear in previous papers of the author] of suchhidden endomorphisms which exhibit a remarkable tendency to be related [forinstance, via some induced action on the arithmetic fundamental group] to somesort ofanabelian result are the following:

(i) the interpretation of theautomorphism group P SL2(R) of the uni-versal covering of a hyperbolic Riemann surface as an object that givesrise to a certain Grothendieck Conjecture-type result in the geometryof categories [cf. [Mzk11], Theorem 1.12];

(ii) the interpretation of the theory ofTeichmuller mappings [a sort ofendomorphism cf. (iii) below] between hyperbolic Riemann surfaces asa Grothendieck Conjecture-type result in the geometry of categories[cf. [Mzk11], Theorem 2.3];

(iii) the use of theendomorphismsconstituted by Frobenius liftings inthe form ofp-adic Teichmuller theory to obtain the absolute anabelianresultconstituted by [Mzk6], Corollary 3.8;

(iv) the use of theendomorphismrings ofLubin-Tate groupsto obtain theabsolute anabelian resultconstituted by [Mzk15], Corollaries 3.8, 3.9.

The main resultsof the present paper in which both themes (A) and (B)play a central role are the following:

(1) In 1, we develop apurely combinatorial approachto the algebro-geometricnotion of astable polycurve[cf. [Mzk2], Definition 4.5]. This approach may


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TOPICS IN ABSOLUTE ANABELIAN GEOMETRY II 3

be thought of as being motivated by the purely combinatorial approach tothe notion of a stable curve given in [Mzk13]. Moreover, in 1, we applythe theory of [Mzk13] to give, in effect, group-theoretic algorithms

for reconstructing the abstract combinatorial analogue of the geome-try of the various divisors in the form of inertiaand decompositiongroups associated to the canonical log structure of a stable polycurve[cf. Theorem 1.7]. These techniques, together with the theory of [MT],give rise to

various relative and absolute anabelian results concerningconfiguration spaces associated to hyperbolic curves

[cf. Corollaries 1.10, 1.11]. Relative to the discussion above of hidden en-domorphisms, we observe that such configuration spaces may be thought

of as representing a sort oftautological endomorphism/correspondenceof the hyperbolic curve in question.

(2) In 2, we study the absolute pro- anabelian geometry of hyperboliccurves over mixed-characteristic local fields, for a set of primes of car-dinality 2 that contains the residue characteristic of the base field. Inparticular, we show that the condition, on an isomorphism of arithmeticfundamental groups, of preservation of decomposition groups of mostclosed points implies that the isomorphism arises from an isomorphism ofschemes i.e., in a word,

point-theoreticityimpliesgeometricity

[cf. Corollary 2.9]. This condition may be removed if one works withpro-curvesobtained by removing from a proper curve some set of closed pointswhich is p-adically dense in a Galois-compatible fashion [cf. Corollary2.10]. The key technical result that underlies these anabelian results is acertainpro-presolution of nonsingularitiestype result [cf. Lemma2.6; Remark 2.6.1; Corollary 2.11] i.e., a result reminiscent of the main[profinite] results of [Tama2]. This technical result allows one to applythe theory of uniformly toral neighborhoods developed in [Mzk15],

3. Relative to the discussion above of hidden endomorphisms, thistechnical result is interesting [cf., e.g., (iii) above] in that one centralstep of the proof of the technical result is quite similar to the well-knownclassical argument that implies the nonexistence of a Frobenius liftingfor stable curves over the ring of Witt vectors of a finite field [cf. Remark2.6.2].

(3) In 3, we re-examine the theory of [Mzk8], 2, for reconstructing thedecomposition groups of closed points from the point of view of the presentseries of developinggroup-theoretic algorithms. In particular, we observethat these group-theoretic algorithms allow one to use

Belyi maps andendomorphisms of elliptic curves to con-struct [not only decomposition groups of closed points, but also]cuspidalizations


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[i.e., the full arithmetic fundamental groups of the open subschemes ob-tained by removing various closed points cf. the theory of [Mzk14]] as-sociated to various types of closed points [cf. Corollaries 3.3, 3.4, 3.7, 3.8].

Relative to the discussion above of hidden endomorphisms, the theoryofBelyiand elliptic cuspidalizationsgiven in3 illustrates quite explicitlyhowendomorphisms[arising from Belyi maps or endomorphisms of ellipticcurves] can give rise to group-theoretic reconstruction algorithms.

Finally, we remarkthat although the algorithmic approach to stating anabelianresults is not carried out very explicitly in1, 2 [by comparison to3 or [Mzk15]],the translation into algorithmic language of the more traditional GrothendieckConjecture-type statements of the main results of1, 2 is quite routine. [Here,it should be noted that the results of 1 that depend on Uchidas theorem i.e., Theorem 1.8, (ii); Corollary 1.11, (iv) constitute a notatable exception tothis remark, an exception that will be discussed in more detail in [Mzk16] cf.,e.g., [Mzk16], Remark 1.9.5.] That is to say, this translation was not carried outexplicitlyby the author solely because of the complexityof the algorithms implicitin 1, 2, i.e., not as a result of any substantive mathematical obstacles.

Acknowledgements:

I would like to thank Akio Tamagawa and Yuichiro Hoshi for many helpfuldiscussions concerning the material presented in this paper and Emmanuel Lepage

for a suggestion that led to Remark 2.11.1, (i).


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Section 0: Notations and Conventions

We shall continue to use the Notations and Conventions of [Mzk15], 0. In

addition, we shall use the following notation and conventions:

Topological Groups:

LetGbe a topologically finitely generated, slim profinite group. Thus,Gadmitsa basis of characteristic open subgroups. Any such basis determines a profinitetopology on the groups Aut(G), Out(G). If : H Out(G) is any continuoushomomorphismof profinite groups, then we denote by

Gout H

the profinite groupobtained by pulling back the natural exact sequenceof profinitegroups 1 G Aut(G) Out(G) 1 via . Thus, we have a natural exact

sequenceof profinite groups 1 G Gout H H 1.

Semi-graphs:

Let be aconnected semi-graph[cf., e.g., [Mzk9],1, for a review of the theoryof semi-graphs]. We shall refer to the [possibly infinite] dimension over Q of the

singular homology module H1(,Q) as the loop-rank lp-rk() of . We shall saythat is loop-ample if for any edge e of , the semi-graph obtained from byremoving e remains connected. We shall say that is untangled if every closededge of abuts to two distinct vertices [cf. [Mzk9], 1]. We shall say that isedge-paired(respectively, edge-even) if is untangled, and, moreover, for any two[not necessarily distinct!] vertices v, v of , the set of edges e of such that eabuts to a vertex w of if and only ifw {v, v}is either empty or of cardinality 2 (respectively, empty or of even cardinality). [Thus, one verifies immediatelythat if is edge-paired (respectively,edge-even), then it isloop-ample(respectively,edge-paired).] We shall refer to as a simple pathin any connected subgraph such that the following conditions are satisfied: (a)is a finite treethat has at leastone edge; (b) given any vertex v of, there exist at most two branches of edges ofthat abut to v. Thus, [one verifies easily that] a simple path has precisely twoverticesv such that there exists precisely one branch of an edge ofthat abuts tov; we shall refer to these two vertices as the terminal verticesof the simple path .If, are simple paths in such that the terminal vertices of, coincide, thenwe shall say that , are co-terminal.

Log Schemes:

We shall often regard a schemeas a log scheme equipped the trivial log struc-ture. Any fiber productof fs [i.e., fine, saturated] log schemes is to be taken in thecategory of fs log schemes. In particular, the underlying scheme of such a productisfiniteover, but not necessarily isomorphicto, the fiber product of the underlyingschemes.


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Curves:

We shall refer to a hyperbolic orbicurveXas semi-elliptic[i.e., of type (1, 1)

in the terminology of [Mzk12], 0] if there exists a finite etale double coveringY X, where Y is a once-punctured elliptic curve, and the covering is given bythe stack-theoretic quotient ofYby the action of1 [i.e., relative to the groupoperation on the elliptic curve given by the canonical compactification ofY].

Fori = 1, 2, letXi be a hyperbolic orbicurve over a field ki. Then we shall saythat X1, X2 are isogenous [cf. [Mzk14], 0] if there exists a hyperbolic orbicurveXover a field k, together with finite etale morphisms XXi, for i= 1, 2. Notethat in this situation, the morphisms X Xi induce finite separable inclusions offieldski k. [Indeed, this follows immediately from the easily verified fact thatevery subgroup G (X, OX) such that G

{0} determines a fieldis necessarily

contained in k

.]We shall use the term stable log curveas it was defined in [Mzk9], 0. Let

Xlog Slog

be a stable log curveover an fs log scheme Slog, where S= Spec(k) for some fieldk; k a separable closureofk . Then we shall refer to as the loop-ranklp-rk(Xlog) [orlp-rk(X)] ofXlog [or X] the loop-rank of the dual graph ofXlog kk [or Xkk ].We shall say that Xlog [or X] is loop-ample(respectively, untangled; edge-paired;edge-even) if the dual semi-graph with compact structure [cf. [Mzk5], Appendix]

of Xlog

k k is loop-ample (respectively, untangled; edge-paired; edge-even) [asa connected semi-graph]. We shall say that Xlog [or X] is sturdy if every thenormalization of irreducible component ofXis of genus 2 [cf. [Mzk13], Remark1.1.5].

Observe that for any prime number l invertible on S, there exist an fs logscheme Tlog over Slog, where T= Spec(k), for some finite separable extension k

of k, and a connected Galois log admissible covering Ylog Xlog Slog Tlog [cf.

[Mzk1],3] of degree a power ofl such thatYlog issturdyand edge-paired[hence, inparticular, untangledand loop-ample]; if, moreover, l = 2, then one may also takeYlog to beedge-even. [Indeed, to verify this observation, we may assume that k = k.

Then note that any hyperbolic curve Uoverk admits a connected finite etale Galoiscovering V Uof degree a power of l such that V is of genus 2 and ramifiedwith ramification index l2 at each of the cusps ofV. Thus, by gluing together suchcoverings at the nodes ofX, one concludes that there exists a connected Galois log

admissible covering Zlog1 Xlog Slog T

log of degree a power of l which is totallyramified over every node ofXwith ramification indexl2 such that every irreducible

component ofZ1 is of genus 2 i.e., Zlog1 is sturdy. Next, observe that there

exists a connected Galois log admissible coveringZlog2 Zlog1 of degree a power ofl

that arises from a covering of the dual graph ofZ1 such thatZlog2 isuntangled[and

still sturdy]. Finally, observe that there exists a connected Galois log admissible

covering Zlog3 Zlog2 of degree a positive power ofl which restricts to a connectedfinite etale covering over every irreducible component ofZ2 [hence isunramifiedat

the nodes] such thatZlog3 isedge-paired[and still sturdy and untangled] for arbitrary

l and edge-evenwhen l= 2. Thus, we may take Ylogdef= Zlog3 .]


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Observe that ifX is loop-ample, then for every point x X(k) which is nota unique cusp ofX [i.e., either x is not a cusp or ifx is a cusp, then it is not theunique cusp ofX], the evaluation map

H0(X, Xlog/Slog) Xlog/Slog |x

issurjective. Indeed, by considering the long exact sequence associated to the shortexact sequence 0 Xlog/Slog OX Ix Xlog/Slog Xlog/Slog |x 0, where

Ix OX is the sheaf of ideals corresponding to x, one verifies immediately that itsuffices to show that the surjection

Hxdef= H1(X, Xlog/Slog OX Ix) H

def= H1(X, Xlog/Slog)

isinjective. Ifx is anode, then the fact thatX isloop-ampleimplies [by computing

via Serre duality] that either dimk(Hx) = dimk(H) = 1 [ if X has no cusps] ordimk(Hx) = dimk(H) = 0 [ifX has cusps]. Thus, we may assume thatx is not anode, so the surjection Hx H is dual to the injection

Mdef= H0(X, OX(D)) Mx

def= H0(X, OX(x D))

where we write D X for the divisor of cusps of X. If D is of degree 2,then dimk(M) = dimk(Mx) = 0. Thus, we may assume thatD is of degree 1,which implies that x is not a cusp. Write C for the irreducible component ofXcontaining x. Then any nonzero element ofMx that is not contained in the image

ofM determines a morphism : XP1

kthat isof degree1 onC i.e., determinesan isomorphismC P1k and constant on the other irreducible components of

X. Since X is loop-ample, it follows that the dual graph of X either has noedgesor admits a loop containing the vertex determined by C. On the other hand,the existence of such a loop contradicts the fact that determines an isomorphismC P1k. Thus, we may assume thatX=C

= P1k. But since D is of degree 1,this contradicts the stabilityofXlog.

Finally, let U be a hyperbolic curveover an algebraically closed field k and l aprime number invertible in k. Suppose that the cardinality r of the set of cusps ofU is 2, and, moreover, that, ifl= 2, then r iseven. Thenobservethat it follows

immediately from the well-known structure of the maximal pro-l quotient of theabelianization of the etale fundamental group ofU that

for every power ln of l, where n is a positive integer, there exists a cycliccoveringV Uof degree ln that is totally ramifiedover the cusps ofU.

Indeed, this observation is an immediate consequence of the elementary fact that,in light of our assumptions on r, there always exist r 1 integers prime to l whosesum is also prime to l. We shall often make use of the assumption that a stablelog curve is edge-paired or, when l = 2, edge-even by applying the aboveobservation to the various connected components of the complement of the cuspsand nodes of the stable log curve.


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Section 1: A Combinatorial Analogue of Stable Polycurves

In the present 1, we apply the theory of [Mzk13] to study a sort of purely

group-theoretic, combinatorialanalogue [cf. Definition 1.5 below] of the notion of astable polycurveintroduced in [Mzk2], Definition 4.5. This allows one to reconstructtheabstract combinatorial analogueof thegeometry of log divisors[i.e., divisorsassociated to the log structure of a stable polycurve] of such a combinatorial objectvia group theory [cf. Theorem 1.7]. Finally, we apply the theory of [MT] to obtainvarious consequences of the theory of the present 1 [cf. Corollaries 1.10, 1.11]concerning the absolute anabelian geometry of configuration spaces.

We begin by recalling the discussion of [Mzk13], Example 2.5.

Example 1.1. Stable Log Curves over a Logarithmic Point (Revisited).

(i) Let k be a separably closed field; a nonempty set of prime numbers in-vertible in k; M Q the monoid of positive rational numbers with denomina-tors invertible in k; Slog (respectively, Tlog) the log scheme obtained by equipping

S def= Spec(k) (respectively,T

def= Spec(k)) with the log structure determined by the

chart N 1 0 k (respectively, M 1 0 k); Tlog Slog the morphismdetermined by the natural inclusion N M;

Xlog Slog

a stable log curve over Slog. Thus, the profinite group ISlogdef= Aut(Tlog/Slog)

admits a natural isomorphismISlog Hom(Q/Z, k) and fits into an natural exact

sequence

1 Xlogdef= 1(X

log Slog Tlog) Xlog

def= 1(X

log) ISlog 1

where we write 1() for the log fundamental group of the log scheme inparentheses [which amounts, in this case, to the fundamental group arising fromthe admissible coveringsofXlog], relative to an appropriate choice of basepoint [cf.[Ill] for a survey of the theory of log fundamental groups]. In particular, if we write

ISlog for the maximal pro- quotient of ISlog , then as abstract profinite groups,

ISlog=Z, where we writeZ for the maximal pro- quotientofZ.

(ii) On the other hand,Xlog determines asemi-graph of anabelioids[cf. [Mzk9],Definition 2.1]of pro-PSC-type[cf. [Mzk13], Definition 1.1, (i)], whose underlyingsemi-graph we denote by G. Thus, for each vertexv[corresponding to an irreduciblecomponent ofXlog] (respectively, edge e [corresponding to a node or cusp ofXlog])ofG, we have a connected anabelioid [i.e., a Galois category] Gv (respectively, Ge),and for each branch b of an edge e abutting to a vertexv , we are given a morphismof anabelioids Ge Gv. Then the maximal pro- completion of Xlog may beidentifiedwith the PSC-fundamental group G associated to G. Also, we recallthat G is slim [cf., e.g., [Mzk13], Remark 1.1.3], and that the groups Aut(G),Out(G) may be equipped with profinite topologiesin such a way that the naturalmorphism

Aut(G) Out(G)


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is acontinuous injection [cf. the discussion at the beginning of [Mzk13], 2], whichwe shall use to identifyAut(G) with its image in Out(G). In particular, we obtaina natural continuous homomorphism ISlog Aut(G). Moreover, it follows imme-

diately from the well-known structure of admissible coverings at nodes [cf., e.g.,[Mzk1], 3.23] that this homomorphism factors through ISlog , hence determines anatural continuous homomorphismI : I

Slog Aut(G). Also, we recall that each

vertex v (respectively, edge e) ofG determines a(n) verticial subgroup v G(respectively,edge-like subgroup e G), which is well-defined up to conjugation cf. [Mzk13], Definition 1.1, (ii). Here, the edge-like subgroups e may be eithernodalor cuspidal, depending on whether e corresponds to a node or to a cusp. Ifan edge e corresponds to a node (respectively, cusp), then we shall simply say thate is a node (respectively, cusp).

(iii) Let e be a nodeofX. Write Me for the stalk of the characteristic sheafof the log scheme Xlog at e; MS for the stalk of the characteristic sheaf of thelog scheme Slog at the tautological S-valued point ofS. Thus, MS = N; we havea natural inclusion MS Me, with respect to which we shall often [by abuse ofnotation] identifyMSwith its image inMe. Write Me for theunique generatorof [the image of] MS. Then there exist elements , Me satisfying the relation

+ =ie

for some positive integer ie, which we shall refer to as the indexof the nodee, suchthat Me is generated by , , . Also, we shall write i

e for the largest positive

integer j such that ie/j is a product of primes and refer to ie as the -indexof the node e. One verifies easily that the set of elements {, } of Me may becharacterized intrisicallyas the set of elements Me\MSsuch that any relation =n + for n a positive integer, Me\MS,

MS implies that n= 1, = 0. In particular, ie, i

e are well-definedand depend only on the isomorphism

classof the pair consisting of the monoid Me and the submonoid Me given bythe image ofMS.

Remark 1.1.1. Of course, in Example 1.1, it is not necessaryto assume that kis separably closed [cf. [Mzk13], Example 2.5]. If k is not separably closed, then

one must also contend with the action of the absolute Galois group of k. Moregenerally, for the theory of the present 1, it is not even necessaryto assume thatan additional profinite group acting on Garises from scheme theory. It is thispoint of view that formed the motivation for Definition 1.2 below.

Definition 1.2. In the notation of Example 1.1:

(i) Let H : H Aut(G) ( Out(G)) be a continuous homomorphism ofprofinite groups; suppose that Xlog is nonsingular [i.e., has no nodes]. Then weshall refer to as a [pro-] PSC-extension [i.e., pointed stable curve extension]

any extension of profinite groups that is isomorphic via an isomorphism whichwe shall refer to as the structure of [pro-] PSC-extension to an extension ofthe form

1 G Hdef= (G

out H) H 1


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[cf. 0 for more on the notation out ], which we shall refer to as the PSC-

extension associated to the construction data(Xlog Slog, , G, H). In this sit-uation, each [necessarily cuspidal] edge e ofG determines [up to conjugation in

G ] a subgroup e G , whose normalizer De def= NH (e) in H we shallrefer to as the decomposition group associated to the cusp e; we shall refer to

Iedef= e = De

G De [cf. [Mzk13], Proposition 1.2, (ii)] as the inertia group

associated to the cusp e. Finally, we shall apply the terminology applied to objectsassociated to 1 G H H 1 to the objects associated to an arbitraryPSC-extension via its structure of PSC-extension isomorphism.

(ii) Let H : H Aut(G) ( Out(G)) be a continuous homomorphism ofprofinite groups; : ISlog H a continuous injection of profinite groups with

normal imagesuch that H = I. Suppose thatXlog isarbitrary[i.e.,Xmay be

singularor nonsingular]. Then we shall refer to as a [pro-] DPSC-extension [i.e.,degenerating pointed stable curve extension] any extension of profinite groupsthat is isomorphic via an isomorphism which we shall refer to as the structureof [pro-] DPSC-extension to an extension of the form

1 G Hdef= (G

out H) H 1

which we shall refer to as the DPSC-extension associated to the constructiondata(Xlog Slog, , G, H, ). In this situation, we shall refer to the image I Hof as the inertia subgroup ofHand to the extension

1 G Idef= (G

out I) I 1

[so I= HHI H] as the[pro-] IPSC-extension[i.e., inertial pointed stablecurve extension] associated to the construction data (Xlog Slog, , G, H, );each vertex v (respectively, edge e) ofG determines [up to conjugation in G] asubgroup v G (respectively, e G), whose normalizer

Dvdef= NH (v) (respectively, De

def= NH (e))

in Hwe shall refer to as the decomposition group associated to v (respectively,e);

for v arbitrary (respectively, e a node), we shall refer to the centralizer

Ivdef= ZI (v) Dv (respectively, Ie

def= ZI (e) De)

as theinertia groupassociated tov (respectively,e). Ifeis acuspofG, then we shall

refer to Iedef= e=De

G De [cf. [Mzk13], Proposition 1.2, (ii)] as the inertia

group associated to the cusp e. Finally, we shall apply the terminology applied toobjects associated to 1 G H H 1 to the objects associated to anarbitrary DPSC-extension via its structure of DPSC-extension isomorphism.

Remark 1.2.1. Note that in the situation of Definition 1.2, (i) (respectively,(ii); (ii)), any open subgroup of H (respectively, I; H) [equipped with theinduced extension structure] admits a structure of [pro-] PSC-extension (respec-tively, IPSC-extension; DPSC-extension) for appropriate construction data that


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may be derived from the original construction data. Here, it is important to notethat even if, for instance, an open subgroup of I surjectsontoI, in order to endowthis open subgroup with a structure of IPSC-extension, it may be necessary to re-

placethe inertia subgroup I ofHby some open subgroup ofI. Such replacementsmay be regarded as a sort of abstract group-theoretic analogue of the operation ofpassing to a finite extension of a discretely valued field in order to achieve a situationin which a given hyperbolic curve over the original field has stable reduction.

Remark 1.2.2. Note that in the situation of Definition 1.2, (ii), the inertiasubgroupI H isnot intrinsically determinedin the sense that any open subgroupofImay also serve as the inertia subgroup ofH cf. the replacement operationdiscussed in Remark 1.2.1.

Remark 1.2.3. Recall that forl , one may construct directly from Ga pro-lcyclotomic characterl : Aut(G) Z

l [cf. [Mzk13], Lemma 2.1]. In particular,

anyHas in Definition 2.1, (i), (ii), determines a pro-l cyclotomic characterl|H :H Zl . The action H is called l-cyclotomically full [cf. [Mzk13], Definition2.3, (ii)] if the image of l|H is open. We shall also apply this terminology l-cyclotomically full to the corresponding PSC-, DPSC-extensions.

Proposition 1.3. (Basic Properties of Inertia and Decomposition Groups)In the notation of Definition 1.2, (ii):

(i) Ife is acusp ofG, then as abstract profinite groups, Ie=Z.(ii) Ife is anode ofG, then we have anatural exact sequence 1 e

Ie I 1; as abstract profinite groups, Ie=Z Z. Ife abuts to verticesv,v,then [for appropriate choices of conjugates of the various inertia groups involved]we have inclusionsIv, Iv Ie, and the natural morphismIv Iv Ie is anopeninjective homomorphism, with image of index equal to ie.

(iii) If v is a vertex ofG, then we have anatural isomorphism Iv I;

Dv

I = Iv v; as abstract profinite groups, Iv =Z. If e is a cusp thatabuts to v, then [for appropriate choices of conjugates of the various inertia and

decomposition groups involved] we have inclusions Ie, Iv DeI, and the nat-ural morphismIe Iv De

I is an isomorphism; in particular, as abstract

profinite groups, De

I =Z Z, and we have a natural exact sequence1 Ie De

I I 1.

(iv) Let v, v be vertices of G. If Dv

Dv

I = {1}, then one of thefollowing three [mutually exclusive] properties holds: (1) v = v; (2) v andv aredistinct, but adjacent [i.e., there exists a nodee that abuts to v, v]; (3) v andv are distinct and non-adjacent, but there exists a vertexv =v, v ofG such thatv is adjacent to v andv. Moreover, in the situation of (2), we haveIv

Iv =

{1}, [for appropriate choices of conjugates of the various inertia and decompositiongroups involved]DvDv I =Ie; in the situation of (3), we havevv =Iv

Iv = Iv

Iv = Iv

Iv = {1}, [for appropriate choices of conjugates ofthe various inertia and decomposition groups involved] Dv

Dv

I = Iv . In

particular,Iv

Iv = {1} implies thatv= v.


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(v) Let v be a vertex of G. Then Dv = CH (Iv) = NH (Iv) is commen-surably terminal in H; Dv

I = CI (Iv) = NI (Iv) = ZI (Iv) is com-

mensurably terminal in I; Dv

G = v iscommensurably terminal in

G.

(vi) Let v be an vertex of G. Then the image of Dv in H is open; on theother hand, ifGhasmore than one vertex [i.e., the curveX issingular], thenDv isnot open inH.

(vii) Lete be anedge ofG. ThenDe=CH (e) =NH (e) iscommensu-rably terminal inH. Ife is anode, thenIe=De

I.

(viii) Lete,e beedgesof G. IfDe

De

I= {1}, then one of the followingtwo [mutually exclusive] properties holds: (1) e = e; (2) e and e are distinct,but abut to the same vertex v, and D

eDe G = {1}. Moreover, in thesituation of (2), [for appropriate choices of conjugates of the various inertia anddecomposition groups involved] we haveIv =De

De

I.

(ix) Let e be anedge ofG. Then the image ofDe inH isopen, but De isnot open inH.

(x) LetI :I Ibe the [outer] homomorphism that arises [by functoriality!]from a log point S X

log(Slog). Let us call I non-verticial (respectively,non-edge-like) if I(I) is not contained in Iv (respectively, Ie) for any vertexv (respectively, edge e) of G. Then if I is non-verticial and non-edge-like, thenthe image of

S is theunique cusp e

ofX such that [for an appropriate choice

of conjugate of De] I(I) De. Now suppose that the image of S is not acusp. Then Isatisfies the condition I(I) =Iv for some vertexv ofG [and anappropriate choice of conjugate ofIv] if and only if the image ofSis anon-nodalpointof the irreducible component ofXcorresponding tov;Iis non-verticial andsatisfies the conditionI(I) Ie for some nodee ofG[and an appropriate choiceof conjugate ofIe] if and only if the image ofS is thenode ofX correspondingto e.

Proof. Assertion (i) follows immediately from the definitions. Next, we consider

assertion (ii). Write(=S) for the closed subscheme ofXdetermined by the nodeofX corresponding to e; log for the result of equipping with the log structure

pulled back from Xlog. Thus, we obtain a natural [outer] homomorphism def=

1(log) 1(X

log) = Xlog I. Now [in the notation of Example 1.1, (iii)] onecomputes easily [by considering the Galois groups of the various Kummer log etale

coverings oflog] that we have natural isomorphismsHom(Mgpe Q/Z, k

),

ISlogHom(MgpS Q/Z, k

) [where gp denotes the groupificationof a monoid].Moreover, if we write

for the maximal pro- quotientof , then one

verifies immediately that the isomorphisms induced on maximal pro- quotients bythese natural isomorphisms are compatible, relative to the surjection

() Hom(Mgpe Q/Z, k

) Z Hom(MgpS Q/Z, k) Z ( ISlog)induced by the inclusion MS Me, with the morphism I

Slog induced by

the composite morphism I I ISlog

= Hom(MgpS Q/Z, k

) Z.


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The kernel of this surjection ISlog may be identified with the profinite group

Hom(Mgpe /MgpS Q/Z, k

) Z, and one verifies immediately [from the definitionofG] that this kernel maps isomorphicallyonto e I. In particular, it follows

that we obtain aninjection Iwhose imagecontainseandsurjects ontoI.Since is abelian, it follows that the image Im(

) of this injection is contained

in Ie; since Ie

G = e [cf. [Mzk13], Proposition 1.2, (ii)], we thus conclude thatIm() = Ie. Now it follows immediately from the definitions that Iv, Iv Ie;moreover, onecomputesimmediately that [in the notation of Example 1.1, (iii)] the

subgroups Iv, Iv Ie correspond to the subgroups of Hom(Mgpe Q/Z, k

) Zconsisting of homomorphisms that vanish on , , respectively. Now the variousassertions contained in the statement of assertion (ii) follow immediately. Thiscompletes the proof of assertion (ii).

Next, we consider assertion (iii). Since v is slim [cf., e.g., [Mzk13], Remark

1.1.3] and commensurably terminal in G [cf. [Mzk13], Proposition 1.2, (ii)], itfollows that Dv

G = v and Iv

G = {1}, so we obtain a natural injection

Iv I. The fact that this injection is, in fact, surjective is immediate from thedefinitions when X is smooth over k and follows from the computation of Ivperformed in the proof of assertion (ii) when X is singular. Next, let us observethat since Iv commutes [by definition!] with v, we obtain a natural morphismIv v Dv

I, which is both injective [since Iv

v = {1}] and surjective

[cf. the isomorphismIv I; the fact that Dv

G = v]. Now suppose that e

is a cusp that abuts to v. Then [for appropriate choices of conjugates] it followsimmediately from the definitions that we have inclusionsIe, Iv De

I, and that

Ie commuteswithIv. Note, moreover, thatDeG =Ie[cf. [Mzk13], Proposition1.2, (ii)]. Thus, the fact that the natural projection Iv I is an isomorphismimplies that we have a natural exact sequence1 Ie De

I I 1, and

that the natural morphism Ie Iv De

I is an isomorphism. This completesthe proof of assertion (iii).

Next, we consider assertion (vii). SinceDe

G = e [cf. [Mzk13], Proposi-tion 1.2, (ii)], it follows that De CH (De) CH (e); on the other hand, by[Mzk13], Proposition 1.2, (i), it follows that CH (e) = NH (e) (= De); thus,De = CH (De) = CH (e) = NH (e), as desired. Now it remains only to con-sider the case where e is a node. In this case, since Ie is abelian [cf. assertion (ii)],

it follows that Ie ZI (Ie) CI (Ie) CI (IeG) =CI (e); thus, the factthatDe

I=CI (e) =CI (Ie) =Ie follows from the fact that Ie surjects onto

I [cf. assertion (ii)], together with the commensurable terminalityof e in G [cf.[Mzk13], Proposition 1.2, (ii)]. This completes the proof of assertion (vii).

Next, we consider assertion (iv). Suppose that (2) holds. Then it follows fromassertions (ii), (iii) [and the definitions] thatIv

Iv = {1},Ie= e Iv = e Iv

Dv

Dv

I ZI (Iv Iv) CI (Ie). On the other hand, by assertion (vii),we have CI (Ie) = Ie. But this implies that Dv

Dv

I = Ie. Next, suppose

that (3) holds, and that v

v = {1}. Then it follows immediately from ourdiscussion of the situation in which (2) holds [cf. also assertion (iii)] that Iv

DvDv I I [so Iv = DvDv I], Iv Iv Iv Iv = Iv Iv ={1}. Thus, to complete the proof of assertion (iv), it suffices to verify, under theassumption that (1) and (2) are false, that (3) holds, and that v

v = {1}.

Write Cv, Cv for the irreducible components of X corresponding to v, v.


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Suppose that both (1) and (2) arefalse. Thus, X issingular, so v is a free pro-group [cf. [Mzk13], Remark 1.1.3], hence torsion-free. Thus, Iv v = Dv

I

[cf. assertion (iii)] is torsion-free, so by replacing Hby an open subgroup of H

[cf. Remark 1.2.1], we may assume without loss of generality that G [i.e., Xlog

]is sturdy [cf. 0], and that G is edge-paired [cf. 0]. Also, by projecting to themaximal pro-l quotients, for some l , of the various pro- groups involved, wemay assume without loss of generality [for the remainder of the proof of assertion(iv)] that = {l}. When l= 2, we may also assume without loss of generality [byreplacing Hby an open subgroup of H] that G isedge-even[cf. 0].

Now I claim that Dv

Dv

G = v

v = {1}. Indeed, suppose thatv

v = {1}. Then one verifies immediately that there exist log admissiblecoverings [cf. [Mzk1],3]Ylog Xlog SlogT

log, corresponding to open subgroupsJ G , which are splitover Cv [so v J, v

v v

J], but determine

arbitrarily small neighborhoods vJ of the identity element in v. [Here, wenote that the existence of such coverings follows immediately from the fact thatXlog is edge-pairedfor arbitrary l and edge-evenwhen l = 2. That is to say, onestarts by constructing the covering over Cv in such a way that the ramificationindicesat the nodes and cusps ofCv are all equal; one then extends the coveringover the irreducible components ofX adjacenttov [by applying the fact that Xlog

is edge-pairedfor arbitrary l and edge-evenwhen l = 2 cf. the discussion of0]in such a way that the covering is unramifiedover the nodes of these irreduciblecomponents that do not abut to Cv; finally, one extends the covering to a splitcovering over the remaining portion ofX [which includes Cv !].] But the existence

of such J implies that vv ={1}, a contradiction. This completes the proofof the claim. Thus, the natural projection DvDv II has nontrivial openimage [since = {l}], which we denote by Iv,v I. Moreover, to complete theproof of assertion (iv), it suffices to derive a contradiction under the assumptionthat (1), (2), and (3) are false. Thus, for the remainder of the proof of assertion(iv), we assume that (1), (2), and (3) are false.

Write C+v X for the unionofCv and the irreducible components ofX thatare adjacentto Cv. We shall refer to a vertex ofGas a C

+v -vertexif it corresponds

to an irreducible component of C+v . We shall say that a nodee is a bridge nodeif it abuts both to a C+v -vertex and to a non-C

+v -vertex. Thus, no bridge node

abuts to v. Now let us write iv for the least common multiple of the indices ieof the bridge nodes e; iv for the largest nonnegative power of l dividing iv. Let

d def

= l iv [I : Iv,v ]; d def

= l iv [I : Iv,v ] [so d is the largest positive power

of l dividing d]. Here, we observe that for any open subgroup J0 I such thatDv

J0 surjectsonto I [cf. assertion (iii)], and Dv

I J0, it holds that

[I :Iv,v ] = [(Dv

J0) : (v

J0) (Dv

Dv

I)]

[where we note that Dv

Dv

I Dv

J0]. Thus, it suffices to constructopen subgroups J J0 I such that Dv

J0 surjects onto I, v

J0 J,

and Dv IJ [which implies that (vJ0) (DvDv I) DvJ], but[Dv

J0 :Dv

J]> [I :Iv,v ].

To this end, let us first observe that the characteristic sheafof the log schemeXlog admits asectionoverXsatisfying the following properties: (a)vanisheson


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the open subscheme ofXgiven by the complement ofC+v [hence, in particular, onCv ]; (b) coincides with iv MS[cf. the notation of Example 1.1, (iii)] at thegeneric points ofC+v ; (c)coincides with either (iv/ie) Me or (iv/ie) Me

[cf. the notation of Example 1.1, (iii)] at each bridge nodee. [Indeed, the existenceof such a section follows immediately from the discussion of Example 1.1, (iii),together with our definition ofiv, and our assumption that Gis edge-paired, henceuntangled.] Thus, by taking the inverse image ofin the monoid that defines thelog structure of Xlog, we obtain a line bundle L on X. Let Y X be a finiteetale cyclic coveringof order a positive power ofl such thatL|Y hasdegree divisiblebyd on every irreducible component ofY, and Y Xrestricts to a connectedcovering over every irreducible component ofXthat is =Cv [e.g., Cv], but splits

over Cv ; Ylog def= Xlog X Y; Cw

def= CvX Y; C

+w

def= C+v X Y. [Note that the

fact that such a covering exists follows immediately from our assumption that G is

sturdy.] Now let Zlog Ylog

be a log etale cyclic covering of degree d satisfying the following properties: (d)Zlog Ylog restricts to an etale covering ofZ Y over the complement ofC+wand splitsover the irreducible components ofY that lie over Cv [cf. (a); the factthat (1), (2), and (3) are assumed to be false!]; (e) Zlog Ylog is ramified, withramification index d/iv , over the generic points of C

+w , but induces the trivial

extension of the function field of Cw [cf. (b)]; (f) for each node f of Y that liesover a bridge node e ofX, the restriction ofZlog Ylog to the branch off thatdoes not abut to an irreducible component of C+w is ramified, with ramification

index d

ie/i

v [cf. (c)]. Indeed, to construct such a covering Z

log

Ylog

,it suffices to construct a covering satisfying (d), (f) over the complement of C+w[which is always possible, by the conditions imposed on Y, together with the factthat (1), (2), and (3) are assumed to be false], and then to glue this covering to

the Kummer log etale covering of (C+w )log def= Ylog Y C

+w [by an fs log scheme!]

obtained by extracting a d-th root ofL|C+w [cf. the divisibilitycondition on the

degrees of L over the irreducible components of C+w ]. [Here, we regard the Gm-torsor determined by L|C+w as a subsheaf of the monoid defining the log structure

of(C+w )log.] Now if we write JZ JY I for the open subgroups defined by the

coveringsZlog Ylog Xlog, then Dv

JY surjectsonto I; Dv

IJY. On

the other hand, vJY JZ [cf. (e)] and Dv I JZ [cf. (d)], while [cf. (e)][Dv

JY :Dv

JZ] =d

/iv >[I :Iv,v ]

[since d = l iv [I : Iv,v ]]. Thus, it suffices to take J0def= JY, J

def= JZ. This

completes the proof of assertion (iv).

Next, we consider assertion (v). First, let us observe that it follows fromassertion (iv) [i.e., by applying assertion (iv) to various open subgroupsof H, I cf. also Remark 1.2.1] that if, for H, Iv

( Iv

1) = {1}, then v = v

1. Thus, we conclude that NH (Iv) CH (Iv) NH (v) = Dv. Onthe other hand, since [by definition] Iv =ZI (v), andI isnormalin H, it followsthat Dv = NH (v) NH (Iv), so NH (Iv) = CH (Iv) = Dv, as desired. Inparticular,Dv

I=NI (Iv) =CI (Iv). Next, let us observe that Dv

G = v

[cf. [Mzk13], Proposition 1.2, (ii)]. Thus,Dv CH (Dv)CH (v). Moreover,


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by [Mzk13], Proposition 1.2, (i), it follows that CH (v) =NH (v) =Dv; thus,we conclude that Dv (respectively,Dv

I;Dv

G) iscommensurably terminalin

H(respectively, I; G). Finally, by assertion (iii), we haveDv

I=Ivv

ZI (Iv) NI (Iv) =DvI, so DvI=ZI (Iv), as desired. This completesthe proof of assertion (v).Next, we consider assertion (vi). The fact that the image ofDv in H is open

follows immediately from the fact that since the semi-graph Gis finite, some opensubgroup ofH necessarilyfixesv. On the other hand, ifGadmits a vertex v =v,then v

v isnot openin v [cf. [Mzk13], Proposition 1.2, (i)]; sinceDv

G =

v [cf. [Mzk13], Proposition 1.2, (ii)], this implies that Dv isnot openin H. Thiscompletes the proof of assertion (vi).

Next, we consider assertion (viii). First, we observe that if property (2) holds,then by assertions (ii), (iii), Iv De

De I I, so Iv = De

De I.

Thus, it suffices to verify that either (1) or (2) holds. Next, let us observe that,by projecting to the maximal pro-l quotients, for some l , of the various pro-groups involved, we may assume without loss of generality [for the remainder of theproof of assertion (viii)] that = {l}. Now if De

De

G = {1}, then [since

= {l}]De

De

G isopeninDe

G(=Z),De G (=Z) [cf. assertions(ii), (iii), (vii)], so we conclude from [Mzk13], Proposition 1.2, (i) thate = e. Thus,to complete the proof of assertion (viii), it suffices to derive a contradictionunderthe further assumption that De

De

G = {1}, and e and e do not abut to

a common vertex. Moreover, by replacing H by an open subgroup of H [cf.Remark 1.2.1], we may assume without loss of generality that G [i.e.,Xlog] issturdy

[cf. 0], and that G isedge-paired[cf. 0] for arbitraryl andedge-even[cf. 0] whenl= 2.

Now if, say, eis acusp that abuts to a vertex v , then one verifies immediatelythat there exist log etale cyclic coveringsYlog Xlog of degree an arbitrarily largepower ofl which are totally ramifiedover e, but unramifiedover the nodesofX, aswell as over thecuspsofXthat abut to vertices=v. [Indeed, the existence of suchcoverings follows immediately from the fact that Xlog is edge-paired for arbitrary land edge-evenwhenl= 2 cf. the discussion of0.] In particular, such coveringsareunramifiedovere, as well as over the generic point of the irreducible componentof X corresponding to v, hence correspond to open subgroups J I such that

De IJ [so DeDe IJDeI], and, moreover, Jmay be chosenso that the subgroupJ

De

I De

I=Ie Iv [cf. assertion (iii)] forms anarbitrarily small neighborhoodofIv. Thus, we conclude that De

De

I Iv.

On the other hand, ife is also acuspthat abuts to a vertexv , then [by symmetry]we conclude thatDe

De

I Iv , hence thatIv

Iv = {1}. But, by assertion

(iv), this implies that v=v , a contradiction. Thus, we may assume that, say, e isa node, so Ie=De

I [cf. assertion (vii)].

Write v1, v2 for the two distinct vertices to which e abuts; C1, C2 for the

irreducible components ofX corresponding to v1, v2; C def

= C1

C2 X; UC C for the open subscheme obtained by removing the nodes that abut to vertices= v1, v2. Let us refer to the nodes and cusps of UC as inner, to the nodes ofX that were removed from C to obtain UC as bridge nodes, and to the nodesand cusps ofXwhich are neither inner nodes/cusps nor bridge nodes as external.[Thus,eis inner;e isexternal.] Observe that the natural projection to Iyields an


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inclusion De

De

I I with open image [since = {l}]; denote the imageof this inclusion by IC. Write iC for the least common multipleof the indices ifof the bridge nodesf; iC for the largest nonnegative power of l dividing iC. Let

d def= l iC [I : IC]; d def= l iC [I : IC] [so d is the largest positive power of ldividing d]. Here, we observe that

[I :IC] = [Ie: e (De

De

I)]

[cf. assertion (ii)]. Then it suffices to construct an open subgroup J Isuch thate J and De

IJ [which implies that e (De

De

I) Ie

J], but

[Ie : Ie

J]> [I :IC].

To this end, let us first observe that the characteristic sheafof the log scheme

Xlog admits asectionoverXsatisfying the following properties: (a)vanishesonthe open subscheme ofXgiven by the complement ofC; (b)coincides withiC

MS [cf. the notation of Example 1.1, (iii)] at the generic points ofC def

= C1

C2;(c) coincides with either (iC/if) Mf or (iC/if) Mf [cf. the notation ofExample 1.1, (iii), where we take e to be f] at each bridge nodef. [Indeed, theexistence of such a sectionfollows immediately from the discussion of Example 1.1,(iii), together with our definition ofiC, and our assumption that G is edge-paired,henceuntangled.] Thus, by taking the inverse image ofin the monoid that definesthe log structure ofXlog, we obtain a line bundle L on X. Let Y X be a finiteetale Galois coveringof order a positive power ofl such that L|Y hasdegree divisible

byd

on every irreducible component ofY, and Y Xrestricts to a connectedcovering over every irreducible component ofX; Ylog

def= Xlog X Y. [Note that

the fact that such a covering exists follows immediately from our assumption thatG is sturdy.] Write CY1 , C

Y2 for the irreducible components of Y lying over C1,

C2, respectively; we shall apply the terms internal, external, and bridge tonodes/cusps ofYthat lie over such nodes/cusps ofX. Now let

Zlog Ylog

be a log etale cyclic covering of degree d satisfying the following properties: (d)

Zlog Ylog restricts to an etale covering of Z Y over the complement ofCY

def= CY1

CY2 [cf. (a)], hence, in particular, over theexternal nodes/cuspsofY;

(e) Zlog Ylog is ramified, with ramification index d/iC, over the generic pointsof CY1 , C

Y2 , and, at each internal node of Y lying over e, determines a covering

corresponding to an open subgroup of Ie that contains e [cf. (b)]; (f) for eachbridge node f of Y, the restriction of Zlog Ylog to the branch of f that doesnot abut to CY is ramified, with ramification index d if/i

C [cf. (c)]. Indeed, to

construct such a covering Zlog Ylog, it suffices to construct a covering satisfying(d), (f) over the complement of CY [which is always possible, by the conditionsimposed on Y], and then to gluethis covering to the Kummer log etale covering

of (CY)log def= Ylog Y CY [by an fs log scheme!] obtained byextracting a d-throot ofL|CY [cf. thedivisibilitycondition on the degrees ofL|CY

1,L|CY

2]. [Here, we

regard the Gm-torsor determined byL|CY as a subsheaf of the monoid defining thelog structure of (CY)log.] Now if we write JZ JY Ifor the open subgroups


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defined by the coverings Zlog Ylog Xlog, then Ie, De

I JY. On theother hand, e JZ [cf. (e)] and De

I JZ [cf. (d)], while [cf. (e)]

[Ie:IeJZ] =d

/iC>[I :IC]

[since d = l iC [I : IC]]. Thus, it suffices to take J def

= JZ. This completes theproof of assertion (viii).

Next, we consider assertion (ix). The fact that the image ofDein Hisopenfol-lows immediately from the fact that since the semi-graph G isfinite, some open sub-group ofHnecessarilyfixese. On the other hand, since De

G =NG (e) = e

[cf. [Mzk13], Proposition 1.2, (ii)] is abelian, hence not openin the slim, nontrivialprofinite group G , it follows that De isnot openin H. This completes the proofof assertion (ix).

Finally, we consider assertion (x). First, let us observe that an easy com-putation reveals that if the image of S is a non-nodal, non-cuspidal pointof theirreducible component ofX corresponding to a vertex v ofG, then I(I) = Iv.Next, let us suppose that the image ofS is the nodeofXcorresponding to somenode e ofG. Then an easy computation [cf. the computations performed in theproof of assertion (ii)] reveals that I(I) Ie, but that I(I) is notcontained inIv for any vertex v

to which e abuts. If, moreover, I(I) Iv for some vertexv to which e does notabut, then [since the very existence of the node e impliesthatX issingular] there exists a node e =ethat abuts tov, soI(I) Iv Ie[cf.assertion (ii)]; but this implies thatI(I) Ie

Ieso, by assertion (viii), it followsthat I(I) Iv for some vertex to which both e and e abut a contradiction.Thus, in summary, we conclude that in this case, I is non-verticial.

Now suppose thatI isnon-verticialandnon-edge-like. Then the observationsof the preceding paragraph imply that the image of S is a cusp ofX. Write efor the corresponding cusp ofG. Thus, one verifies immediately that I(I) De.Theuniquenessofethen follows from assertion (viii) [and the fact that I isnon-verticial]. Thus, for the remainder of the proof of assertion (x), we may assume thatthe image ofS is nota cusp. Now the remainder of assertion (x) follows formally,in light of what of we have done so far, from assertions (iv), (viii). This completesthe proof of assertion (x).

Corollary 1.4. (Graphicity of Isomorphisms of (D)PSC-Extensions)Let l be a prime number. For i = 1, 2, let 1 Gi Hi Hi 1 be anl-cyclotomically full [cf. Remark 1.2.3]DPSC-extension (respectively, PSC-

extension), associated to construction data (Xlogi Slogi , i, Gi, Hi , i) (re-

spectively,(Xlogi Slogi , i, Gi, Hi)) such thatl i; in the non-respd case, write

Ii Hi for theinertia subgroup. Let

H :H1 H2; : G1

G2

becompatible[i.e., with the respective outer actions ofHionGi]isomorphismsof profinite groups; in the non-respd case, suppose further that H(I1) = I2.Then1 = 2; isgraphic [cf. [Mzk13], Definition 1.4, (i)], i.e., arises from

anisomorphism of semi-graphs of anabelioids G1 G2.


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Proof. This follows immediately from [Mzk13], Corollary 2.7, (i), (iii). Here, asin the proof of [Mzk13], Corollary 2.8, we first apply [Mzk13], Corollary 2.7, (i)[which suffices to complete the proof of Corollary 1.4 in the respd case and allows

one to reduce to the noncuspidalcase in the non-respd case], then apply [Mzk13],Corollary 2.7, (iii), to the compactifications of corresponding sturdy finite etalecoverings of the Gi.

We are now ready to define a purely group-theoretic, combinatorialanalogue ofthe notion of a stable polycurvegiven in [Mzk2], Definition 4.5.

Definition 1.5. We shall refer to an extension of profinite groups as a PPSC-extension [i.e., poly-PSC-extension] if, for some positive integer n and somenonempty set of primes , it admits a structure of pro- PPSC-extension ofdimension n. Here, for na positive integer, a nonempty set of primes, and

1 H 1

an extension of profinite groups, we define the notion of a structure of pro-PPSC-extension of dimensionnas follows [inductivelyon n]:

(i) Astructure of pro- PPSC-extension of dimension1 on the extension 1 H 1 is defined to be a structure of pro- PSC-extension. Suppose thatthe extension 1 H1 is equipped with a structure of pro- PPSC-

extension of dimension 1. Thus, we have an associated semi-graph of anabelioidsG, together with a continuous action of H on G, and a compatible isomorphism

G . We define the [horizontal] divisors of this PPSC-extension to be the

cuspsof the PSC-extension 1 H 1. Thus, each divisor c of thePPSC-extension 1 H 1 has associated inertiaand decompositiongroupsIc Dc [cf. Definition 1.2, (i)]. Moreover, by [Mzk13], Proposition 1.2,(i), a divisor is completely determinedby [the conjugacy class of] its inertia group,as well as by [the conjugacy class of] itsdecomposition group. Finally, we shall referto the extension 1 H1 [itself] as the fiber extensionassociated tothe PPSC-extension 1 H 1 of dimension 1.

(ii) Astructure of pro- PPSC-extension of dimensionn + 1 on the extension1 H 1 is defined to be a collection of data as follows:

(a) a quotient such that def= Ker( ) ; thus, the image

of in determines an extension

1 H 1

which we shall refer to as the associated base extension; the subgroup determines an extension

1 1

which we shall refer to as the associated fiber extension;


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(b) astructure of pro-PPSC-extension of dimensionn on thebase extension1 H 1;

(c) astructure of pro-PPSC-extension of dimension1 on thefiber extension1 1;

(d) for each base divisor [i.e., divisor of the base extension] c, a structureofDPSC-extensionon the extension

1 cdef= Dc Dc 1

which we shall refer to as the extension at c which is compatiblewith the PSC-extension structure on the fiber extension [cf. (c)], in thesense that both structures yield thesame cuspidal inertia subgroups ;

also, we require that theinertia subgroup ofDc [i.e., that arises from thisstructure of DPSC-extension] be equal to Ic .

In this situation, we shall refer to as the fiber prime setof the PPSC-extension1 H 1; we shall refer to as a divisor of the PPSC-extension1 H 1 any element of the union of the set ofcusps which weshall refer to ashorizontaldivisors of the PSC-extension 1 1and, for each base divisor c, the set ofverticesof the DPSC-extension 1 c Dc 1 which we shall refer to as vertical divisors [lying overc

]. Thus,each divisor c of the PPSC-extension 1 H 1 has associated inertia

and decomposition groupsIc Dc . In particular, whenever c is verticalandlies over a base divisor c, we have Ic Dc c .

Remark 1.5.1. Thus, [the collection offiber extensionsarising from] any struc-ture ofPPSC-extension of dimension n on an extension 1 H 1determine two compatible sequences of surjections

ndef= n1 . . . 1 0

def= {1}

ndef= n1 . . . 1 0

def= H

such that each [extension determined by a] surjection m m1, for m =1, . . . , n, is a fiber extension [hence equipped with a structure ofPSC-extension];m = Ker(m H). If c = cn is a divisor of [the extension determined by] = n, then [cf. Definition 1.5, (ii)] there exists a uniquely determined sequenceof divisors

cn cn1 . . . cnc1 cnc

where nc n is a positive integer; for m = nc, . . . , n, cm is a divisor of m;cnc is a horizontaldivisor; the notation denotes the relation oflying over[so cm+1 is a verticaldivisor that lies over cm, for nc m < n] together with

sequences of [conjugacies classes of] inertia and decomposition groups

Icn Icn1 . . . Icnc1 Icnc

Dcn Dcn1 . . . Dcnc1 Dcnc


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[i.e., for nc m < n, Icm+1 m+1 maps into Icm m, and Dcm+1 m+1maps into Dcm m].

Remark 1.5.2. Let 1 H1 be a PPSC-extensionof dimension n[wheren is a positive integer]. Then one verifies immediately that if is anyopen subgroupof , then there exists an open subgroup of that [when equippedwith the induced extension structure] admits a structure of PPSC-extension ofdimension n cf. Remark 1.2.1. Here, we note that one must, in general, pass tosome open subgroup of in order to achieve a situation in which all of the fiber[PSC-]extensions havestable reduction[cf. Remark 1.2.1; Definition 1.5, (ii), (d)].

Remark 1.5.3. For l a prime number, we shall say that a PPSC-extension1 H 1 of dimension n is l-cyclotomically full if each of its nassociated fiber extensions [cf. Remark 1.5.1] is l-cyclotomically full as a PSC-extension [cf. Remark 1.2.3].

Remark 1.5.4. Let k be a field; k a separable closure of k; Gkdef= Gal(k/k);

S def= Spec(k);

Zlog S

the log scheme determined by a stable polycurve over S i.e., Zlog admits asuccessive fibrationby generically smooth stable log curves [cf. [Mzk2], Definition

4.5, for more details]; UZ Z the interiorofZlog; DZdef

= Z\UZ [with the reducedinduced structure]; n the dimension of the scheme Z;

1 Zdef= 1(UZkk) Z

def= 1(UZ) Gk 1

the exact sequence of etale fundamental groups[well-defined up to inner automor-phism] associated to the structure morphism UZ S. Then by repeated appli-cation of the discussion of Example 1.1 to the fibers of the successive fibration[mentioned above] ofZlog by stable log curves, one verifies immediately that:

(i) If k is of characteristic zero, then the structure of stable polycurveonZlog determines a structure of profinite PPSC-extensionof dimension non the extension 1 Z Z Gk 1.

Moreover, one verifies immediately that:

(ii) In the situation of (i), the Z-orbits of divisors ofZ [in the sense ofDefinition 1.5] are in natural bijective correspondencewith the irreducibledivisorsofDZin a fashion that is compatiblewith the inertiaanddecom-position groupsof divisors of Z [in the sense of Definition 1.5] and ofirreducible divisors ofDZ [in the usual sense].

Finally, even ifk is not necessarily of characteristic zero, depending on the structureofZlog [cf., e.g., Corollary 1.10 below], variousquotientsof the extension 1 ZZ Gk 1 may be equipped with a structure ofpro-PPSC-extension[induced


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by the structure of stable polycurve on Z], for various nonempty sets of primenumbers that arenotequal to the set of all prime numbers; a similar observationto (ii) concerning a natural bijective correspondence of divisors then applies to

such quotients. When considering such quotients 1 H 1 of theextension 1 Z ZGk 1, it is useful to observe that the slimnessof [cf. Proposition 1.6, (i), below] implies that such a quotient Z is completelydeterminedby the induced quotients Z, Gk H [cf. the discussion of the

notation out in 0]; we shall refer to such a quotient 1 H1 as

a PPSC-extension arising from Zlog,k where we writek k for the subfieldfixed by Ker(Gk H).

Proposition 1.6. (Basic Properties of PPSC-Extensions) Let

1 H 1

be a pro- PPSC-extension of dimensionn [wheren is a positive integer]; 1 1 the associatedfiber extension; c, c divisors of. Then:

(i) isslim. In particular, ifH isslim, then so is.

(ii) Dc is commensurably terminal in.

(iii) We have: C(Ic) =Dc. As abstract profinite groups, Ic=Z.(iv) Dc is not open in . The divisor c ishorizontal if and only if Dc

projects to anopen subgroup of. Ifc isvertical and lies over abase divisorc, thenDc projects onto anopen subgroup ofDc.

(v) IfDc

Dc isopen inDc, Dc , thenc= c. In particular, a divisor of

is completely determined by its associateddecomposition group.

(vi) IfIc

Ic isopen inIc, Ic , thenc= c. In particular, a divisor of is

completely determined by its associatedinertia group.

Proof. Assertion (i) follows immediately from the slimness ofG discussed inExample 1.1, (ii) [cf. Definition 1.5, (i); Definition 1.5, (ii), (c)]. Next, we considerassertion (ii). We apply induction on n. Ifc is horizontal, then assertion (ii) followsfrom Proposition 1.3, (vii) [cf. also Definition 1.5, (ii), (c)]. Ifc is vertical, then clies over some base divisorc, and we are in the situation of Definition 1.5, (ii), (d).By Proposition 1.3, (vi), it follows that Dcsurjectsonto someopensubgroup ofDc ,hence thatC(Dc) maps into C(Dc); by the induction hypothesis, C(Dc) =Dc , so C(Dc) c . Thus, the fact that C(Dc) =Dc follows from Proposition1.3, (v). This completes the proof of assertion (ii).

Next, we consider assertion (iii). Again we apply induction on n. Ifc is hor-izontal, then assertion (iii) follows from Proposition 1.3, (i), (vii). Ifc is vertical,then c lies over some base divisorc, and we are in the situation of Definition 1.5,(ii), (d). By Proposition 1.3, (iii) [cf. Definition 1.5, (ii), (d)], we have isomor-

phismsIcIc =Z. In particular, C(Ic) maps into C(Ic); by the induction


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hypothesis, C(Ic) = Dc . Thus, C(Ic) c , so the fact that C(Ic) = Dcfollows from Proposition 1.3, (v). This completes the proof of assertion (iii).

Next, we consider assertion (iv). Again we apply induction on n. If c is

horizontal, then by Proposition 1.3, (ix), Dc is not open in , but Dc projects toan opensubgroup of . Ifc is vertical, then c lies over some base divisorc, andwe are in the situation of Definition 1.5, (ii), (d); Dc c . By Proposition 1.3,(vi), Dc projects onto an opensubgroup ofDc . By the induction hypothesis, Dcisnot openin , so c isnot openin ; thus, Dc isnot openin , and its imagein is not openin . This completes the proof of assertion (iv).

Next, we consider assertion (v). Again we apply induction on n. By assertion(iv), c is horizontal if and only if c is. If c, c are horizontal, then the fact thatc = c follows from Proposition 1.3, (i), (viii). Thus, we may suppose that c, c

are vertical and lie over respective base divisors c, (c). By assertion (iv), itfollows that Dc D(c) is open in Dc , D(c) ; by the induction hypothesis, thisimplies that c = (c). Thus, by intersecting with G [cf. Proposition 1.3, (v)]and applying [Mzk13], Proposition 1.2, (i), we conclude that c = c. This completesthe proof of assertion (v). Finally, we observe that assertion (vi) is an immediateconsequence of assertions (iii), (v).

We are now ready to state and prove the main resultof the present 1.

Theorem 1.7. (Graphicity of Isomorphisms of PPSC-Extensions) Let l

be a prime number; na positive integer. For =, , let be a nonempty set ofprimes; 1 H 1 an l-cyclotomically full [cf. Remark 1.5.3]pro- PPSC-extension of dimensionn;

ndef= n1 . . .

1

0

def= H

the sequence ofsuccessive fiber extensions associated to [cf. Remark 1.5.1].Let

:

be anisomorphism of profinite groupsthat induces isomorphismsm:

m

m,form= 0, 1, . . . , n [so = n]. Then:

(i) We have = .

(ii) Form {1, . . . , n}, m induces abijection between the set ofdivisorsofm and the set of divisors of

m.

(iii) Form {1, . . . , n}, suppose thatc, c are divisors ofm, m, respec-

tively, that correspond via the bijection of (ii). Then m(Ic) = Ic , m(Dc) =Dc . That is to say, m is compatible with the inertia and decomposition

groups of divisors.(iv) Form {0, . . . , n 1}, the isomorphism

Ker(m+1 m)

Ker(m+1

m)


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induced by m+1 isgraphic [i.e., compatible with the semi-graphs of anabelioidsthat appear in the respective collections of construction data of the PSC-extensionsm+1

m, for =, ].

(v) For m {1, . . . , n 1}, c, c corresponding divisors of m, m, the

isomorphism

Ker((m+1)c Dc) Ker((m+1)c Dc )

induced by m+1 isgraphic [i.e., compatible with the semi-graphs of anabelioidsthat appear in the respective collections of construction data of the DPSC-extensions(m+1)c Dc , for =, ].

Proof. All of the assertions of Theorem 1.7 follow immediately from [the variousdefinitions involved, together with]repeated applicationof Corollary 1.4 to thePSC-extensions m+1

m [cf. Definition 1.5, (ii), (c)] and the DPSC-extensions

(m+1)c Dc [cf. Definition 1.5, (ii), (d)], for =, .

Remark 1.7.1. In Theorem 1.7, instead of phrasing the result as an asser-tion concerning the preservation of structures via some isomorphismbetween twoPPSC-extensions, one may instead phrase the result as an assertion concerning theexistence of anexplicit group-theoretic algorithmfor reconstructing, from asinglegiven PPSC-extension, the various structures corresponding to graphicity,divisors,

and inertiaand decomposition groupsof divisors i.e., in the fashion of [Mzk15],Lemma 4.5, for cuspidal decomposition groups; a similar remark may be madeconcerning Corollary 1.4. [We leave the routine details to the interested reader.]Indeed, both Corollary 1.4 and Theorem 1.7 are, in essence, formal consequencesof thegraphicity theoryof [Mzk13], which [just as in the case of [Mzk15], Lemma4.5] consists precisely of such explicit group-theoretic algorithms for reconstruct-ing the various structures corresponding to graphicity in the case of semi-graphs ofanabelioids of PSC-type.

Before proceeding, we observe the following result, which is, in essence, inde-

pendentof the theory of the present 1.

Theorem 1.8. (PPSC-Extensions over Galois Groups of Arithmetic

Fields) For = , , let k be a field ofcharacteristic zero;k a solvablyclosed [cf. [Mzk15], Definition 1.4] Galois extension ofk; H

def= Gal(k/k);(Z)

log the log scheme determined by astable polycurveoverk; a nonempty

set of primes; 1 H 1 apro- PPSC-extension associatedto (Z)

log,k [cf. Remark 1.5.4];

n

def=

n1 . . .

1

0

def= H

the sequence ofsuccessive fiber extensions associated to [cf. Remark 1.5.1].Let

:


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be anisomorphism of profinite groupsthat induces isomorphismsm: m

m,

form= 0, 1, . . . , n [so = n]. Then:

(i)(Relative Version of the Grothendieck Conjecture for Stable Poly-curves over Generalized Sub-p-adic Fields) Suppose that for = , , kisgeneralized sub-p-adic [cf. [Mzk4], Definition 4.11] for some prime number

p

, and that the isomorphism of Galois groups 0 : H H arises

from a pair ofisomorphisms of fieldsk k, k k. Then = ; thereexists aunique isomorphism of log schemes (Z)

log (Z)log that gives rise

to .

(ii) (Absolute Version of the Grothendieck Conjecture for StablePolycurves over Number Fields) Suppose that for =, , k is anumberfield. Then = ; there exists a unique isomorphism of log schemes

(Z)log (Z)log that gives rise to .

Proof. Assertions (i), (ii) follow immediately from repeated applicationof [Mzk4],Theorem 4.12, together with [in the case of assertion (ii)] Uchidas theorem [cf.,e.g., [Mzk10], Theorem 3.1].

Finally, we study the consequences of the theory of the present 1 in the caseof configuration spaces. We refer to [MT] for more details on the theory of config-uration spaces.

Definition 1.9. Let l be a prime number; a set of primes which is either ofcardinality oneor equal to the set of all primes; Xa hyperbolic curveof type (g, r)over afieldk of characteristic ;k a separable closureofk;n 1 an integer; Xnthe n-th configuration spaceassociated to X [cf. [MT], Definition 2.1, (i)]; E theindex set[i.e., the set of factors cf. [MT], Definition 2.1, (i)] ofXn;

1(Xn kk)

themaximal pro-quotientof1(Xnkk); aproduct-theoretic open subgroup[cf. [MT], Definition 2.3, (ii)]; 1 H 1 an extension of profinitegroups.

(i) We shall refer to as a labelingon Ea bijection : {1, 2, . . . , n} E. Thus,

for each labeling onE, we obtain astructure of hyperbolic polycurve[a collectionof data exhibitingXnas ahyperbolic polycurve cf. [Mzk2], Definition 4.6] on Xn,arising from the various natural projection morphismsassociated to Xn [cf. [MT],Definition 2.1, (ii)], by projecting in the order specified by . In particular, foreach labeling on E, we obtain a structure of PPSC-extensionon [the extension1 {1} 1 associated to] some open subgroup [which maybe taken to be arbitrarily small cf. Remark 1.5.2].

(ii) Let be a labeling on E. Then we shall refer to a structure of PPSC-extension on [the extension 1 H 1 arising from 1 H1 by intersecting with] an open subgroup as -admissible if itinduces the structure of PPSC-extension on discussed in (i).


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(iii) We shall refer to as a structure of [pro-] CPSC-extension [of type(g, r)and dimensionn, with index setE][i.e., configuration (space) pointed stable curveextension] on the extension 1 H 1 any collection of data as follows:

for each labeling on E, a -admissible structure of PPSC-extensionon some opensubgroup [which may be taken to bearbitrarily small cf. Remark 1.5.2].We shall refer to a structure of CPSC-extension on as l-cyclotomically fullif, foreach labeling onE, the -admissible structure of PPSC-extension that constitutesthe given structure of CPSC-extension is l-cyclotomically full. We shall refer to astructure of CPSC-extension on as strictif = [in which case one may alwaystake = ]. We shall refer to 1 H 1 as a [pro-] CPSC-extension [of type(g, r) and dimensionn, with index setE]if it admits a structureof CPSC-extension; if this structure of CPSC-extension may be taken to be l-cyclotomically full (respectively, strict), then we shall refer to the CPSC-extensionitself as l-cyclotomically full (respectively, strict). If 1 H 1 is aCPSC-extension, then we shall refer to (,X,k, ) as construction data for thisCPSC-extension.

(iv) Let k kbe asolvably closed[cf. [Mzk15], Definition 1.4] Galois extensionofk; suppose that

Z Xn

is a finite etale coveringsuch that Zkk Xnkk is the [connected] coveringdetermined by the open subgroup [so we have a natural surjection 1(Zkk) ]. Then [cf. the discussion of Remark 1.5.4] we shall refer to a [structure of]

CPSC-extension [on] 1 H 1 as arising fromZ,k/k if there exist

a surjection 1(Z) and an isomorphism Gal(k/k) H that are compatiblewith one another as well as with the natural surjections 1(Zkk) ,1(Z)

Gal(k/k) Gal(k/k) and, moreover, satisfies the property that the structure ofCPSC-extensionon 1 H1 is induced by the various structures ofhyperbolic polycurve on Z, Xn, associated to the various labelingsofE [cf. (i)].

Corollary 1.10. (Cominatorial Configuration Spaces) Let l be aprimenumber. For =, , let

1

H

1

be anextension of profinite groupsequipped with some [fixed!] l-cyclotomicallyfull structure of CPSC-extension of type(g, r) / {(0, 3), (1, 1)} and dimen-sionn, with index setE. If the CPSC-extension1

H 1 isnot strict foreither = or =, then we assume thatboth g, g are 2.Let

:

be anisomorphism of profinite groups such that() = . Then:

(i) The isomorphism determines abijection E E of index sets. In

particular,n =n, so we writendef= n=n.

(ii) For each pair ofcompatible[i.e., relative to the bijection of (i)]labelings = (, ) of E, E, there exist open subgroups

[for = , ]


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such that the following properties hold: (a) () = ; (b) for =, , the open

subgroup admits an -admissible structure of PPSC-extension; (c) ifwe write

( )ndef= ( ) (

)n1 . . . (

)1 (

)0

def= H

for the sequence of successive fiber extensions associated to the structures ofPPSC-extension of (b) [cf. Remark 1.5.1], then inducesisomorphisms

()m ()m

[form= 0, . . . , n]. In particular, satisfies the hypotheses of Theorem 1.7.

Proof. By [MT], Corollaries 4.8, 6.3 [cf. our hypotheses on (g, r)], inducesa bijectionE E between the respective index sets, together with compatible

isomorphisms between the various fiber subgroups of , . [Note that eventhough these results of [MT] are stated only in the case where the field appearingin the construction data is ofcharacteristic zero, the results generalize immediatelyto the case where this field is ofcharacteristic invertible in, since any hyperboliccurve in positive characteristic may be lifted to a hyperbolic curve in characteristiczero in a fashion that is compatible with the maximal pro- quotients of theetale fundamental groups of the associated configuration spaces cf., e.g., [MT],Proposition 2.2, (iv).] To obtain open subgroups

satisfying the desiredproperties, it suffices to argue by induction on n, by applying Theorem 1.7, (iii).Indeed, the casen = 1 is immediate. To derive the casen = n0 +1 from the casen= n0, it suffices to apply Theorem 1.7, (iii), to the case n = n0, which impliesthat is compatiblewith theinertia groupsof lower dimensional divisors, so onemay apply a well-known group-theoretic criterion for stable reduction[in terms ofthe action of the inertia group cf., e.g., [BLR], 7.4, Theorem 6] to constructopen subgroups

[cf. Remark 1.5.2] satisfying the desired properties.

Remark 1.10.1. A similar remark to Remark 1.7.1 may be made for Corollary1.10.

Corollary 1.11. (Configuration Spaces over Arithmetic Fields)For =

, , letk be aperfect field;k asolvably closed [cf. [Mzk15], Definition 1.4]Galois extension ofk; X ahyperbolic curve of type(g, r) / {(0, 3), (1, 1)}overk; n a positive integer;

Z (X)n

a geometrically connected [overk]finite etale covering of then-thconfigu-ration space ofX;

a nonempty set of primes;

1 H 1

an extension of profinite groups equipped with some [fixed!] structure of

pro- CPSC-extension arising fromZ,k [cf. Definition 1.9, (iv)]. If the


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CPSC-extension 1 H 1 isnot strict foreither = or =, then we assume thatboth g, g are 2. Let

:

be anisomorphism of profinite groups. Then:

(i) (Relative Version of the Grothendieck Conjecture for Configura-tion Spaces over Generalized Sub-p-adic Fields) Suppose that for =, ,kisgeneralized sub-p-adic[cf. [Mzk4], Definition 4.11] for some prime number

p

, and that lies over an isomorphism of Galois groups0 :H H

that arises from a pair of isomorphisms of fieldsk k, k k. Then = ; there exists aunique isomorphism of schemes Z

Z that gives

rise to .

(ii) (Strictly Semi-absoluteness) Suppose, for=, , thatk is eitheran FF, an MLF, or an NF [cf. [Mzk15], 0]. Then () = [i.e., isstrictly semi-absolute].

(iii) (Absolute Version of the Grothendieck Conjecture for Configu-ration Spaces over MLFs) Suppose, for = , , that k is an MLF, thatn 2, thatn 3 ifX isproper, and that

is the set ofall primes. Thenthere exists aunique isomorphism of schemes Z

Z that gives rise to .

(iv) (Absolute Version of the Grothendieck Conjecture for Config-uration Spaces over NFs) Suppose, for = , , that k is an NF. Then

= ; there exists aunique isomorphism of schemes Z Z that gives

rise to .

Proof. Assertion (i) (respectively, (iv)) follows immediately from Corollary 1.10,(ii), and Theorem 1.8, (i) (respectively, Theorem 1.8, (ii)) [applied to the coveringsofZ, Z determined by the open subgroupsof Corollary 1.10, (ii)]. Assertion (ii)follows immediately from [Mzk15], Corollary 2.8, (ii). Note that in the situation ofassertion (ii), assertion (ii) implies that = [since may be characterized

as the unique minimal set of primes such that is a pro- group]; moreover,in light of our assumptions onk, it follows immediately that

isl-cyclotomicallyfullfor any l

= = .

Finally, we consider assertion (iii). First, let us observe that by Corollary1.10, (ii), and Theorem 1.8, (i) [applied to the coverings ofZ, Z determined bytheopen subgroupsof Corollary 1.10, (ii)], it suffices to verify that the isomorphism

H :H H induced by [cf. assertion (ii)] arises from an isomorphism of fields

k k [where we recall that in the present situation,kis necessarily analgebraic

closureofk]. To this end, let us observe that by Corollary 1.10, (ii), we may applyTheorem 1.7 to the present situation. Also, by Corollary 1.10, (i), n = n = nis always 2; moreover, if either of the X is proper, then n 3. Next, letus observe that if X is proper(respectively, affine), then the stable log curvethat appears in the logarithmic compactification of the fibration (X)3 (X)2(respectively, (X)2 (X)1) over the generic point of the diagonal divisor of


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(X)2(respectively, over any cusp ofX) contains an irreducible component whoseinterior is a tripod[i.e., a copy of the projective line minus three marked points]. Inparticular, if we apply Theorem 1.7, (iii), to thevertical divisordetermined by such

an irreducible component, then we may conclude that induces an isomorphismbetween the decomposition groups of these vertical divisors. In particular, [afterpossibly replacing the givenk by corresponding finite extensions ofk] we obtain,for = , , a hyperbolic curves C over k, together with an isomorphism ofprofinite groups

C :1(C kk) 1(Ck k)

induced by [so the 1(Ckk) correspond to the respective vs of the

vertical divisors under consideration] that is compatible with the outer action of

H on 1(Ckk) and the isomorphism H; moreover, here we may assume

that, say, C is a finite etale covering of a tripod. On the other hand, since theabsolutep-adic version of the Grothendieck Conjecture is known to hold in thissituation [cf. [Mzk14], Corollary 2.3], we thus conclude that H does indeed arise

from an isomorphism of fieldsk k , as desired. This completes the proof of

assertion (iii).

Remark 1.11.1. At the time of writing Corollary 1.11, (iii), constitutes theonlyabsolute isomorphism versionof the Grothendieck Conjecture over MLFs [to theknowledge of the author] that may be applied to arbitrary hyperbolic curves.


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Section 2: Geometric Uniformly Toral Neighborhoods

In the present 2, we prove a certain resolution of nonsingularities type re-sult [cf. Lemma 2.6; Remark 2.6.1; Corollary 2.11] i.e., a result reminiscent ofthe main results of [Tama2] [cf. also the techniques applied in the verification ofobservation (iv) given in the proof of [Mzk9], Corollary 3.11] that allows usto apply the theory ofuniformly toral neighborhoodsdeveloped in [Mzk15], 3, toprove a certain conditional absolute p-adic version of the Grothendieck Conjec-ture namely, that point-theoreticity implies geometricity [cf. Corollary 2.9].This condition of point-theoreticity may be removed if, instead of starting witha hyperbolic orbicurve, one starts with a pro-curve obtained by removing froma proper curve some [necessarily infinite] set of closed points which is p-adicallydense in a Galois-compatible fashion [cf. Corollary 2.10].

First, we recall the following positive slope version of Hensels lemma [cf.[Serre], Chapter II, 2.2, Theorem 1, for a discussion of a similar result].

Lemma 2.1. (Positive Slope Version of Hensels Lemma) Let k be acomplete discretely valued field;Ok k the ring of integers ofk [equipped withthe topology determined by the valuation]; mk the maximal ideal ofOk; mk a

uniformizer ofOk; A def= Ok[[X1, . . . , X m]]; B

def= Ok[[Y1, . . . , Y n]]. Let us suppose

that A (respectively, B) is equipped with the topology determined by its maximal

ideal; writeX def

= Spf(A) (respectively,Y def= Spf(B)), KA (respectively,KB) for the

quotient field ofA (respectively, B), andA (respectively, B) for the module ofcontinuous differentials ofA (respectively, B) overOk [so A (respectively, B) isa freeA- (respectively, B-) module of rankm (respectively, n)]. Let: B A bethe continuousOk-algebra homomorphism induced by an assignment

B Yj fj(X1, . . . , X m) A

[wherej = 1, . . . , n]; let us suppose that the induced morphismd : BBA Asatisfies the property that theimage ofd AKA is aKA-subspace of rankn inAAKA [son m]. Then there exists a point0 Y(Ok)and a positive integer

r satisfying the following property: Let k be a finite extension of k, with ring ofintegersOk ; writeB(0, k, r) for the ball of points Y(Ok) such that, 0map to the same point ofY(Ok/(r)). Then theimage of the map

X(Ok) Y(Ok)

induced by contains the ball B(0, k, r).

Proof. First, let us observe that by Lemma 2.2 below, after possibly re-orderingthe Xis, the differentials dXi A, where i = n+ 1, . . . , m, together with thedifferentials dfj A, where j = 1, . . . , n, form a KA-basisof AAKA. Thus,by adding indeterminatesYn+1, . . . , Y m to B and extending by sendingYi Xifori = n + 1, . . . , m, we may assume without loss of generality thatn = m,A = B ,X = Y, i.e., that the morphism Spf() : X Y= X isgenerically formal ly etale.


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Write Mfor the n by n matrix with coefficients A given by {dfi/dXj}i,j=1,... ,n;g Afor thedeterminantofM. Thus, by elementary linear algebra, it follows thatthere exists annbynmatrixNwith coefficients Asuch thatMN=NM=g I

[where we write I for the n byn identity matrix]. By our assumption concerningthe image ofdAKA, it follows thatg = 0, hence, by Lemma 2.3 below, that there

exist elements xi mk, where i = 1, . . . , n, such that g0def= g(x1, . . . , xn) mk is

nonzero. By applying appropriate affine translations to the domain and codomainof , we may assume without loss of generality that, for i = 1, . . . , n, we have

xi =fi(0, . . . , 0) = 0 Ok. Write M0def= M(0, . . . , 0), N0

def= N(0, . . . , 0) [so M0,

N0 are n bynmatrices with coefficients Ok].

Next, suppose that g0 msk\ms+1k . In the remainder of the present proof, all

vectors are to be understood as column vectors with coefficients Ok , wherek is as in the statement of Lemma 2.1. Then I claim that for every vector y =(y1, . . . , yn) 0 (mod 3s), there exists a vector x= (x1, . . . , xn) 0 (mod 2s)

such that f(x)def= {f1(x

topics in absolute anabelian geometry ii

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