the profinite grothendieck conjecture for closed hyperbolic curves over number fields

8/13/2019 The Profinite Grothendieck Conjecture for Closed Hyperbolic Curves Over Number Fields

1/47

The Pronite Grothendieck Conjecture for ClosedHyperbolic Curves over Number Fields

by Shinichi Mochizuki

Section 0: Introduction

In [Tama], a proof of the Grothendieck Conjecture (reviewed below) was given forsmooth affine hyperbolic curves over nite elds (and over number elds). The purposeof this paper is to show how one can derive the Grothendieck Conjecture for arbitrary(i.e., not necessarily affine) smooth hyperbolic curves over number elds from the resultsof [Tama] for affine hyperbolic curves over nite elds.

We obtain three types of results: one over number elds, one over nite elds, andone over local elds. We remark here that when this paper was rst written (October1995), Theorems A and C below were the strongest known results of their respective kinds.Since then, the author wrote [Mzk2] (November 1995), which gives rise to much strongerresults than Theorems A or C of the present paper. Moreover, the proofs of [Mzk2] arecompletely different from (and, in particular, do not rely on) the proofs of the presentpaper. Nevertheless, it seems to the author that the present paper still has some marginalinterest, partly because most of the present paper is devoted to the proof of Theorem Bbelow (which is not implied by any result of [Mzk2]), and partly because it is in some senseof interest to see how Theorems A or C can be derived within the context of the theory of [Tama].

Our main result over number elds (Theorem 10.2 in the text) is as follows:

Theorem A: Let K be a nite extension of Q ; let K be an algebraic closure of K . LetX K Spec(K ) and X K Spec(K ) be smooth, geometrically connected, proper curves over K , of genus 2. Let X K (respectively, X K ) be the geometric fundamental group of X K (respectively, X K ). Then the natural map

Isom K (X K , X K ) Out ( X K , X K )

is bijective . Here, Out refers to outer isomorphisms that respect the natural outer representations of Gal (K/K ) on X K and X K .

The statement of this Theorem is commonly referred to as the Grothendieck Conjecture.In [Tama], a theorem similar to Theorem A, except that X is replaced by a hyperbolicaffine curve, is proven. It is a simple exercise to derive the affine case from the proper

1


2/47

case. On the other hand, to derive the proper case from the affine case is by no meansstraightforward; in this paper, we derive the proper case over number elds from the affinecase over nite elds .

In fact, to be more precise, we shall derive the proper case over number elds from

a certain logarithmic Grothendieck Conjecture for singular (proper) stable curves overnite elds. This logarithmic Grothendieck conjecture, which is our main result overnite elds (Theorem 7.4), is as follows:

Theorem B: Let S log be a log scheme such that S is the spectrum of a nite eld k,and the log structure is isomorphic to the one associated to the chart N k given by the zero map. Let X log S log and (X )log S log be stable log-curves such that at least one of X or X is not smooth over k. Let X log (respectively, (X ) log ) be the geometric fundamental group of X log (respectively, (X )log ) obtained by considering log admissible coverings of X log (respectively, (X )log ) (as in [Mzk], 3). Then the natural map

Isom S log (X log , (X )log ) Out D ( X log , (X ) log )

is bijective . Here, the D stands for degree one (outer isomorphisms).

This Theorem is derived directly from Tamagawas results on affine hyperbolic curves overnite elds. Its proof occupies the bulk of the present paper.

Finally, by supplementing Theorem B with various arguments concerning the funda-mental groups of curves over local elds, we obtain the following local result (Theorem9.8):

Theorem C: Let K be a nite extension of Q p ; let A K be its ring of integers; and letk be its residue eld. Let X K be a smooth, geometrically connected, proper curve of genus g 2 over K . Assume that X K admits a stable extension X Spec(A) such that the abelian variety portion of P ic0(X k ) (where X k = X A k) is ordinary . Then the natural outer representation

X : K Out ( X K )

of the absolute Galois group of K on the geometric fundamental group of X K completely determines the isomorphism class of X K .

Even though this is a rather weak version of the Grothendieck Conjecture (compared tothe results we obtain over nite and global elds), this sort of result is interesting in thesense that it shows that curves behave somewhat differently from abelian varieties (cf. theRemark following Theorem 9.8).

2


3/47

Now we discuss the contents of the paper in more detail. Sections 1 through 7 aredevoted to deriving Theorem B from the results of [Tama]. In Section 1, we show how torecover the set of irreducible components of a stable curve from its fundamental group. InSection 2, we review various facts from [Mzk] concerning log admissible coverings, and showhow one can dene an admissible fundamental group of a stable log-curve. In Section3, we show how one can group-theoretically characterize the quotient of the admissiblefundamental group corresponding to etale coverings. In Section 4, we show that the tamefundamental group of each connected component of the smooth locus of a stable log-curveis contained inside the admissible fundamental group of the stable log-curve. In Section5, we show how to recover the set of nodes (including the information of which irreduciblecomponents each node sits on) of a stable log-curve from its admissible fundamental group.In Section 6, we show how the log structure at a node of a stable log-curve can be recoveredfrom the admissible fundamental group of the stable log-curve. In Section 7, we put allof this information together and show how one can derive Theorem B from the results of [Tama].

In Sections 8 and 9, we shift from studying curves over nite elds to studying curvesover local elds. In order to do this, it is necessary rst to characterize (group-theoretically)the quotient of the (characteristic zero) geometric fundamental group of a curve over a localeld which corresponds to admissible coverings. This is done in Section 8. In Section 9,we rst show (Lemma 9.1) that the degree of an isomorphism between the arithmeticfundamental groups of two curves over a local eld is necessarily one. This is importantbecause one cannot apply Theorem 7.4 to an arbitrary isomorphism of fundamental groups:one needs to know rst that the degree is equal to one. Then, by means of a certain trickwhich allows one to reduce the study of curves over local elds with smooth reduction tothe study of curves over local elds with singular reduction, we show (Theorem 9.2) that

one can recover the reduction (over the residue eld) of a given smooth, proper, hyperboliccurve over a local eld group-theoretically. The rest of Section 9 is devoted to curves withordinary reduction, culminating in the proof of Theorem C. Finally, in Section 10, weobserve that Theorem A follows formally from Theorem 9.2.

The author would like to thank A. Tamagawa for numerous fruitful discussions con-cerning the contents of [Tama], as well as the present paper. In some sense, the presentpaper is something of a long appendix to [Tama]: That is to say, several months after theauthor learned of the results of [Tama], it dawned upon the author that by using admissiblecoverings, Theorem A follows trivially from the results of [Tama]. On the other hand,since many people around the author were not so familiar with admissible coverings or logstructures, it seemed to the author that it might be useful to write out a detailed versionof this trivial argument. The result is the present paper.

Finally, the author would also like to thank Prof. Y. Ihara for his encouragement andadvice during the preparation of this paper.

3


4/47

Section 1: The Set of Irreducible Components

Let k be the nite eld of q = pf . Fix an algebraic closure k of k. Let be theabsolute Galois group of k. Let X Spec(k) be a morphism of schemes.

Denition 1.1: We shall call X a multi-stable curve of genus g if dim k (H 1 (X, OX )) = g,and X k

def = X k k is a nite disjoint union of stable curves over k of genus 2. If X is multi-stable, then we shall call X sturdy if every irreducible component of the normalizationof X k has genus 2.

Thus, in particular, a curve is stable if and only if it is geometrically connected and multi-stable. Moreover, a nite etale covering of a multi-stable (respectively, sturdy) curve ismulti-stable (respectively, sturdy).

Suppose that X is stable of genus g 2. Fix a base-point x X (k). Then wemay form the (algebraic) fundamental group def = 1 (X, x k ) of X . Let X kdef = X k k;

def = 1 (X k , xk ). Then we have a natural exact sequence of groups

1 1

induced by the structure morphism X Spec(k).

The purpose of this Section is to show how the set of irreducible components of X can be canonically recovered from the morphism . Fix a prime l different from p.

Let us consider the etale cohomology group H e def

= H 1et (X k , Z l ). Let : X X be thenormalization of X . Then we can also consider H n def = H 1et ( X k , Z l ). By considering thelong exact cohomology sequence in etale cohomology associated to

0 Z l Z l (Z l )/ Z l 0

we obtain a surjection H e H n . Let us write H c for the kernel of this surjection. (Here,e (respectively, n; c) stands for etale (respectively, normalization; combinatorial).)Note that H c is a free Z l -module of rank N X I X + 1, where N X (respectively, I X ) isthe number nodes (respectively, irreducible components) of X k . Moreover, H

n is a freeZ l -module of rank equal to twice the sum of the genera of the connected components of X k .

Thus, we obtain a natural exact sequence of -modules

0 H c H e H n 0

Let be the automorphism of k given by raising to the q th power. Then one seeseasily that some nite power of acts trivially on H c . On the other hand, by the Weil

4


5/47

conjectures (applied to the various geometric connected components of X ), no power of acts with eigenvalue 1 on H n . We thus obtain the followingProposition 1.2: The natural exact sequence 0 H c H e H n 0 can be

recovered entirely from .

Proof: Indeed, H e = Hom ( , Z l ), while H c can be recovered by looking at the maximalZ l -submodule of H e on which some power of acts trivially.

Let Le = H e F l ; Lc = H c F l ; Ln = H n F l . Thus, Le = H 1et (X k , F l ), and wehave an exact sequence of -modules

0 Lc Le Ln 0

Moreover, elements of Le correspond to etale, abelian coverings of X k of degree l. LetLLe be the subset of elements whose image in Ln is nonzero .

Suppose that L. Let Y X k be the corresponding covering. Then N Y = lN X .Thus, we obtain a morphism : L Z that maps I Y . Since L is a nite set,the image of is nite. Let M L be the subset of elements on which attains itsmaximum. Let us dene a pre-equivalence relation on M as follows:

If , M , then we write if, for every , F l for which + L, we have + M .

Now we have the following result:

Proposition 1.3: Suppose that X is stable and sturdy . Then is, in fact, anequivalence relation, and moreover, C X

def = M/ is naturally isomorphic to the set of irreducible components of X k .

Proof: First, let us observe, that I Y is maximal (equal to l(I X 1) + 1) if and only if there exists a unique irreducible component Z of

X k over which the covering Y X k is

nontrivial. Now, if Z is a connected component of X k , let LZ def = H 1et (Z, F l ). Thus,Ln =

Z

LZ

where the direct sum is over the connected components of X k . Then it follows immediatelyfrom the denitions that M consists precisely of those elements L whose image inLn has (relative to the above direct sum decomposition) exactly one nonzero component

5


6/47

(namely, in LZ ). Moreover, is equivalent to Z = Z . Finally, that every Z appears as a Z follows from the sturdiness assumption. This completes the proof.

Remark: Note that although at rst glance the set C X = M/ appears to depend on thechoice of prime l, it is not difficult to see that in fact, if one chooses another prime l, andhence obtains a resulting C X = M / , one obtains a natural isomorphism C X = C X(compatible with the isomorphisms just obtained of C X and C X to the set of irreduciblecomponents of X k ) as follows: If M and M , let us consider the productY = Y X Y . Thus, we have a cyclic etale covering Y X of degree l l . Thenone checks easily that and correspond to the same irreducible component if and onlyif (Y )k has precisely l l (I X 1) + 1 irreducible components.

Proposition 1.4: Suppose that X is stable and sturdy . Then the set of irreducible components of X

k (together with its natural -action) can be recovered entirely from

.

Proof: Indeed, it follows from Proposition 1.2 that L can be recovered from .Moreover, we claim that M can be recovered, as well. Indeed, the maximality of I Y isequivalent to the minimality of N Y I Y + 1 = l N X I Y + 1, which is equal to thedimension over F l of the Lc of Y . Once one has M , it follows that one can also recover, hence by Proposition 1.3, one can recover the set of irreducible components of X k .Finally, by the above Remark, the set that one recovers is independent of the choice of l.

Corollary 1.5: Suppose that X is stable and sturdy. Let H be an open subgroup.Let Y H X be the corresponding etale covering. Then the set of irreducible components of Y H can be recovered from and H .

Proof: Let k be the (nite) extension of k which is the subeld of k stabilized by theimage of H in . Then Y H is geometrically connected, hence stable and sturdy over k .Thus, we reduce to the case H = , Y H = X . But then the set of irreducible componentsof X is the set of -orbits of the set of irreducible components of X k . Thus, the Corollaryfollows from Proposition 1.4.

Looking back over what we have done, one sees that in fact, we have proven a strongerresult that what is stated in Corollary 1.5. Indeed, x an irreducible component I X .Then let J l (I ) be the set of k-valued l-torsion points of the Jacobian of the normalizationof I k . Then not only have we recovered set of all irreducible components I , we have alsorecovered, for each I , the set J l (I ) (with its natural Frobenius action). We state this as aCorollary:

6


7/47

Corollary 1.6: Suppose that X is stable and sturdy, and l is a prime number differentfrom p. Then for each irreducible component I of X , the set J l (I ) (with its natural Frobenius action) can be recovered naturally from .

Section 2: The Admissible Fundamental Group

Let r and g be nonnegative integers such that 2 g 2+ r 1. If (C Mg,r ; 1 , . . . , r :M g,r C ) is the universal r -pointed stable curve of genus g over the moduli stack, then C and M g,r have natural log structures dened by the respective divisors at innity and theimages of the i . Denote the resulting log morphism by C log M

logg,r . Let X S be the

underlying curve associated to an r -pointed stable curve of genus g over a scheme S (whereS is the underlying scheme of some log scheme S log ). Suppose that X is equipped with thelog structure (call the resulting log scheme X log ) obtained by pulling back C log M logg,r

via some log morphism S log

Mlog

g,r whose underlying non-log morphism S M g,r isthe classifying morphism of X (equipped with its marked points). In this case, we shallcall X log S log an r-pointed stable log-curve of genus g. Similarly, we have r-pointed multistable log-curves of genus g: that is, X log S log such that over some nite etalecovering S S , X log S S becomes a nite union of stable pointed log-curves.

Let k be as in the preceding section. Let S log be a log scheme whose underlying schemeis Spec(k) and whose log structure is (noncanonically!) isomorphic to the log structureassociated to the morphism N k, where 1 N 0 k. Let X log S log be a stablelog-curve of genus g 2.

Next, we would like to consider liftings of X log S log . Let A be a complete discretevaluation ring which is nite over Z p and has residue eld equal to k. Let T log be a logscheme whose underlying scheme is Spec(A) and whose log structure is that dened by thespecial point S = Spec(k) T . Also, let us assume that S log is equal to the restriction of the log structure of T log to S = Spec(k) T . Let Y log T log be a stable log-curve of genus g whose restriction to S log is X log S log . In this case, we shall say that Y log T loglifts X log S log . It is well-known (from the log-smoothness of the moduli stack of stablecurves equipped with its natural log structure) that such log-curves Y log T log alwaysexist.

Next, we would like to consider log admissible coverings

Z log Y log

of Y log . We refer to [Mzk], 3.5, for the rather lengthy and technical denition and rstproperties of such coverings. It follows in particular from the denition that Z is a stablecurve over T . In fact, (as is shown in [Mzk], Proposition 3.11), one can dene such coveringswithout referring to log structures. That is, there is a notion of an admissible covering ([Mzk]. 3.9) Z Y (which can be dened without using log structures). Moreover,Z Y is admissible if and only if Z admits a log structure such that Z log Y log is log

7


8/47

admissible. In [Mzk], we dealt strictly with the case where Z is geometrically connectedover T . Here, we shall call Z Y multi-admissible if Z is a disjoint union of connectedcomponents Z i such that each Z i Y is admissible.

Let be the generic point of T . Let Y = Y T . If Z Y is multi-admissible, then

it will always be the case that the restriction Z Y of this covering to the generic beris nite etale. Now suppose that : Z Y is a nite etale covering. If extends toan multi-admissible covering Z Y , then this extension is unique ([Mzk], 3.13).

Denition 2.1: We shall call pre-admissible if it extends to an multi-admissible covering : Z Y . We shall call potentially pre-admissible if it becomes pre-admissible after a tamely ramied base-change (i.e., replacing A by a tamely ramied extension of A).

Thus, in particular, if A is a tamely ramied extension of A, then Y A A Y ispotentially pre-admissible. If is potentially pre-admissible and Z is geometricallyconnected over , then it is pre-admissible if and only if Z has stable reduction over A.

Lemma 2.2 : Suppose that Z Y and Z Y are pre-admissible. Let Z def =

Z Y Z . Then Z Y is pre-admissible.

Proof: Let Z Y and Z Y be the respective multi-admissible extensions. Let Z be the normalization of Y in Z . Thus, we have a natural morphism Z Z Y Z whichis an isomorphism at height one primes. In particular, Z is etale over Y at all height oneprimes. It thus follows from Lemma 3.12 of [Mzk] that Z Y is multi-admissible.

Lemma 2.3 : Suppose that Z Y is pre-admissible, and that Z Y factors through nite etale surjections Z Z and Z Y . Then Z Y is pre-admissible.

Proof: Similar to that of Lemma 2.2.

Let K be the quotient eld of A. Fix an algebraic closure K of K . Suppose that Y is equipped with a base-point y Y (A) such that the corresponding morphism T Y avoids the nodes of the special ber of Y . Write Y for 1 (Y K , yK ). Thus, we have anatural surjection Y Gal (K/K ), whose kernel is a group Y Y .

Denition 2.4: We shall call an open subgroup H Y co-admissible if the corre-sponding nite etale covering Z Y is potentially pre-admissible. Let admY be the quotient of Y by the intersection H of all co-admissible H Y .

8


9/47

Remark: The admissible fundamental group admY has already been dened and studiedby K. Fujiwara ([Fuji]). Moreover, the author learned much about admY (as well as aboutthe theory of log structures in general) by means of oral communication with K. Fujiwara.

It is easy to see that the intersection H of Denition 2.4 is a normal subgroup of Y . Thus, admY is a group. Moreover, by Lemmas 2.2 and 2.3, it follows that an opensubgroup H Y is co-admissible if and only if Ker (Y admY ) H . Finally, it isimmediate from the denitions that the subeld of K stabilized by the image of H inGal (K/K ) is the maximal tamely ramied extension K of K . Thus, we have a surjection

admY Gal (K /K )

whose kernel admY admY is a quotient of Y .

Denition 2.5: We shall refer to as orderly coverings of Y those coverings Z Y which are Galois and factor as Z Y T U Y , where the rst morphism is pre-admissible; the second morphism is the natural projection; U = Spec(B ); and B is a tamely ramied nite extension of A. We shall refer to as orderly quotients of admY those quotients of admY that give rise to orderly coverings of Y .

It is easy to see that orderly quotients of admY are conal among all quotients of admY .

Let A K be the normalization of A in K . Let k (respectively, m ) be theresidue eld (respectively, maximal ideal) of A . Let T = Spec(A ), and let us endow

T with the log structure given by the multiplicative monoid OT { 0} (equipped withthe natural morphism into OT ). We call the resulting log scheme T log . Let S log bethe log scheme whose underlying scheme is Spec(k ) and whose log structure is pulledback from T log . Thus, the log structure on S log is (noncanonically!) isomorphic to the logstructure dened by the zero morphism ( Z ( p) ) 0 k . (Here ( Z ( p) ) 0 denotes the set of nonnegative rational numbers whose denominators are prime to p.) Note that Gal (K /K )induces S log -automorphisms of S log . In fact, it is easy to see that this correspondencedenes a natural isomorphism

Gal (K /K ) = Aut S log (S log )

Now we have the following important

Lemma 2.6 : Suppose that we are given:

(1) another lifting (Y )log (T )log (where T = Spec(A )) of X log S log

(2) an algebraic closure K of K = Q(A ) (hence a resulting K K ;(S )log );

9


10/47

(3) an S log -isomorphism : S log = (S )log ;

(4) a base-point y Y (A ) such that y|S = y |S in X (k).

Then there is a natural isomorphism between the surjections admY Gal (K /K ) and admY Gal (K /K ).

Proof: Let us rst consider coverings of Y obtained by pulling back tamely ramiedGalois coverings of K . Thus, if U = Spec(B ) T is nite, Galois, and tamely ramied(obtained from some eld extension K L), let U log be the log scheme obtained byequipping U with the log structure dened by the special point u of U . Thus, we obtain anite, log etale morphism U log T log . By base-changing to S log , we then obtain a nite,log etale morphism V log S log . On the other hand, by the denition of log etaleness,this morphism then necessarily lifts to a nite, log etale morphism ( U )log (T )log ,whose underlying morphism U

T is a Galois, tamely ramied nite extension. Thus,

if we pass to the limit, and apply this construction to the extension L = K of K , weend up with some maximal tamely ramied extension L of K . By the functoriality of this construction, we have a natural isomorphism Gal (L/K ) = Gal (L /K ). Now observethat there is a unique K -isomorphism L = K which (relative to this construction) iscompatible with . Thus, we get an isomorphism Gal (L /K ) = Gal (K /K ), hence anisomorphism Gal (K /K ) = Gal (K /K ), as desired.

Now let us consider orderly coverings Z Y . Thus, we have a factorizationZ Y T U Y . Let Z be the normalization of Y in Z . Then Z Y T U ismulti-admissible. Thus, Z admits a log structure such that we have a log multi-admissible

covering Z log

Y log

T log U log

. Base-changing, we obtain a log multi-admissible cov-ering Z log |S log X log S log V log . But since log multi-admissible coverings are logetale, it thus follows that this covering lifts uniquely to a log multi-admissible covering(Z )log (Y )log (T ) log (U )log . Similarly, we obtain a bijective correspondence betweenY -automorphisms of Z and Y -automorphisms of Z . Now observe further that K -valued points of Z over yK dene A -valued points of Z over y (since Z is proper overA). Moreover, these points dene T log -valued points of Z log over yT log , hence (by reduc-ing modulo m ) S log -valued points of Z log |S log over yS log , hence (using ) (S )log -valuedpoints of Z log |S log over yS log = yS log , hence (by log etaleness) ( T )

log -valued points of

(Z )log over yT log , which, nally, give rise to K -valued points of Z over yK .

Thus, in summary, we have dened a natural equivalence of categories between orderlycoverings of Y and orderly coverings of Y . Moreover, this equivalence is compatiblewith the ber functors dened by the base-points yK and yK . Thus, we obtain anisomorphism between the surjections admY Gal (K /K ) and admY Gal (K /K ), asdesired.

Now let us interpret Lemma 2.6. In summary, what Lemma 2.6 says is the following:Suppose we start with the following data:

10


11/47

(1) a log scheme S log , where S = Spec(k), and the log structure is (non-canonically!) isomorphic to the log structure associated to the zeromorphism N k;

(2) a log scheme S log over S log which is (noncanonically!) k-isomorphic toSpec(k) equipped with the log structure associated to the zero morphism(Z ( p) ) 0 k (where the N (Z ( p) ) 0 is pulled back from a chart forS log as in (1));

(3) a stable log-curve X log S log of genus g;

(4) a base-point x X (k) which is not a node.

Then, to this data, we can naturally associate an admissible fundamental group admXwith augmentation admX S log

def = Aut S log (S log ). That is to say, by choosing a lifting

of the above data, we may take admX =

admY , and the augmentation to be

admY Gal (K /K ). Then Lemma 2.6 says that, up to canonical isomorphism, admX and its

augmentation do not depend on the choice of lifting.

Denition 2.7: We shall refer to the data (1) through (4) above as admissible data of genus g. Given admissible data as above, we shall write 1 (X log , xS log ) for

admX and

1 (S log , S log ) for S log . We shall refer to admX as the admissible fundamental group of X .Write X log admX for the kernel of the augmentation. We shall refer to X log as the geometric admissible fundamental group of X .

Next, let us observe that S log admits a natural surjection onto S def = Gal (k/k ). We

shall denote the kernel of this surjection by I S S log , and refer to I S as the inertia subgroup of S log . Note that I S is isomorphic to the inverse limit of the various ( k ) (fornite extensions k k of k), where the transition morphisms in the inverse limit are givenby taking the norm. Or, in other words, I S = Z (1), where Z is the inverse limit of thequotients of Z of order prime to p, and the (1) is a Tate twist. Thus, we have a naturalexact sequence

1 I S = Z

(1) S log

S = Gal (k/k ) 1

Suppose that we are given a continuous action of S log on a nite set . Then we canassociate a geometric object to as follows. Without loss of generality, we can assumethat the action on is transitive. If we choose a lifting T log of S log (where T = Spec(A)),then corresponds to some nite, tamely ramied extension L of K . Let B be thenormalization of A in L. Equip U def = Spec(B ) with the log structure dened by the specialpoint u of U . Thus, we obtain U log . Equip Spec(k(u)) with the log structure induced bythat of U log . Then the geometric ob ject associated to is the nite, log etale morphism

11


12/47

Spec(k(u)) log S log . We shall call such morphisms nite, tamely ramied coverings of S log .

Now suppose that we have an open subgroup H admX that surjects onto S log . Let H

def = H X log . In terms of liftings, H corresponds to a nite etale covering Z Y ,where Z is geometrically connected over . Note that by a well-known criterion ([SGA7]),Z has stable reduction over A if and only if I S acts unipotently on Hom ( H , Z l ) (for someprime l distinct from p). But, as noted above, in this situation, Z has stable reduction if and only if Z Y is pre-admissible. If Z Y is pre-admissible, it extends to someZ log Y log , which we can base-change via S log T log to obtain a log admissible coveringZ log |S log X log . Conversely, every log admissible covering of X log can be obtained inthis manner. Thus, in summary, we have the following result:

Corollary 2.8: The open subgroups of admX that correspond to orderly coverings can be characterized entirely group-theoretically by means of admX S log (and S log S ).Moreover, these subgroups can be interpreted in terms of geometric coverings of X log(namely, base-change via a nite, tamely ramied covering of S log , followed by a log admissible covering).

Remark: Note that this Corollary thus allows us to speak of orderly coverings of X log ,i.e., coverings that arise from orderly quotients (Denition 2.5) of admX = admY .

Before continuing, let us make the following useful technical observation:

Lemma 2.9 : Given any stable X over k, there always exists an admissible covering Z X such that Z is (multi-stable and) sturdy .

Indeed, this follows from the denition of an admissible covering, plus elementary com-binatorial considerations. Moreover, an admissible covering of a sturdy curve is alwayssturdy. Thus, if one wishes to work only with sturdy curves, one can always pass to sucha situation by replacing our original X by some suitable admissible covering of X .

Finally, although most of this paper deals with the case of nonpointed stable curves,it turns out that we will need to deal with pointed stable curves a bit later on. In fact, itwill suffice to consider pointed smooth curves. Thus, let g and r be nonnegative integerssuch that 2 g 2+ r 1. Let S log S log be as above, and let X log S log be an r -pointedstable log-curve of genus g such that X is k-smooth. Also, let x X (k) be a nonmarkedpoint. Then it is easy to see that, just as above, we can dene (by considering variousliftings to some A, then showing that what we have done does not depend on the lifting)an admissible fundamental group admX (with base-point xS log ), together with a naturalsurjection

admX S log

12


13/47

Moreover, the kernel X log admX of this surjection is naturally isomorphic to the tamefundamental group of X k (with base-point xk ). Unlike the singular case, we dont particu-lar gain anything new by doing this, but what will be important is that we still nonethelessobtain a natural surjection admX S log which arises functorially from the same frame-work as the nontrivial admX S log that appears in the case of singular curves.

Section 3: Characterization of the Etale Fundamental Group

We maintain the notation of the preceding Section. Thus, in particular, we have astable log-curve X log S log , together with a choice of S log , and a base-point x X (k)(which is not a node). Then note that we have a natural morphism of exact sequences:

1 X log admX S log 1

1 X X S 1

Here the vertical arrows are all surjections. The goal of this Section is to show how onecan recover the quotient admX X group-theoretically from admX S log .

Let Y log X log be a log admissible covering which is abelian, with Galois groupequal to F l , where l is a prime number (which is not necessarily distinct from p). Let usconsider the following condition on this covering:

(*) Over k, there is an innite log admissible covering Z log X logk whichis abelian with Galois group Z l such that the intermediate coveringcorresponding to Z l F l is Y logk X

logk

.

Here, by innite log admissible covering, we mean an inverse limit of log admissiblecoverings in the usual nite sense. Suppose that Y log X log satises (*). Then we claimthat Y X is, in fact, etale . Indeed, if p = l, then every abelian log admissible coveringof degree l is automatically etale, so there is nothing to prove. If p = l, then we can x anode X , and consider the ramication over the two branches of X at . Consideringthis ramication gives rise to an inertia subgroup H Z l . If Y log X log is ramiedat , then H surjects onto F l , so H = Z l . On the other hand, by the denition of alog admissible covering, in order to have innite ramication occuring over the geometricbranches of X at , we must also have innite ramication over the base S log . But, by(*), Z log X log is already log admissible over S log k k (which is, of course, etale overS log ). This contradiction shows that Z log , and hence Y log , are unramied over X log at .Thus, we see that (*) implies that Y X is etale, as claimed. Note that conversely, if weknow that Y X is etale to begin with, then it is easy to see that (*) is satised. Thus,for an abelian log admissible covering Y log X log of prime degree l, (*) is equivalent tothe etaleness of Y X .

13


14/47

Now observe that the kernel of X log X is normal not just in X log , but also inadmX . Let X = admX /Ker ( X log X ). Let X be the kernel of X S . Thus, X X X . Then we have the following result:

Proposition 3.1: The quotient admX X can be recovered entirely group theoreti-cally from admX S log .

Proof: It suffices to characterize subgroups H admX of nite index that containKer ( X log X ). Without loss of generality, we may assume that H is normal inadmX and (by Corollary 2.8) corresponds to an orderly covering. Since an orderly coveringmay be factored as a composite of a log multi-admissible covering followed by a tamelyramied covering of S log , one sees immediately that we may reduce to the case whereH corresponds to a log admissible covering. Let G = admX /H . Let Y log X log bethe corresponding covering. For every subgroup N G, denote by Y log Y logN thecorresponding intermediate covering. By considering ramication at the nodes, one seesimmediately that Y X is etale if and only for every cyclic N G of prime order,Y Y N is etale. But for such N , the etaleness of Y Y N is equivalent to the condition(*) discussed above. Moreover, it is clear that (*) can be phrased in entirely group theoreticterms, using only admX S log (and S log S ). This completes the proof.

Now suppose that Y log X log is an abelian orderly covering of prime order l obtainedfrom a quotient of X such that Y is geometrically connected. Assume l = p. Considerthe following condition on Y log X log :

() There do not exist any innite abelian orderly coverings Z log X logk

with Galois group Z l that satisfy both of the following two conditions:(i) the intermediate covering corresponding to Z l F l is Y logk X

logk

;(ii) some nite power M of the Frobenius morphism = S stabilizes Z log X log

kand acts on the Galois group Z l with eigenvalue

q M .

Because Y log X log is abelian of prime order, it follows that one of the following holds:

(1) Y log

X log

is obtained from an etale covering Y X (where Y isgeometrically connected) base-changed by S log S .

(2) Y log X log is obtained by base-change via X log S log from sometotally (tamely) ramied covering of S log .

Suppose that ( ) is satised. Then we claim that (1) holds. Indeed, if this were false, then(2) would hold, but it is clear that if (2) holds, then one can easily construct Z log X log

kthat contradict ( ) (by pulling back via X log S log an innite ramied covering of S log ).

14


15/47

This proves the claim. Now suppose that (1) holds. Then we claim that ( ) is satised.To prove this, suppose that there exists an offending Z log X log

k. This offending covering

denes an injection Z l Hom ( X , Z l ) Hom ( X , Z l ) = H 1et (X k , Z l ). On the otherhand, as we saw in Section 1, by the Weil conjectures, no power M of acts with eigenvalueq M on H 1

et(X

k, Z

l). This contradiction completes the proof of the claim. Thus, in summary,

(1) is equivalent to ( ). In other words, we have essentially proven the following result:

Proposition 3.2: The quotient admX X can be recovered entirely group theoreti-cally from admX S log .

Proof: It suffices to characterize nite index subgroups H X that contain Ker (X X ). Without loss of generality, we may assume that H is normal in X and (by Corol-lary 2.8) corresponds to an orderly covering. Let G = X /H . Let Y log X log be thecorresponding covering. Again, without loss of generality, we may assume that Y is geo-

metrically connected over k. For every normal subgroup N G, denote by Y log

N X logthe corresponding intermediate covering. Next, we observe the following: Y log X logarises from an etale covering of X if and only if, for every normal subgroup N G suchthat Y logN X log is orderly and G/N is cyclic of prime order, Y N X arises from anetale covering of X . (This equivalence follows immediately from the denitions and thefact that X / X is abelian.) But for such Y

logN X log , we can apply the criterion ( )

discussed above. Moreover, it is clear that ( ) can be phrased in entirely group theoreticterms, using only X S log (and S log S ). This completes the proof.

Section 4: The Decomposition Group of an Irreducible Component

We maintain the notation of the preceding Section. Fix an irreducible componentI X . Then, corresponding to I , there is a unique (up to conjugacy) decomposition subgroup

admI admX

which may be dened as follows. Let Z log X log be the log scheme obtained by takingthe inverse limit of the various Y logH X log corresponding to open orderly subgroups

H adm

X . Choose an irreducible component J Z that maps down to I X . Here,by irreducible component of Z , we mean a compatible system of irreducible componentsI H Y H . Then admI admX is the subgroup of elements that take the irreduciblecomponent J to itself. We can also dene an inertia subgroup

inI admI

as follows: Namely, we let inI be the subgroup of elements of admI that act trivially onJ . (That is to say, elements of inI will, in general, act nontrivially on the log structure of

15


16/47

J , but trivially on the underlying scheme J .) By Proposition 3.2 and Corollaries 1.5 and1.6, we thus obtain the following:

Proposition 4.1: Suppose that X is stable and sturdy . Then one can recover the

set of irreducible components of X from adm

X S log

. Moreover, for each irreducible component I of X , one can recover the corresponding inertia and decomposition subgroups inI admI admX entirely from admX S log .

Proof: That one can recover admI follows formally from Proposition 3.2 and Corollary1.5. Now observe that (as is well known see, e.g., [DM]) any automorphism of I that actstrivially on J l (I ) (where l 5) is the identity. This observation, coupled with Corollary1.6, allows one to recover inI .

Now let us suppose that the base-point x X (k) is contained in I . Let

I be the

normalization of I . Then one can also dene admI as follows. Let I I be the opensubset which is the complement of the nodes. We give I a log structure by restricting to I the log structure of X log . Denote the resulting log scheme by I log . Now let us regard thepoints of I that map to nodes of I as marked points of I . This gives I the structure of asmooth, pointed curve over k. Because I Spec(k) is smooth, it follows that there existsa unique multistable pointed log-curve I log S log whose underlying curve is I and whosemarked points are as just specied. Since x I (k), by using S log S log , we can dene(as in the discussion as the end of Section 2) the admissible fundamental group admI of I log . Moreover, we have natural log morphisms

I log X log

I log

where the vertical morphism is an open immersion. Now observe that if we restrict (say,orderly) coverings of X log to I log , such a covering extends naturally to an orderly coveringof I log . Thus, we obtain a natural morphism

I : admI

admX

It is immediate from the denitions that the subgroup I (admI ) admX is a admI .

Proposition 4.2: Suppose that X is stable and sturdy . Then the morphism I is injective.

Proof: Let I S log be the image of admI in S log . Then let us note that we have a

commutative diagram of exact sequences:

16


17/47

1 I log admI I S log 1

1 X log admX

S log 1

where the vertical arrow on the right is the natural inclusion. Thus, it suffices to provethat I log X log is injective. In particular, we are always free to replace S

log by anite, tamely ramied covering of S log .

Now it suffices to show that (up to base-changing, when necessary, by nite, tamelyramied coverings of S log ) we can obtain every log admissible covering of I log by pullingback a log admissible covering of X log . Let us call X untangled at I if every node of X that lies on I also lies on an irreducible component of X distinct from I . In general, wecan form a (combinatorial) etale covering of X as follows: Write X = I J , where J isthe union of the irreducible components of X other than I . Let I 1 and I 2 (respectively,J 1 and J 2) be copies of I (respectively, J ). For i = 1 , 2, let us glue I i to J i at every nodeof I that also lies on J . If is a node of I that only lies on I , let and be the pointsof I that lie over . Then glue 1 I 1 to 2 I 2 , and 1 I 1 to 2 I 2 . With thesevarious gluings, I 1 I 2 J 1 J 2 forms a curve Y which is nite etale over X . Moreover,Y is untangled at I 1 and I 2 , and admI = admI i (for i = 1 , 2). Thus, it suffices to prove theProposition under the assumption that X is untangled at I . Therefore, for the remainderof the proof, we shall assume that X is untangled at I .

Now we would like to construct another double etale covering of X . For convenience,

we will assume that p 3. (The case p = 2 is only combinatorially a bit more difficult.)Write X = I J , as above. Since X is sturdy, it follows that (after possibly enlarging k),there exists an etale covering J J of degree two such that for any irreducible componentC J , the restriction of J J to C is nontrivial. Let I 1 and I 2 be copies of I . If isa node on I and J , let (respectively, ) be the corresponding point on I (respectively,J ). (After possibly enlarging k) we may assume that J has two k-rational points 1 and 2 over J . Now, for i = 1 , 2, glue i I i to i J . We thus obtain a double etalecovering Y = I 1 I 2 J X . Endow Y with the log structure obtained by pulling backthe log structure of X log . One can then dene various log structures on the irreduciblecomponents of Y , analogously to the way in which various log structures were dened on

an irreducible component I of X above. We will then use similar notation for the logstructures thus obtained on irreducible components of Y .

Now let Llog I log be a Galois log admissible covering of I log def = I log (recall thatI = I ) of degree d. Let M log J log be an abelian log multi-admissible covering of degreed with the following property:(*) For each node on I and J , suppose that over the corresponding I , L has n geometric points, each ramied with index e over I .

17


18/47

Then, we stipulate that for i = 1 , 2, over i J , M has n geometricpoints, each ramied with index e over I .Note that such an M log J log exists precisely because the s on J come in pairs . Nowlet Llog1 and Llog2 be copies of Llog . Then, for each i = 1 , 2, let us glue the geometricpoints of Li | i to those of M | i . This gives us (after possibly replacing S log by a tamelyramied covering of S log ) a log admissible covering Z log Y log , where Z = L1 L2 M .Moreover, the restriction of Z log Y log to I logi (for i = 1 , 2) is Llog I log . This completesthe proof of the Proposition.

Section 5: The Set of Nodes

We continue with the notation of the preceding Section. Thus, X log S log is a stablelog-curve of genus g. Let us also assume that X is sturdy . In this Section, we would liketo show how (by a technique similar to, but slightly more complicated than that employedin Section 1) we can recover the set of nodes of X . This, in turn, will allow us to recoverthe decomposition group of a node. In the following Section, we shall then show how thelog structure at a node can be reconstructed from the decomposition group at the node.

Let l and n be prime numbers distinct from each other and from p. We assumemoreover that l 1 (mod n ). This means that all nth roots of unity are contained in F l .Let us write Gn F l for the subgroup of nth roots of unity. Next, let us x a Gn -torsorover X k

Y X

which is nontrivial over the generic point of every irreducible component of X k . (Here,by Gn -torsor, we mean a cyclic etale covering of X of degree n whose Galois groupis equipped with an isomorphism with Gn .) Equip Y with the log structure pulledback from that of X log . Let us consider the admissible fundamental group admY of Y log . Let H 1adm (Y

logk

, F l ) def = Hom ( Y log , F l ). Note that we have a natural injection

Le def = H 1et (Y k , F l ) La def = H 1adm (Y

logk

, F l ). Let us write Lr for the cokernel of this injection. (Here, e (respectively, a; r ) stands for etale (respectively, admissible;ramication). Thus, we have an exact sequence

0 Le La L r 0

which may (by Proposition 3.2) be recovered from admX S log and the subgroup of admX that denes Y X . Note, moreover, that Gn acts on the above exact sequence.Let L rG Lr be the subset of elements on which Gn acts via the character Gn F

l . Let

LLa be the subset of elements that map to nonzero elements of LrG .

18


19/47

We would like to analyze L rG . First of all, let us consider Lr . For each node Y (k),write Y for the completion of Y at , and let and be the two irreducible componentsof Y . Let DY (respectively, E Y ) be the free F l -module which is the direct sum of copies of F l ( 1) (where the ( 1) is a Tate twist) generated by the symbols , (respectively, I ),as (respectively, I ) ranges over all the nodes of Y (respectively, irreducible componentsof Y k ). Let DY DY be the submodule generated by ( ) F l ( 1) (where rangesover all the nodes of Y ). Note that we have a natural morphism DY E Y given byassigning to the symbol (respectively, ) the unique irreducible component I in which (respectively, ) is contained. In particular, restricting to DY , we obtain a morphism

DY E Y

Let K Y DY be the kernel of this morphism. Now let us note that we have a naturalmorphism

: Lr DY

given by restricting an admissible covering of Y k to the various and . It followsimmediately from the denition of an admissible covering that Im () DY . Moreover,by considering the Leray-Serre spectral sequence in etale cohomology for the morphismI I (where I Y k is the normalization of an irreducible component of Y k , and I is thecomplement of the points that map to nodes), plus the denition of an admissible covering,one sees easily that, in fact, (Lr ) = K Y . Finally, by counting dimensions, we see that is injective. Thus, we see that denes a natural isomorphism of Lr with K Y . In thefollowing, we shall identify L r and K Y by means of .

Now let us consider the subset LrG Lr . Let X (k) be a node. For each such ,let us x a node Y (k) over . If Gn (regarded as the Galois group of Y X ), weshall write a F l for regarded as an element of F

l . Fix a generator F l ( 1). Let

def =

G n

(a 1 )(( ) ( )) D Y

One checks easily that is, in fact, an element of K Y = Lr . Moreover, by calculating ( ) (for Gn ), one sees that is manifestly an element of LrG Lr . Finally, it isroutine to check that, in fact, Lr

G is freely generated by the (as ranges over the nodes

of X (k) but is xed). This completes our analysis of LrG .

Suppose that L. Let Z Y k be the corresponding covering. Let : L Z

be the morphism that maps to N Z (i.e., the number of nodes of Z ). Let M L bethe subset of elements on which attains its maximum. Let us dene a pre-equivalencerelation on M as follows:

If , M , then we write if, for every , F l for which + L, we have + M .

19


20/47

Now we have the following result:

Proposition 5.1: Suppose that X is stable and sturdy . Then is, in fact, anequivalence relation, and moreover, M/ is naturally isomorphic to the set of nodes of

X k .

Proof: Suppose that L maps to a linear combination (with nonzero coefficients) of precisely c 1 of the elements L rG . Then one calculates easily that Z has precisely() = l(N Y cn)+ cn = l N Y + cn(1 l) nodes. Thus, ( ) attains its maximum precisely

when c = 1. Thus, M L consists of those which map to a nonzero multiple of one of the s. It is thus easy to see (as in the proof of Proposition 1.3) that M/ is naturallyisomorphic to the set of nodes X (k).

Remark: Note that at rst glance the set M/ appears to depend on the choice of n, l,and Y X . However, it is not difficult to see that in fact, if one chooses different datan = n, l = l, and Y X , and hence obtains a resulting M / , then there is a naturalisomorphism ( M/ ) = (M / ) (compatible with the isomorphisms just obtained of M/ and M / to the set of nodes of X k ) as follows: If M and M , let usconsider the product Z = Z X Z . Thus, we have an admissible covering Z X of degree (ln )( l n ). Then one checks easily that and correspond to the same node if and only if (Z )k has precisely nn {l l (N X 1) + 1 } nodes.

Proposition 5.2: Suppose that X is stable and sturdy . Then the set of nodes of X k(together with its natural S -action) can be recovered entirely from admX S log . More-over, (relative to Proposition 1.4) for each node of X k , the set of irreducible components of X k containing can also be recovered entirely from

admX S log .

Proof: Indeed, (after possibly replacing k by a nite extension of k) one can alwayschoose l, n, and Y X as above. Then one can recover Le and La from admX S logand the subgroup of admX dened by Y X . Thus, one can also recover Lr . We saw inSection 1 that for any Z , N Z I Z , as well as I Z , may be recovered group-theoretically.In particular, N Z can also be recovered group-theoretically. Thus, M and can also berecovered group-theoretically. Moreover, by the above Remark, M/ is independent of the choice of n, l, and Y X . (That is to say, the isomorphism ( M/ ) = (M / ) of the above Remark can clearly be recovered group-theoretically.) This completes the proof that the nodes can be recovered group-theoretically.

Now let us consider the issue of determining which irreducible components of X k (rel-ative to the reconstruction of the set of irreducible components of X k given in Proposition1.4) lies on. To this end, note rst that (by Corollary 1.6) the genus gI of each irreduciblecomponent I of X k can also be recovered group-theoretically. (Indeed, J l (I ) has preciselyl2gI elements.) Thus, if M corresponds to the node , the irreducible components I

20


21/47

of X k containing are precisely those that are the image of irreducible components J of Z such that ( gJ 1) > l n (gI 1).

Corollary 5.3: Suppose that X is stable and sturdy. Let H admX be an open orderly

subgroup. Let Y log

H X log

be the corresponding covering. Then the sets of nodes N Y Hand N X of Y H and X , respectively, as well as the natural morphism N Y H N X can be recovered from admX S log and H . Moreover, the set of irreducible components of (Y H )k on which each node of N Y H lies can also be recovered from

admX S log and H .

Proof: In this case, in order to obtain the morphism N Y H N X it is useful to choose land n prime to the index of H in admX , and to choose the Gn -torsor Y over Y H to bethe pull-back of a Gn -torsor on X H . The rest of the proof is straightforward.

Now let be a node of X . Let I X be an irreducible component of X on which sits. Then there is a unique (up to conjugacy) decomposition subgroup

adm admX

which may be dened as follows. Let Z log X log be the log scheme obtained by takingthe inverse limit of the various Y logH X log corresponding to open orderly subgroupsH admX . Choose a node Z that maps down to X . Here, by node of Z ,we mean a compatible system of nodes H Y H . Then adm admX is the subgroup of elements that take the node to itself. If sits on an irreducible component J of Z which

lies over I , then we can also form admI , inI admX , hence adm,I def = admI adm , and in,I

def = inI adm . Up to conjugacy, adm,I and in,I are independent of all the choicesmade. Now, it follows formally from Corollary 5.3 that

Corollary 5.4: Given and I as above, one can recover adm ; adm,I ; in,I admXentirely from admX S log .

Section 6: The Log Structure at a Node

We maintain the notation of the preceding Section. In addition to assuming that X is stable and sturdy, let us assume that it is untangled (i.e., every node lies on two distinctirreducible components). Let X (k) be a node of X . Let I , I be the two irreduciblecomponents of X on which lies. Let Z log X log be the covering corresponding to thetrivial subgroup of admX . Let be a node of Z lying over . Let J (respectively, J ) be theirreducible component of Z that touches and lies over I (respectively, I ). Thus, as in thepreceding Section, we have various subgroups, such as adm , admI , admI admX . Notethat (since X is untangled) elements of adm x J and J . Thus, adm admI , admI .

21


22/47

Next, consider the natural morphism adm admX S . Since is k-rational, thismorphism is surjective. Let us denote the kernel of this morphism by in . Thus, we havean exact sequence

1 in adm S 1

Moreover, by sorting through the denitions, it is clear that inI , inI in .

Next, let us consider the natural morphism inI admX S log . It is clear that theimage of this morphism is contained in the inertia subgroup I S S log . Thus, we obtaina natural morphism inI I S . By using the fact that the restriction of the log structureof X log to the generic point of I is the pull-back (to the generic point of I ) of the logstructure of S log , it is then easy to see that this morphism inI I S is an isomorphism.Thus, we see that we obtain a natural isomorphism

inI = I S = Z (1)In the sequel, we shall identify inI with Z (1) via this isomorphism. Similarly, we have inI = Z (1).

In order to understand these various groups better, it is helpful to think in terms of alocal model, as follows: If e 1 is an integer, let R def = k[[x,y,t ]]/ (xy te ), A def = k[[t]] R.Let X def = Spec(R), T def = Spec(A). Endow T with the log structure dened by the divisort = 0. Let N node be the monoid given by taking the quotient of the free monoid on thesymbols log(x),log(y),log(t) by the relation log(x) + log(y) e log(t). Map N node Rby letting log(?) ?, for ? = x,y,t . Endow X with the log structure associated toN node R. Thus, we obtain a morphism of log schemes X log T log . Let us denoteby : Spec(k) T the special point of T . Moreover, there is a unique e such that thecompletion of X k at k is equal to X

log | . We shall call this e the order of the node .

Let I = V (y, t ) = Spec(k[[x]]); I = V (x, t ) = Spec(k[[y]]); X = X T . Let U = X I I . Let us denote by admX the quotient of the fundamental group of U which is tamely ramied over the divisors I and I . Note that admX is abelian (thus

eliminating the need to choose a base point). Indeed, this follows by noting that admX for e 1 injects into admX for e = 1; but when e = 1, I I is a divisor with normal

crossings, so admX = Z (1) Z (1). Let us denote by admT the tame fundamental groupof T . Thus, admT = Z (1). Note (for instance, by reducing to the case e = 1) thatthe decomposition groups of the divisors I and I are both equal to admX . Let us denoteby inI , inI admX the inertia groups of the divisors I and I . As above, it is easyto see that the natural map inI admX admT is an isomorphism. Thus, we haveisomorphisms inI = Z (1); inI = Z (1). Observe, moreover, (for instance, by reducing tothe case e = 1) that inI inI = {1}.

22


23/47

Let def = V (x,y,t ) X . Let us denote by admI the tame fundamental group of I .Thus, admI = Z (1). Similarly, we have admI = Z (1). Note, moreover, that we have anatural morphism admX / inI admI . Thus, we obtain a morphism

Z (1) = inI admX / inI admI = Z (1)This morphism corresponds to multiplication by some element Z . We claim that = e.Indeed, this follows from the fact that taking roots of the function x| I on I correspondsto taking roots of te = y 1 x over I (since y is invertible on I ). Thus, we seethat we have injections

Z (1) Z (1) = inI inI admX admI admI = Z (1) Z (1)Here the composite Z (1) Z (1) Z (1) Z (1) is given by multiplication by e.Now let us translate what we have learned locally back into information concerning ouroriginal X . First of all, let us observe that admX (respectively, inI ; inI ) corresponds to in (respectively, inI ; inI ). Thus, in particular, we obtain that inI inI = {1}. Next,let us observe that admI / inI may be identied with admI , i.e., the tame fundamentalgroup of I . Let us denote by in [I ] admI =

admI / inI the subgroup of elements that

x and act trivially on the residue eld of . That is to say, in [I ] is the inertia groupof in admI . In particular, we have a natural isomorphism

in [I ]= Z (1)Similarly, we have an isomorphism in [I ] = Z (1). Then in [I ] (respectively, in [I ])corresponds to admI (respectively, admI ).

Now let us observe that, by the theory developed thus far, all the subgroups and iso-morphisms of the above discussion may be recovered from admX S log , except (possibly)the injections

: Z (1) = in [I ]

admI ; : Z (1) =

in [I ]

admI

(or, equivalently, except (possibly) the isomorphisms Z (1) = in [I ], Z (1) = in [I ]).Thus, we have the following resultProposition 6.1: Suppose that X is stable, sturdy, and untangled, and that X (k)is a node. Then the order e of the node may be recovered entirely from admX S log , , and .

23


24/47

Next we would like to consider the issue of reconstructing the log structure of X at group-theoretically. Let S log be the log scheme whose underlying scheme is S = Spec(k),and whose log structure is that obtained by pulling back the log structure of X log via :S X . Thus, we have a natural structure morphism S log S log . Let M S (respectively,M ) be the monoid dening the log structure of S log (respectively, S log

). Thus, we have

k M S ; k M . Moreover, M S /k = N ; M /k = N node (where N node is the monoidintroduced above). The structure morphism S log S log denes a morphism M S M (which is the identity on k ), hence an inclusion N N node . Let us denote by N nodethe image of e N in N node . Let us denote by Le M S (respectively, L M ) theinverse image of e N (respectively, N node ) in M S (respectively, M ). Thus, we obtainan isomorphism of k -torsors

: Le L

We would like to reconstruct group-theoretically.

First, recall that we may naturally regard k as a quotient of Z (1). Now it followsfrom Kummer theory thatLemma 6.2 : There is a natural one-to-one correspondence between elements of L and morphisms : adm k whose restriction to Z (1) Z (1) = inI inI adm is the composite of the morphism (e, e) : Z (1) Z (1) Z (1) (i.e., multiplication by e on bothfactors) with the natural quotient Z (1) k .Note that here, the k -torsor structure on the set of such is given by observing that thedifference between two such is a morphism H om (S , k ), which may be identied withk by means of the Frobenius element = S .

Similarly, we have

Lemma 6.3 : There is a natural one-to-one correspondence between elements of Leand morphisms : S log k whose restriction to Z (1) = I S S log is the composite of the morphism e : Z (1) Z (1) (i.e., multiplication by e) with the natural quotient Z (1) k .Moreover, the correspondence induced by : Le = L between the of Lemma 6.2and the of Lemma 6.3 is the correspondence obtained by composing S log k with adm admX S log .

We thus conclude the following

Proposition 6.4: Suppose that X is stable, sturdy, and untangled, and that X (k)is a node. Then the morphism : Le = L may be recovered entirely from admX S log , , and .

24


25/47

Section 7: The Main Result over Finite Fields

We are now ready to put everything together and prove the main result over niteelds. The point is that the theory developed thus far in this paper will allow us to

reduce the Grothendieck conjecture for singular stable log-curves over nite elds to theGrothendieck conjecture for smooth, affine, hyperbolic curves over nite elds (which isalready proven in [Tama]).

Let S log S log be as in Denition 2.7. Let X log S log and ( X )log S log bestable log-curves of genus g (equipped with base-points x X (k) and x X (k)), suchthat neither X nor X is smooth over k. Let us assume that we are given a commutativediagram of continuous group homomorphisms:

admX admX

S logid S log

where the vertical morphisms are the natural ones, and the horizontal morphisms are iso-morphisms. The goal of this Section is to show that (under a certain technical assumptionon the RT-degree) arises (up to conjugation by an element of X log ) from a geometricS log -isomorphism of X log with ( X )log .

We begin by proving the result under the following simplifying assumption on X andX :

(*) X and X are sturdy and untangled, and their nodes are rationalover k.

By Proposition 4.1, induces a natural isomorphism between the sets of irreduciblecomponents of X and X . Let I X be an irreducible component. Then there is acorresponding irreducible component I X . Moreover, by Proposition 4.1, necessar-ily maps decomposition (respectively, inertia) subgroups of admX to similar subgroups of admX . Thus, we may choose admI admX such that maps admI onto admI admX .

Moreover, we also have

(inI ) =

inI . Thus, since

admI

def

= admI /

inI , we get a naturalisomorphism

I : admI

= admI

Now as we saw at the beginning of Section 6, the natural morphism inI I S is anisomorphism. Hence, the quotient of adm

I induced by admX S log is simply admI

S . Thus, we see that I ts into a diagram:

25


26/47

admI

I admI

S id S

Let I = Ker (admI S ). Since I is a smooth affine hyperbolic curve over k, we are

now in a position to apply the theory of [Tama]. The only two consequences of the theoryof [Tama] that we will use in this paper are the following:

(1) I

induces a commutative diagram

I I I

S I S

of morphisms of schemes. Here the vertical morphisms are the naturalones, and the horizontal morphisms are isomorphisms. The morphism I , however, need not be the identity.

(2) For each node of X lying on I , the morphisms : Z (1) admI (well-dened up to conjugation by an element of I ) of Proposition6.1 are taken to each other by , up to multiplication by some unit

I (

Z ) . Here, the automorphism induced by

I on the quotient

Z (1) k is equal to the automorphism of k induced by I .It follows, in particular, that I pZ Q (i.e., I is a rational number which is a powerof p). In fact, as we shall see below (Lemma 7.1), I is independent of I . Thus, we shallwrite

degRT ( ) def = I pZ Q

and we shall refer to this number as the RT-degree of on I (where RT stands for

ramication-theoretic, as opposed to another type of degree that will be discussed inSection 9).

Lemma 7.1 : The rational number I is independent of the irreducible component I .

Proof: Suppose that there is another irreducible component J of X that touches I . LetJ X be the corresponding component of X . Let us focus our attention at a node lying on I and J . By Corollary 5.3, there is a corresponding node X (k). Let e

26


27/47

(respectively, e ) be the order of the node (respectively, ). Now we have a commutativediagram:

Z (1)

Z (1) = in [I ] in [J ]

(e,e )

Z (1)

Z (1) = inJ inI

(1 ,1)

Z (1) = I S

I J id

Z (1) Z (1) = in [I ] in [J ] (e ,e )

Z (1) Z (1) = inJ inI (1 ,1) Z (1) = I S

where the vertical map in the middle is that induced by , and I and J are themorphisms obtained from the theory of [Tama] (cf., item (2) in the list given above). Itfollows immediately from the commutativity of this diagram that I = e e 1 = J . (Thus,we also obtain a new proof of the rationality of I .) This completes the proof.

Let us assume henceforth that

() degRT ( ) = 1

Thus, I is an S -isomorphism, and the are taken to each other precisely (not justup to some multiple) by . By the theory of [Tama], this means that I induces anS -isomorphism I = I that respects nodes. We thus obtain an S -isomorphism

X : X = X

Thus, it remains to consider log structures. Let X X be the normalization of X . Letus equip X with the log structure dened by the divisor consisting of the points of X thatmap to nodes of X . Thus, we obtain a log scheme X log whose log structure is genericallytrivial. Similarly, we have ( X )log . Note that X already induces an S -isomorphism X log : X log = (X )log

Now let us concentrate on a single node X (k) (which corresponds to X (k)). Itis not difficult to see that given X log , in order to recover the S log -log scheme X log in aneighborhood of , it suffices to know the isomorphism

: Le = L

But by Proposition 6.4, this morphism may be recovered from admX S log , plus item(2) of the above review of the theory of [Tama] (now that we know/are assuming that allthe I = 1). Thus, we see that X extends naturally to a morphism

X log : X log = (X )log

27


28/47

as desired.

The next step is to check that the morphism induced by logX between admX and

admX agrees (up to conjugation by an element of X log ) with the original . But thisfollows by using a similar argument to that of [Tama]: Namely, rst we observe that

it is clear from the construction of X log that if we start with an

that arises fromsome : X log = (X )log , then = X log . Then we note that for each orderly coveringY log X log , there is a corresponding orderly covering ( Y )log (X )log , together withan isomorphism admY = admY induced by . Moreover, this isomorphism gives rise toan isomorphism Y log that is compatible with X log . It thus follows formally from thegeneral theory of the algebraic fundamental group that the isomorphism of admX withadmX induced by X log differs from by conjugation by some element admX . Onthe other hand, since the automorphism of admX given by conjugation by must inducethe identity on S log , and S log clearly has trivial center, it thus follows that the imageof in S log is trivial. Thus, X log , as desired.

Now let us denote by Isom S log (X log

, (X )log

) the set of S log

-isomorphisms of logschemes between X log and ( X )log . Next, let us consider the set Isom S log (admX , admX )

of isomorphisms admX = admX that preserve and induce the identity on the quotient S log .Then given , Isom S log (

admX , admX ), we regard if and only if there exists

an X log such that

( 1) = ()

for all admX . The resulting set of equivalence classes will be denoted

Isom GO S log (admX ,

admX )

Here, GO stands for geometrically outer. Now observe that it is clear that innerautomorphisms induced by elements of X log do not affect the RT-degree. Thus, it makessense to consider

Isom GORT S log (admX ,

admX ) Isom

GO S log (

admX ,

admX )

that is, the classes of isomorphisms whose RT-degree is equal to 1.

Thus, in summary, we have proven (at least under the assumption (*)) the followingresult:

Theorem 7.2: Let S log S log be as in Denition 2.7. Let X log S log and (X )log S log be stable log-curves (equipped with base-points x X (k) and x X (k)), such thatat least one of X or X is not smooth over k. Then the natural map

Isom S log (X log , (X )log ) Isom GORT S log (admX ,

admX )

28


29/47

is bijective .

Proof: It remains to deal with the case where the simplifying assumption (*) is notsatised. But let us note that given an arbitrary X log as in the statement of the Theorem,

there is always a nite orderly covering Y log

X log

such that Y satises (*). Moreover,any admissible covering (with rational nodes) of Y will clearly still satisfy (*). Thus, it isclear that we may take Y log X log such that the corresponding (via : admX = admX )(Y )log (X )log is such that Y also satises (*). Then one concludes the Theorem bydescent.

Just as in [Tama], if X log is center-free, then one can rewrite Theorem 7.2 in termsof outer automorphisms. For this, we need the following

Lemma 7.3 : The group X log is center-free.

Proof: The proof is formally the same as that given in [Tama], 1, for the case where X is smooth over k. The only facts that one needs to check are:

(1) The pro- p-quotient pX log of X log is free.

(2) There exists an open subgroup H X log for which H p is nonabelian.

We begin by checking (1). First note that H 2et (X k , F p) = 0. Indeed, this follows immedi-ately from writing out the long exact sequence associated to

0 F p G a1 F G a 0

(where F is the Frobenius morphism). Thus, for all nite etale coverings Y X k ,we also have H 2et (Y, F p) = 0. It thus follows that H 2 (

pX log , F p) = H

2( pX , F p) =H 2et (X k , F p) = 0. (Here we use the elementary fact that the natural surjection

pX log

pX is an isomorphism.) On the other hand, it is a well-known fact from group-theory([Shatz], Chapter III, 3, Proposition 2.3) that this implies that pX log is free. This com-pletes the verication of (1). As for (2), since the smooth case is already discussed in[Tama], 1, we shall concentrate here on the case when X is singular. Then it sufficesto note the existence of an orderly covering of X log

kwhose dual graph has a nonabelian

fundamental group. But the existence of such a covering follows immediately from simplecombinatorial considerations.

If G1 and G2 are two topological groups, let us denote by Isom (G1 , G2) the set of continuous isomorphisms G1 = G2 . Let us denote by Out (G1 , G2) the set of equivalenceclasses of Isom (G1 , G2 ), where we consider two isomorphisms equivalent if they differ byan inner automorphism. If G1 = G2 , then we shall denote Out (G1 , G2 ) by Out (G1 ).

29


30/47

If Isom (G1 , G2), then induces an isomorphism Out () : Out (G1 ) = Out (G2 ).Moreover, Out () depends only the class [ ]Out (G1 , G2) dened by .

Suppose that X log and ( X )log are as in Theorem 7.2. Let

X : S log Out ( X log )

be the representation arising from the extension 1 X log admX S log 1. Notethat X is independent of the choice of base point x. Similarly, we have X . Let us denoteby

Out ( X log , (X ) log )

the set of [ ] Out ( X log , (X ) log ) such that Out () X = X . Now note that we have

a natural map

Isom GO S log (admX ,

admX ) Out ( X log , (X ) log )

Then it follows group-theoretically (cf. [Tama]) from the fact that X log has trivial centerthat this map is a bijection. Let us denote by

Out D ( X log , (X ) log )

the image under this bijection of

Isom GORT S log (admX ,

admX ) Isom

GO S log (

admX ,

admX )

(Here, the D stands for degree one.) Thus, one can rephrase Theorem 7.2 in thefollowing (seemingly weaker) form:

Theorem 7.4: Let S log S log be as in Denition 2.7. Let X log S log and (X )log S log be stable log-curves such that at least one of X or X is not smooth over k. Thenthe natural map

Isom S log (X log , (X )log ) Out D ( X log , (X ) log )

is bijective .

30


31/47

Section 8: Characterization of Admissible Coverings

So far, we have been working with stable curves over a nite eld. In this Sectionand the next, we shift gears and consider stable curves over local elds. The purpose of

this Section is to show that given such a curve, one can characterize the quotient of thefundamental group (of the generic curve) corresponding to admissible coverings in entirelygroup-theoretic terms. The technique of proof is similar to that of Proposition 3.1, wherewe characterized etale coverings among all admissible coverings.

Let K be a nite extension of Q p . Let K be an algebraic closure of K ; let K def =

Gal (K/K ). Let K unr K be the maximal unramied extension of K in K ; let unrK K be the corresponding closed subgroup. Let A K (respectively, Aunr K unr ) be thering of integers; k be the residue eld of A; S = Spec(A). Let us endow S with the logstructure dened by the closed point. Let X log S log be a stable log-curve of genus gwhich is generically smooth. Thus, X K Spec(K ) is a smooth curve of genus g. Choose

a base-point x X (K ). Let X K def = 1(X K , xK ) be the resulting fundamental group.Thus, we have an exact sequence

1 X K X K K 1

where X Kdef = 1(X K , xK ). Let X K

admX be the quotient (as in Denition 2.4) of

X K by the intersection H of all the co-admissible open subgroups H X K .

Let Y K X K be an abelian etale covering of degree p. Let us assume that Y K isgeometrically connected over K . Now let us consider the following condition on Y K X K :

(*) Over K unr , there is an innite abelian etale covering Z K unr X K unr = X K K K unr with Galois group Z p such that the interme-diate covering corresponding to Z p F p is Y K unr X K unr .

In the next few paragraphs, we would like to show that condition (*) is equivalent to thestatement that Y K X K extends to a nite abelian etale covering Y X . Indeed, thenecessity of condition (*) follows easily from well-known facts concerning the fundamentalgroup of X k = X A k. Thus, it remains to show that condition (*) is sufficient.

To prove the sufficiency of (*), we will need to review certain basic facts from [FC]

concerning the p-adic Tate module of a semi-abelian scheme over S . Let J K Spec(K )be the Jacobian of X K . We shall always regard J K as equipped with its usual principalpolarization. By [FC], Chapter I, Theorem 2.6 and Proposition 2.7, it follows that J K extends uniquely to a semi-abelian scheme J S over S . Let V J

def = Hom (Q p / Z p , J (K ))be the p-adic Tate module of J K . Thus, V J is a free Z p-module of rank 2 g, equippedwith a natural K -action. In Chapter III of [FC], one nds a theory of degenerations of semi-abelian varieties. According to this theory (more precisely: the equivalence M pol of Corollary 7.2 of [FC], Chapter III), there exists an abelian scheme G S , togetherwith a torus T S , and an extension

31


32/47

0 T J G 0over S , such that roughly speaking, J is obtained as a rigid analytic quotient of J by anetale sheaf P of free abelian groups of rank r

def

= dim(T/S ) on S . (Caution: Our choiceof notation differs somewhat from that of [FC].) Let V J def = Hom (Q p / Z p , J (K )) be the

p-adic Tate module of J K . Thus, if d = dim (G/S ), then V J is a free Z p-module of rankr + 2d. Note that the etale sheaf P may also be regarded as a free abelian group of rankr equipped with a K -action. Let P Z p

def = P Z Z p . Then, according to Corollary 7.3 of [FC], Chapter III, we have an exact sequence of K -modules:

0 V J V J P Z p 0

Let V Gdef = Hom (Q p / Z p , G(K )) (respectively, V T

def = Hom (Q p / Z p , T (K ))) be the p-adic

Tate module of GK (respectively, T K ). Then V J itself ts into another exact sequence of K -modules:

0 V T V J V G 0

Next let us consider the K -module V G . Let a be the p-rank of the abelian varietyGk

def = GA k over k. Then, as is well-known, there is a free Z p-module V G ord of rank aequipped with an unramied K -action such that we have exact sequences of K -modules:

0 V G V G V G ord 0

and

0 V G ord (1) V G V G ss 0

Here, the (1) is a Tate twist, and by unramied action we mean that unrK acts trivially.Thus, V G ss is the supersingular part of the representation V G . Let us denote by V J etthe quotient of V J by Ker (V

J V

Gord ) V

J V J . Thus, V J et is a free Z p-module of

rank r + a, equipped with a natural unramied K -action. Let V J mt V J V J be theinverse image of V G ord (1) (V G ) under the projection V J V G . Note that the submoduleV J mt V J is dual to the quotient V J V J et under the bilinear form on V J arising fromthe canonical polarization of J .

Now we would like to take a closer look at the K -module V G ss . First, recall that theK -module V G is crystalline . This fact is well-known from the general theory of Galoisrepresentations arising from p-adic etale cohomology groups of varieties (see, e.g., [FC],Chapter VI, 6, for a brief review of this theory, as well as a list of further references). Let

32


33/47

K 0 K be the maximal unramied extension of Q p in K . Then V G is (say, covariantly)associated to a ltered module with Frobenius action ( M, F (M ), M ), where M is aK 0 -vector space, M : M M is a semilinear automorphism, and F (M ) is a ltrationof M K 0 K . It then follows from the above exact sequences that V G ss is also crystalline,and is associated to a ltered module with Frobenius action ( M ss , F (M ss ), M ss ) whichis a subquotient of ( M, F (M ), M ). Moreover, since V G ss was constructed as the su-persingular part of V G , it follows that the Frobenius action M ss on M ss is topologically nilpotent . Now we have the following crucial

Lemma 8.1 : Any unrK -equivariant Z p-linear morphism : V J Z p (where Z p is equipped with the trivial unrK -action) factors through the quotient V J V G ord .

Proof: First, note that since T is a torus, and K unr contains only nitely many pth powerroots of unity, the restriction of to V T must be zero. Thus, factors through V G . Denote

the resulting morphism V G Z p by . By the same argument, the restriction of toV G ord (1) must be zero. Thus, we obtain a unrK -equivariant morphism G ss : V G ss Z p .

But since V G ss is crystalline , it follows from the basic theory of crystalline representationsthat G ss denes a Frobenius-equivariant, K unr -linear morphism M ss : M ss K 0 K unr K unr . (Here, K unr is equipped with the trivial Frobenius action.) On the other hand, itfollows from the topological nilpotence of M ss that M ss must be zero. Thus, G ss isalso zero. This completes the proof.

Now let us return to the abelian covering Y K X K of degree p discussed above.Suppose that this covering satises condition (*). Then there exists a covering Z K unr X

K unr as in (*). Moreover, since V

J = Hom (

X K, Z

p(1)), it follows that Z

K unr X

K unr

denes a Z p-linear morphism : Z p(1) V J which is unrK -equivariant. By taking thedual to (and using the fact that the polarization of J gives an isomorphism of V J withits Cartier dual), we obtain a unrK -equivariant morphism

: V J Z p . Thus, by Lemma8.1, it follows that factors through V J et . In particular, factors through V J mt . On theother hand, it is clear that Z p-coverings arising from morphisms Z p(1) V J mt extend toetale coverings of X A unr . Thus, in particular, it follows that Y K X K extends to a niteetale covering Y X . Thus, we have proven the following result:

Lemma 8.2 : Let Y K X K be an abelian etale covering of degree p. Let us assume that Y K is geometrically connected over K . Then Y K X K extends to a nite etale covering Y X if and only if Y K X K satises condition (*) above.

Next, we would like to consider more general coverings Y K X K . We begin with thefollowing

Lemma 8.3 : Let K : Y K X K be a nite etale covering such that Y K is geometrically connected over K . Suppose, moreover, that Y K arises from a stable curve Y S over S .Then K extends to a proper surjective morphism : Y X .

33


34/47

Proof: Note that if A K = K , then will automatically be proper and domi-nant, hence surjective. Thus, it suffices to show the existence of such a . Let us rstshow the existence of such a under the additional assumption that X k is sturdy . Nowit follows from the general theory of stable curves that there exists a semi-stable curveY S equipped with a birational (blow-up) morphism Y Y such that Y is regu-lar . Moreover, it follows from the general theory of elimination of indeterminacy forregular two-dimensional schemes ([Lipman],

the profinite grothendieck conjecture for closed hyperbolic curves over number fields

Documents