The Profinite Grothendieck Conjecture for Closed Hyperbolic Curves over Number Fields

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1 J. Math. Sci. Univ. Toyo 3 (1996), The Profinite Grothendiec Conjecture for Closed Hyperbolic Curves over Number Fields By Shinichi Mochizui Abstract. n this paper, we extend the results of [Tama] on the Grothendiec Conjecture for affine hyperbolic curves over finite fields to obtain a Grothendiec Conjecture-type result for singular, proper, stable log-curves over finite fields. Using this result, we derive a strong Grothendiec Conjecture-type result for smooth, proper hyperbolic curves over number fields, and a wea Grothendiec Conjecture-type result for smooth, proper, hyperbolic curves over local fields with ordinary reduction. 0. ntroduction n [Tama], a proof of the Grothendiec Conjecture (reviewed below) was given for smooth affine hyperbolic curves over finite fields (and over number fields). The purpose of this paper is to show how one can derive the Grothendiec Conjecture for arbitrary (i.e., not necessarily affine) smooth hyperbolic curves over number fields from the results of [Tama] for affine hyperbolic curves over finite fields. We obtain three types of results: one over number fields, one over finite fields, and one over local fields. We remar here that when this paper was first written (October 1995), Theorems A and C below were the strongest nown results of their respective inds. Since then, the author wrote [Mz2] (November 1995), which gives rise to much stronger results than Theorems A or C of the present paper. Moreover, the proofs of [Mz2] are completely different from (and, in particular, do not rely on) the proofs of the present 1991 Mathematics Subject Classification. Primary 11G30; Secondary 14H25, 14H30, 11G20, 11G

2 572 Shinichi Mochizui paper. Nevertheless, it seems to the author that the present paper still has some marginal interest, partly because most of the present paper is devoted to the proof of Theorem B below (which is not implied by any result of [Mz2]), and partly because it is in some sense of interest to see how Theorems A or C can be derived within the context of the theory of [Tama]. Our main result over number fields (Theorem 10.2 in the text) is as follows: Theorem A. Let be a finite extension of Q; let be an algebraic closure of. Let Spec() and Spec() be smooth, geometrically connected, proper curves over, of genus 2. Let (respectively, ) be the geometric fundamental group of (respectively, ). Then the natural map som (, ) Out ρ (, ) is bijective. Here, Out ρ refers to outer isomorphisms that respect the natural outer representations of Gal(/) on and. The statement of this Theorem is commonly referred to as the Grothendiec Conjecture. n [Tama], a theorem similar to Theorem A, except that is replaced by a hyperbolic affine curve, is proven. t is a simple exercise to derive the affine case from the proper case. On the other hand, to derive the proper case from the affine case is by no means straightforward; in this paper, we derive the proper case over number fields from the affine case over finite fields. n fact, to be more precise, we shall derive the proper case over number fields from a certain logarithmic Grothendiec Conjecture for singular (proper) stable curves over finite fields. This logarithmic Grothendiec conjecture, which is our main result over finite fields (Theorem 7.4), is as follows: Theorem B. Let S log be a log scheme such that S is the spectrum of a finite field, and the log structure is isomorphic to the one associated to the chart N given by the zero map. Let log S log and ( ) log S log be stable log-curves such that at least one of or is not smooth over

3 Profinite Grothendiec Conjecture 573. Let log (respectively, ( ) log) be the geometric fundamental group of log (respectively, ( ) log ) obtained by considering log admissible coverings of log (respectively, ( ) log ) (as in [Mz], 3). Then the natural map som S log( log, ( ) log ) Out D ρ ( log, ( ) log) is bijective. Here, the D stands for degree one (outer isomorphisms). This Theorem is derived directly from Tamagawa s results on affine hyperbolic curves over finite fields. ts proof occupies the bul of the present paper. Finally, by supplementing Theorem B with various arguments concerning the fundamental groups of curves over local fields, we obtain the following local result (Theorem 9.8): Theorem C. Let be a finite extension of Q p ; let A be its ring of integers; and let be its residue field. Let be a smooth, geometrically connected, proper curve of genus g 2 over. Assume that admits a stable extension Spec(A) such that the abelian variety portion of Pic 0 ( ) (where = A ) is ordinary. Then the natural outer representation ρ :Γ Out( ) of the absolute Galois group of on the geometric fundamental group of completely determines the isomorphism class of. Even though this is a rather wea version of the Grothendiec Conjecture (compared to the results we obtain over finite and global fields), this sort of result is interesting in the sense that it shows that curves behave somewhat differently from abelian varieties (cf. the Remar following Theorem 9.8). Now we discuss the contents of the paper in more detail. Sections 1 through 7 are devoted to deriving Theorem B from the results of [Tama]. n Section 1, we show how to recover the set of irreducible components of a stable curve from its fundamental group. n Section 2, we review various facts from [Mz] concerning log admissible coverings, and show how one can define an admissible fundamental group of a stable log-curve. n Section 3, we show how one can group-theoretically characterize the quotient of the admissible fundamental group corresponding to étale coverings. n Section

4 574 Shinichi Mochizui 4, we show that the tame fundamental group of each connected component of the smooth locus of a stable log-curve is contained inside the admissible fundamental group of the stable log-curve. n Section 5, we show how to recover the set of nodes (including the information of which irreducible components each node sits on) of a stable log-curve from its admissible fundamental group. n Section 6, we show how the log structure at a node of a stable log-curve can be recovered from the admissible fundamental group of the stable log-curve. n Section 7, we put all of this information together and show how one can derive Theorem B from the results of [Tama]. n Sections 8 and 9, we shift from studying curves over finite fields to studying curves over local fields. n order to do this, it is necessary first to characterize (group-theoretically) the quotient of the (characteristic zero) geometric fundamental group of a curve over a local field which corresponds to admissible coverings. This is done in Section 8. n Section 9, we first show (Lemma 9.1) that the degree of an isomorphism between the arithmetic fundamental groups of two curves over a local field is necessarily one. This is important because one cannot apply Theorem 7.4 to an arbitrary isomorphism of fundamental groups: one needs to now first that the degree is equal to one. Then, by means of a certain tric which allows one to reduce the study of curves over local fields with smooth reduction to the study of curves over local fields with singular reduction, we show (Theorem 9.2) that one can recover the reduction (over the residue field) of a given smooth, proper, hyperbolic curve over a local field group-theoretically. The rest of Section 9 is devoted to curves with ordinary reduction, culminating in the proof of Theorem C. Finally, in Section 10, we observe that Theorem A follows formally from Theorem 9.2. The author would lie to than A. Tamagawa for numerous fruitful discussions concerning the contents of [Tama], as well as the present paper. n some sense, the present paper is something of a long appendix to [Tama]: That is to say, several months after the author learned of the results of [Tama], it dawned upon the author that by using admissible coverings, 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 log structures, it seemed to the author that it might be useful to write out a detailed version of this trivial argument. The result is the present paper. Finally, the author would also lie to than Prof. Y. hara for his en-

5 Profinite Grothendiec Conjecture 575 couragement and advice during the preparation of this paper. 1. The Set of rreducible Components Let be the finite field of q = p f. Fix an algebraic closure of. Let Γ be the absolute Galois group of. Let Spec() be a morphism of schemes. Definition 1.1. We shall call a multi-stable curve of genus g if dim (H 1 def (, O )) = g, and = is a finite disjoint union of stable curves over of genus 2. f is multi-stable, then we shall call sturdy if every irreducible component of the normalization of has genus 2. Thus, in particular, a curve is stable if and only if it is geometrically connected and multi-stable. Moreover, a finite étale covering of a multi-stable (respectively, sturdy) curve is multi-stable (respectively, sturdy). Suppose that is stable of genus g 2. Fix a base-point x (). Then we may form the (algebraic) fundamental group Π def = π 1 (, x )of def. Let = ; def = π 1 (,x ). Then we have a natural exact sequence of groups 1 Π Γ 1 induced by the structure morphism Spec(). The purpose of this Section is to show how the set of irreducible components of can be canonically recovered from the morphism Π Γ. Fix a prime l different from p. Let us consider the étale cohomology group H e def = Hét 1 (, Z l). Let ψ : be the normalization of. Then we can also consider H n def = Hét 1 (, Z l ). By considering the long exact cohomology sequence in étale cohomology associated to 0 Z l ψ ψ Z l (ψ ψ Z l )/Z l 0 we obtain a surjection H e H n. Let us write H c for the ernel of this surjection. (Here, e (respectively, n ; c ) stands for étale (respectively, normalization; combinatorial).) Note that H c is a free Z l -module of ran N +1, where N (respectively, ) is the number nodes (respectively, irreducible components) of. Moreover, H n is a free Z l -module of ran equal to twice the sum of the genera of the connected components of.

6 576 Shinichi Mochizui Thus, we obtain a natural exact sequence of Γ-modules 0 H c H e H n 0 Let φ Γ be the automorphism of given by raising to the q th power. Then one sees easily that some finite power of φ acts trivially on H c. On the other hand, by the Weil conjectures (applied to the various geometric connected components of ), no power of φ acts with eigenvalue 1 on H n. We thus obtain the following Proposition 1.2. The natural exact sequence 0 H c H e H n 0 can be recovered entirely from Π Γ. Proof. ndeed, H e = Hom(, Z l ), while H c can be recovered by looing at the maximal Z l -submodule of H e on which some power of φ Γ acts trivially. Let L e = H e F l ; L c = H c F l ; L n = H n F l. Thus, L e = H 1 ét (, F l), and we have an exact sequence of Γ-modules 0 L c L e L n 0 Moreover, elements of L e correspond to étale, abelian coverings of of degree l. Let L L e be the subset of elements whose image in L n is nonzero. Suppose that α L. Let Y α be the corresponding covering. Then N Yα = l N. Thus, we obtain a morphism ɛ : L Z that maps α Yα. Since L is a finite set, the image of ɛ is finite. Let M L be the subset of elements α on which ɛ attains its maximum. Let us define a pre-equivalence relation onm as follows: f α, β M, then we write α β if, for every λ, µ F l λ α + µ β L, we have λ α + µ β M. for which Now we have the following result: Proposition 1.3. Suppose that is stable and sturdy. Then is, def in fact, an equivalence relation, and moreover, C = M/ is naturally isomorphic to the set of irreducible components of.

7 Profinite Grothendiec Conjecture 577 Proof. First, let us observe, that Yα is maximal (equal to l( 1) + 1) if and only if there exists a unique irreducible component Z α of over which the covering Y α is nontrivial. Now, if Z is a connected component of, let L Z def = H 1 ét (Z, F l). Thus, L n = Z L Z where the direct sum is over the connected components of. Then it follows immediately from the definitions that M consists precisely of those elements α L whose image in L n has (relative to the above direct sum decomposition) exactly one nonzero component (namely, in L Zα ). Moreover, α β is equivalent to Z α = Z β. Finally, that every Z appears as a Z α follows from the sturdiness assumption. This completes the proof. Remar. Note that although at first glance the set C = M/ appears to depend on the choice of prime l, it is not difficult to see that in fact, if one chooses another prime l, and hence obtains a resulting C = M /, one obtains a natural isomorphism C = C (compatible with the isomorphisms just obtained of C and C to the set of irreducible components of ) as follows: f α M and α M, let us consider the product Y αα = Y α Y α. Thus, we have a cyclic étale covering Y αα of degree l l. Then one checs easily that α and α correspond to the same irreducible component if and only if (Y αα ) has precisely l l ( 1) + 1 irreducible components. Proposition 1.4. Suppose that is stable and sturdy. Then the set of irreducible components of (together with its natural Γ-action) can be recovered entirely from Π Γ. Proof. ndeed, it follows from Proposition 1.2 that L can be recovered from Π Γ. Moreover, we claim that M can be recovered, as well. ndeed, the maximality of Yα is equivalent to the minimality of N Yα Yα +1 = l N Yα + 1, which is equal to the dimension over F l of the L c ofy. 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. Finally, by the above Remar, the set that one recovers is independent of the choice of l.

8 578 Shinichi Mochizui Corollary 1.5. Suppose that is stable and sturdy. Let H Π be an open subgroup. Let Y H be the corresponding étale covering. Then the set of irreducible components of Y H can be recovered from Π Γ and H. Proof. Let be the (finite) extension of which is the subfield of stabilized by the image of H in Γ. Then Y H is geometrically connected, hence stable and sturdy over. Thus, we reduce to the case H = Π, Y H =. But then the set of irreducible components of is the set of Γ-orbits of the set of irreducible components of. Thus, the Corollary follows from Proposition 1.4. Looing bac over what we have done, one sees that in fact, we have proven a stronger result that what is stated in Corollary 1.5. ndeed, fix an irreducible component. Then let J l () be the set of -valued l-torsion points of the Jacobian of the normalization of. Then not only have we recovered set of all irreducible components, we have also recovered, for each, the set J l () (with its natural Frobenius action). We state this as a Corollary: Corollary 1.6. Suppose that is stable and sturdy, and l is a prime number different from p. Then for each irreducible component of, the set J l () (with its natural Frobenius action) can be recovered naturally from Π Γ. 2. The Admissible Fundamental Group Let r and g be nonnegative integers such that 2g 2+r 1. f (C M g,r ; σ 1,...,σ r : M g,r C) is the universal r-pointed stable curve of genus g over the moduli stac, then C and M g,r have natural log structures defined by the respective divisors at infinity and the images of the σ i. Denote the resulting log morphism by C log M log g,r. Let S be the underlying curve associated to an r-pointed stable curve of genus g over a scheme S (where S is the underlying scheme of some log scheme S log ). Suppose that is equipped with the log structure (call the resulting log scheme log ) obtained by pulling bac C log M log g,r via some log morphism S log M log g,r whose underlying non-log morphism S M g,r is the classifying morphism

9 Profinite Grothendiec Conjecture 579 of (equipped with its mared points). n this case, we shall call 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, log S log such that over some finite étale covering S S, log S S becomes a finite union of stable pointed log-curves. Let be as in the preceding section. Let S log be a log scheme whose underlying scheme is Spec() and whose log structure is (noncanonically!) isomorphic to the log structure associated to the morphism N, where 1 N 0. Let log S log be a stable log-curve of genus g 2. Next, we would lie to consider liftings of log S log. Let A be a complete discrete valuation ring which is finite over Z p and has residue field equal to. Let T log be a log scheme whose underlying scheme is Spec(A) and whose log structure is that defined by the special point S = Spec() T. Also, let us assume that S log is equal to the restriction of the log structure of T log to S = Spec() T. Let Y log T log be a stable log-curve of genus g whose restriction to S log is log S log. n this case, we shall say that Y log T log lifts log S log. t is well-nown (from the log-smoothness of the moduli stac of stable curves equipped with its natural log structure) that such log-curves Y log T log always exist. Next, we would lie to consider log admissible coverings Z log Y log of Y log. We refer to [Mz], 3.5, for the rather lengthy and technical definition and first properties of such coverings. t follows in particular from the definition that Z is a stable curve over T. n fact, (as is shown in [Mz], Proposition 3.11), one can define such coverings without referring to log structures. That is, there is a notion of an admissible covering ([Mz]. 3.9) Z Y (which can be defined 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 admissible. n [Mz], we dealt strictly with the case where Z is geometrically connected over T. Here, we shall call Z Y multi-admissible if Z is a disjoint union of connected components Z i such that each Z i Y is admissible. Let η be the generic point of T. Let Y η = Y T η. f Z Y is multiadmissible, then it will always be the case that the restriction Z η Y η of this covering to the generic fiber is finite étale. Now suppose that ψ η : Z η

10 580 Shinichi Mochizui Y η is a finite étale covering. f ψ η extends to an multi-admissible covering Z Y, then this extension is unique ([Mz], 3.13). Definition 2.1. We shall call ψ η pre-admissible if it extends to an multi-admissible covering ψ : Z Y. We shall call ψ η potentially preadmissible if it becomes pre-admissible after a tamely ramified base-change (i.e., replacing A by a tamely ramified extension of A). Thus, in particular, if A is a tamely ramified extension of A, then Y η A A Y η is potentially pre-admissible. f ψ η is potentially pre-admissible and Z η is geometrically connected over η, then it is pre-admissible if and only if Z η has stable reduction over A. Lemma 2.2. Let Z η Suppose that Z η Y η and Z η Y η are pre-admissible. def = Z η Yη Z η. Then Z η Y η 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 η. Thus, we have a natural morphism Z Z Y Z which is an isomorphism at height one primes. n particular, Z is étale over Y at all height one primes. t thus follows from Lemma 3.12 of [Mz] that Z Y is multi-admissible. Lemma 2.3. Suppose that Z η Y η is pre-admissible, and that Z η Y η factors through finite étale surjections Z η Z η and Z η Y η. Then Z η Y η is pre-admissible. Proof. Similar to that of Lemma 2.2. Let be the quotient field of A. Fix an algebraic closure of. 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 fiber of Y. Write Π Y for π 1 (Y,y ). Thus, we have a natural surjection Π Y Gal(/), whose ernel is a group Y Π Y. Definition 2.4. We shall call an open subgroup H Π Y co-admissible if the corresponding finite étale covering Z η Y η is potentially preadmissible. Let Π adm Y be the quotient of Π Y by the intersection H of all co-admissible H Π Y.

11 Profinite Grothendiec Conjecture 581 Remar. The admissible fundamental group Π adm Y has already been defined and studied by. Fujiwara ([Fuji]). Moreover, the author learned much about Π adm Y (as well as about the theory of log structures in general) by means of oral communication with. Fujiwara. t is easy to see that the intersection H of Definition 2.4 is a normal subgroup of Π Y. Thus, Π adm Y is a group. Moreover, by Lemmas 2.2 and 2.3, it follows that an open subgroup H Π Y is co-admissible if and only if er(π Y Π adm Y ) H. Finally, it is immediate from the definitions that the subfield of stabilized by the image of H in Gal(/)isthe maximal tamely ramified extension of. Thus, we have a surjection whose ernel adm Y Π adm Y Gal( /) Π adm Y is a quotient of Y. Definition 2.5. We shall refer to as orderly coverings of Y η those coverings Z η Y η which are Galois and factor as Z η Y η T U Y η, where the first morphism is pre-admissible; the second morphism is the natural projection; U = Spec(B); and B is a tamely ramified finite extension of A. We shall refer to as orderly quotients of Π adm Y those quotients of Π adm Y that give rise to orderly coverings of Y η. t is easy to see that orderly quotients of Π adm Y are cofinal among all quotients of Π adm Y. Let A be the normalization of A in. Let (respectively, m ) be the residue field (respectively, maximal ideal) of A. Let T = Spec(A ), and let us endow T with the log structure given by the multiplicative monoid O T {0} (equipped with the natural morphism into O T ). We call the resulting log scheme T log. Let S log be the log scheme whose underlying scheme is Spec( ) and whose log structure is pulled bac from T log. Thus, the log structure on S log is (noncanonically!) isomorphic to the log structure defined by the zero morphism (Z (p) ) 0. (Here (Z (p) ) 0 denotes the set of nonnegative rational numbers whose denominators are prime to p.) Note that Gal( /) induces S log -automorphisms of S log. n fact, it is easy to see that this correspondence defines a natural isomorphism Gal( /) = Aut S log(s log )

12 582 Shinichi Mochizui 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 log S log (2) an algebraic closure of = Q(A ) (hence a resulting ; (S ) log ); (3) an S log -isomorphism γ : S log = (S ) log ; (4) a base-point y Y (A ) such that y S = y S in (). Then there is a natural isomorphism between the surjections Π adm Y Gal( /) and Π adm Y Gal( / ). Proof. Let us first consider coverings of Y η obtained by pulling bac tamely ramified Galois coverings of. Thus, if U = Spec(B) T is finite, Galois, and tamely ramified (obtained from some field extension L), let U log be the log scheme obtained by equipping U with the log structure defined by the special point u of U. Thus, we obtain a finite, log étale morphism U log T log. By base-changing to S log, we then obtain a finite, log étale morphism V log S log. On the other hand, by the definition of log étaleness, this morphism then necessarily lifts to a finite, log étale morphism (U ) log (T ) log, whose underlying morphism U T is a Galois, tamely ramified finite extension. Thus, if we pass to the limit, and apply this construction to the extension L = of, we end up with some maximal tamely ramified extension L of. By the functoriality of this construction, we have a natural isomorphism Gal(L/) = Gal(L / ). Now observe that there is a unique - isomorphism L = which (relative to this construction) is compatible with γ. Thus, we get an isomorphism Gal(L / )=Gal( / ), hence an isomorphism Gal( /) = Gal( / ), as desired. Now let us consider orderly coverings Z η Y η. Thus, we have a factorization Z η Y η T U Y η. Let Z be the normalization of Y η in Z. Then Z Y T U is multi-admissible. Thus, Z admits a log structure such that we have a log multi-admissible covering Z log Y log T log U log. Basechanging, we obtain a log multi-admissible covering Z log S log log S log V log. But since log multi-admissible coverings are log étale, it thus follows that this covering lifts uniquely to a log multi-admissible covering

13 Profinite Grothendiec Conjecture 583 (Z ) log (Y ) log (T ) log (U ) log. Similarly, we obtain a bijective correspondence between Y η -automorphisms of Z η and Y η -automorphisms of Z η.now observe further that -valued points of Z η over y define A -valued points of Z over y (since Z is proper over A). Moreover, these points define T log -valued points of Z log over y T log, hence (by reducing modulo m ) S log - valued points of Z log S log over y S log, hence (using γ) (S ) log -valued points of Z log S log over y S log = y, hence (by log étaleness) (T ) log S log -valued points of (Z ) log over y, which, finally, give rise to T -valued points of Z log η over y. Thus, in summary, we have defined a natural equivalence of categories between orderly coverings of Y η and orderly coverings of Y η. Moreover, this equivalence is compatible with the fiber functors defined by the base-points y and y. Thus, we obtain an isomorphism between the surjections Π adm Y Gal( /) and Π adm Y Gal( / ), as desired. Now let us interpret Lemma 2.6. n summary, what Lemma 2.6 says is the following: Suppose we start with the following data: (1) a log scheme S log, where S = Spec(), and the log structure is (noncanonically!) isomorphic to the log structure associated to the zero morphism N ; (2) a log scheme S log over S log which is (noncanonically!) -isomorphic to Spec() equipped with the log structure associated to the zero morphism (Z (p) ) 0 (where the N (Z (p) ) 0 is pulled bac from a chart for S log as in (1)); (3) a stable log-curve log S log of genus g; (4) a base-point x () which is not a node. Then, to this data, we can naturally associate an admissible fundamental group Π adm with augmentation Πadm Π def S log = Aut S log(s log ). That is to say, by choosing a lifting of the above data, we may tae Π adm =Πadm Y, and the augmentation to be Π adm Y Gal( /). Then Lemma 2.6 says that, up to canonical isomorphism, Π adm and its augmentation do not depend on the choice of lifting. Definition 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 π 1 ( log,x S log) for Πadm and π 1(S log,s log ) for Π S log. We shall refer to Π adm

14 584 Shinichi Mochizui as the admissible fundamental group of. Write log Π adm for the ernel of the augmentation. We shall refer to log as the geometric admissible fundamental group of. def Next, let us observe that Π S log admits a natural surjection onto Π S = Gal(/). We shall denote the ernel of this surjection by S Π S log, and refer to S as the inertia subgroup of Π S log. Note that S is isomorphic to the inverse limit of the various ( ) (for finite extensions of ), where the transition morphisms in the inverse limit are given by taing the norm. Or, in other words, S = Ẑ (1), where Ẑ is the inverse limit of the quotients of Z of order prime to p, and the (1) is a Tate twist. Thus, we have a natural exact sequence 1 S = Ẑ (1) Π S log Π S = Gal(/) 1 Suppose that we are given a continuous action of Π S log on a finite set Σ. Then we can associate a geometric object to Σ as follows. Without loss of generality, we can assume that the action on Σ is transitive. f we choose a lifting T log of S log (where T = Spec(A)), then Σ corresponds to some finite, tamely ramified extension L of. Let B be the normalization of A in L. Equip U def = Spec(B) with the log structure defined by the special point u of U. Thus, we obtain U log. Equip Spec((u)) with the log structure induced by that of U log. Then the geometric object associated to Σ is the finite, log étale morphism Spec((u)) log S log. We shall call such morphisms finite, tamely ramified coverings of S log. Now suppose that we have an open subgroup H Π adm that surjects def onto Π S log. Let H = H log. n terms of liftings, H corresponds to a finite étale covering Z η Y η, where Z η is geometrically connected over η. Note that by a well-nown criterion ([SGA7]), Z η has stable reduction over A if and only if S acts unipotently on Hom( H, Z l ) (for some prime l distinct from p). But, as noted above, in this situation, Z η has stable reduction if and only if Z η Y η is pre-admissible. f Z η Y η is pre-admissible, it extends to some Z log Y log, which we can base-change via S log T log to obtain a log admissible covering Z log S log log. Conversely, every log admissible covering of log can be obtained in this manner. Thus, in summary, we have the following result:

15 Profinite Grothendiec Conjecture 585 Corollary 2.8. The open subgroups of Π adm that correspond to orderly coverings can be characterized entirely group-theoretically by means of Π adm Π S log (and Π S log Π S ). Moreover, these subgroups can be interpreted in terms of geometric coverings of log (namely, base-change via a finite, tamely ramified covering of S log, followed by a log admissible covering). Remar. Note that this Corollary thus allows us to spea of orderly coverings of log, i.e., coverings that arise from orderly quotients (Definition 2.5) of Π adm =Πadm Y. Before continuing, let us mae the following useful technical observation: Lemma 2.9. Given any stable over, there always exists an admissible covering Z such that Z is (multi-stable and) sturdy. ndeed, this follows from the definition of an admissible covering, plus elementary combinatorial considerations. Moreover, an admissible covering of a sturdy curve is always sturdy. Thus, if one wishes to wor only with sturdy curves, one can always pass to such a situation by replacing our original by some suitable admissible covering of. 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. n fact, it will suffice to consider pointed smooth curves. Thus, let g and r be nonnegative integers such that 2g 2+r 1. Let S log S log be as above, and let log S log be an r-pointed stable log-curve of genus g such that is -smooth. Also, let x () bea nonmared point. Then it is easy to see that, just as above, we can define (by considering various liftings to some A, then showing that what we have done does not depend on the lifting) an admissible fundamental group Π adm ), together with a natural surjection (with base-point x S log Π adm Π S log Moreover, the ernel log Π adm of this surjection is naturally isomorphic to the tame fundamental group of (with base-point x ). Unlie the singular case, we don t particular gain anything new by doing this, but what

16 586 Shinichi Mochizui will be important is that we still nonetheless obtain a natural surjection Π adm Π S log which arises functorially from the same framewor as the nontrivial Π adm Π Slog that appears in the case of singular curves. 3. Characterization of the Étale Fundamental Group We maintain the notation of the preceding Section. Thus, in particular, we have a stable log-curve log S log, together with a choice of S log, and a base-point x () (which is not a node). Then note that we have a natural morphism of exact sequences: 1 log Π adm Π S log 1 1 Π Π S 1 Here the vertical arrows are all surjections. The goal of this Section is to show how one can recover the quotient Π adm Π group-theoretically from Π adm Π S log. Let Y log log be a log admissible covering which is abelian, with Galois group equal to F l, where l is a prime number (which is not necessarily distinct from p). Let us consider the following condition on this covering: (*) Over, there is an infinite log admissible covering Z log log which is abelian with Galois group Z l such that the intermediate covering corresponding to Z l F l is Y log log. Here, by infinite log admissible covering, we mean an inverse limit of log admissible coverings in the usual finite sense. Suppose that Y log log satisfies (*). Then we claim that Y is, in fact, étale. ndeed, if p = l, then every abelian log admissible covering of degree l is automatically étale, so there is nothing to prove. f p l, then we can fix a node ν, and consider the ramification over the two branches of at ν. Considering this ramification gives rise to an inertia subgroup H Z l. f Y log log is ramified at ν, then H surjects onto F l,soh = Z l. On the other hand, by the definition of a log admissible covering, in order to have infinite ramification occuring over the geometric branches of at ν, we must also

17 Profinite Grothendiec Conjecture 587 have infinite ramification over the base S log. But, by (*), Z log log is already log admissible over S log (which is, of course, étale over S log ). This contradiction shows that Z log, and hence Y log, are unramified over log at ν. Thus, we see that (*) implies that Y is étale, as claimed. Note that conversely, if we now that Y is étale to begin with, then it is easy to see that (*) is satisfied. Thus, for an abelian log admissible covering Y log log of prime degree l, (*) is equivalent to the étaleness of Y. Now observe that the ernel of log is normal not just in log, but also in Π adm. Let Π =Πadm /er( log ). Let be the ernel of Π Π S. Thus, Π. Then we have the following result: Proposition 3.1. The quotient Π adm group theoretically from Π adm Π S log. Π can be recovered entirely Proof. t suffices to characterize subgroups H Π adm of finite index that contain er( log ). Without loss of generality, we may assume that H is normal in Π adm and (by Corollary 2.8) corresponds to an orderly covering. Since an orderly covering may be factored as a composite of a log multi-admissible covering followed by a tamely ramified covering of S log, one sees immediately that we may reduce to the case where H corresponds to a log admissible covering. Let G =Π adm /H. Let Y log log be the corresponding covering. For every subgroup N G, denote by Y log Y log N the corresponding intermediate covering. By considering ramification at the nodes, one sees immediately that Y is étale if and only for every cyclic N G of prime order, Y Y N is étale. But for such N, the étaleness of Y Y N is equivalent to the condition (*) discussed above. Moreover, it is clear that (*) can be phrased in entirely group theoretic terms, using only Π adm Π S log (and Π S log Π S). This completes the proof. Now suppose that Y log log is an abelian orderly covering of prime order l obtained from a quotient of Π such that Y is geometrically connected. Assume l p. Consider the following condition on Y log log :

18 588 Shinichi Mochizui ( ) There do not exist any infinite abelian orderly coverings Z log log 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 log log ; (ii) some finite power φm of the Frobenius morphism φ Γ=Π S stabilizes Z log log and acts on the Galois group Z l with eigenvalue q M. Because Y log log is abelian of prime order, it follows that one of the following holds: (1) Y log log is obtained from an étale covering Y (where Y is geometrically connected) base-changed by S log S. (2) Y log log is obtained by base-change via log S log from some totally (tamely) ramified covering of S log. Suppose that ( ) is satisfied. Then we claim that (1) holds. ndeed, if this were false, then (2) would hold, but it is clear that if (2) holds, then one can easily construct Z log log that contradict ( ) (by pulling bac via log S log an infinite ramified covering of S log ). This proves the claim. Now suppose that (1) holds. Then we claim that ( ) is satisfied. To prove this, suppose that there exists an offending Z log log. This offending covering defines an injection Z l Hom(, Z l) Hom(, Z l )=Hét 1 (, Z l). On the other hand, as we saw in Section 1, by the Weil conjectures, no power φ M of φ acts with eigenvalue q M on Hét 1 (, Z l). This contradiction completes the proof of the claim. Thus, in summary, (1) is equivalent to ( ). n other words, we have essentially proven the following result: Proposition 3.2. The quotient Π adm group theoretically from Π adm Π S log. Π can be recovered entirely Proof. t suffices to characterize finite index subgroups H Π that contain er(π Π ). Without loss of generality, we may assume that H is normal in Π and (by Corollary 2.8) corresponds to an orderly covering. Let G =Π /H. Let Y log log be the corresponding covering. Again, without loss of generality, we may assume that Y is geometrically connected over. For every normal subgroup N G, denote by Y log N log the corresponding intermediate covering. Next, we observe the following: Y log log arises from an étale covering of if and only if, for every normal subgroup N G such that Y log N log is orderly and G/N is cyclic

19 Profinite Grothendiec Conjecture 589 of prime order, Y N arises from an étale covering of. (This equivalence follows immediately from the definitions and the fact that / is abelian.) But for such Y log N log, we can apply the criterion ( ) discussed above. Moreover, it is clear that ( ) can be phrased in entirely group theoretic terms, using only Π Π S log (and Π S log Π S). This completes the proof. 4. The Decomposition Group of an rreducible Component We maintain the notation of the preceding Section. Fix an irreducible component. Then, corresponding to, there is a unique (up to conjugacy) decomposition subgroup adm Π adm which may be defined as follows. Let Z log log be the log scheme obtained by taing the inverse limit of the various Y log H log corresponding to open orderly subgroups H Π adm. Choose an irreducible component J Z that maps down to. Here, by irreducible component of Z, we mean a compatible system of irreducible components H Y H. Then adm Π adm is the subgroup of elements that tae the irreducible component J to itself. We can also define an inertia subgroup in adm as follows: Namely, we let in be the subgroup of elements of adm that act trivially on J. (That is to say, elements of in will, in general, act nontrivially on the log structure of J, but trivially on the underlying scheme J.) By Proposition 3.2 and Corollaries 1.5 and 1.6, we thus obtain the following: Proposition 4.1. Suppose that is stable and sturdy. Then one can recover the set of irreducible components of from Π adm Π Slog. Moreover, for each irreducible component of, one can recover the corresponding inertia and decomposition subgroups in adm Π adm entirely from Π adm Π S log. Proof. That one can recover adm follows formally from Proposition 3.2 and Corollary 1.5. Now observe that (as is well nown see, e.g.,

20 590 Shinichi Mochizui [DM]) any automorphism of that acts trivially on J l () (where l 5) is the identity. This observation, coupled with Corollary 1.6, allows one to recover in. Now let us suppose that the base-point x () is contained in. Let Ĩ be the normalization of. Then one can also define adm as follows. Let Ĭ be the open subset which is the complement of the nodes. We give Ĭ a log structure by restricting to Ĭ the log structure of log. Denote the resulting log scheme by Ĭlog. Now let us regard the points of Ĩ that map to nodes of as mared points of Ĩ. This gives Ĩ the structure of a smooth, pointed curve over. Because Ĩ Spec() is smooth, it follows that there exists a unique multistable pointed log-curve Ĩlog S log whose underlying curve is Ĩ and whose mared points are as just specified. Since x Ĩ(), by using Slog S log, we can define (as in the discussion as the end of Section 2) the admissible fundamental group Π adm of Ĩlog. Moreover, we have natural log morphisms Ĭ log log Ĩ log where the vertical morphism is an open immersion. Now observe that if we restrict (say, orderly) coverings of log to Ĭlog, such a covering extends naturally to an orderly covering of Ĩlog. Thus, we obtain a natural morphism ζ :Π adm Π adm t is immediate from the definitions that the subgroup ζ (Π adm a adm. Suppose that is stable and sturdy. Then the mor- Proposition 4.2. phism ζ is injective. ) Π adm Proof. Let Π be the image of Π adm S log in Π S log. Then let us note that we have a commutative diagram of exact sequences: 1 Ĩlog Π adm Π 1 S log 1 log Π adm Π S log 1 is

21 Profinite Grothendiec Conjecture 591 where the vertical arrow on the right is the natural inclusion. Thus, it suffices to prove that Ĩlog log is injective. n particular, we are always free to replace S log by a finite, tamely ramified covering of S log. Now it suffices to show that (up to base-changing, when necessary, by finite, tamely ramified coverings of S log ) we can obtain every log admissible covering of Ĩlog by pulling bac a log admissible covering of log. Let us call untangled at if every node of that lies on also lies on an irreducible component of distinct from. n general, we can form a ( combinatorial ) étale covering of as follows: Write = J, where J is the union of the irreducible components of other than. Let Ĩ1 and Ĩ 2 (respectively, J 1 and J 2 ) be copies of Ĩ (respectively, J). For i =1, 2, let us glue Ĩi to J i at every node of that also lies on J. fν isanodeof that only lies on, let α and β be the points of Ĩ that lie over ν. Then glue α 1 Ĩ1 to β 2 Ĩ2, and β 1 Ĩ1 to α 2 Ĩ2. With these various gluings, Ĩ 1 Ĩ2 J1 J2 forms a curve Y which is finite étale over. Moreover, Y is untangled at Ĩ1 and Ĩ2, and Π adm =Π adm i (for i =1, 2). Thus, it suffices to prove the Proposition under the assumption that is untangled at. Therefore, for the remainder of the proof, we shall assume that is untangled at. Now we would lie to construct another double étale covering of. For convenience, we will assume that p 3. (The case p = 2 is only combinatorially a bit more difficult.) Write = J, as above. Since is sturdy, it follows that (after possibly enlarging ), there exists an étale covering J J of degree two such that for any irreducible component C J, the restriction of J J to C is nontrivial. Let 1 and 2 be copies of. fν isanodeon and J, let α (respectively, β) bethe corresponding point on (respectively, J). (After possibly enlarging ) we may assume that J has two -rational points β 1 and β 2 over β J. Now, for i =1, 2, glue α i i to β i J. We thus obtain a double étale covering Y = 1 2 J. Endow Y with the log structure obtained by pulling bac the log structure of log. One can then define various log structures on the irreducible components of Y, analogously to the way in which various log structures were defined on an irreducible component of above. We will then use similar notation for the log structures thus obtained on irreducible components of Y. Now let L log log be a Galois log admissible covering of log def = Ĩlog

22 592 Shinichi Mochizui (recall that = Ĩ) of degree d. Let M log J log be an abelian log multiadmissible covering of degree d with the following property: (*) For each node ν on and J, suppose that over the corresponding α, L has n geometric points, each ramified with index e over. Then, we stipulate that for i =1, 2, over β i J, M has n geometric points, each ramified with index e over. Note that such an M log J log exists precisely because the β s on J come in pairs. Now let L log 1 and L log 2 be copies of L log. Then, for each i =1, 2, let us glue the geometric points of L i αi to those of M βi. This gives us (after possibly replacing S log by a tamely ramified covering of S log ) a log admissible covering Z log Y log, where Z = L 1 L2 M. Moreover, the restriction of Z log Y log to log i (for i =1, 2) is L log log. This completes the proof of the Proposition. 5. The Set of Nodes We continue with the notation of the preceding Section. Thus, log S log is a stable log-curve of genus g. Let us also assume that is sturdy. n this Section, we would lie to show how (by a technique similar to, but slightly more complicated than that employed in Section 1) we can recover the set of nodes of. This, in turn, will allow us to recover the decomposition group of a node. n the following Section, we shall then show how the log 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 assume moreover that l 1(mod n). This means that all n th roots of unity are contained in F l. Let us write G n F l for the subgroup of n th roots of unity. Next, let us fix a G n -torsor over Y which is nontrivial over the generic point of every irreducible component of. (Here, by G n -torsor, we mean a cyclic étale covering of of degree n whose Galois group is equipped with an isomorphism with G n.) Equip Y with the log structure pulled bac from that of log. Let us consider

23 Profinite Grothendiec Conjecture 593 the admissible fundamental group Π adm Y of Y log. Let Hadm 1 log (Y, F l ) def = Hom( Y log, F l ). Note that we have a natural injection L e def = Hét 1 (Y, F l) L a def = Hadm 1, F l ). Let us write L r for the coernel of this injection. (Here, e (respectively, a ; r ) stands for étale (respectively, admissible ; ramification ). Thus, we have an exact sequence (Y log 0 L e L a L r 0 which may (by Proposition 3.2) be recovered from Π adm Π Slog and the subgroup of Π adm that defines Y. Note, moreover, that G n acts on the above exact sequence. Let L r G Lr be the subset of elements on which G n acts via the character G n F l. Let L L a be the subset of elements that map to nonzero elements of L r G. We would lie to analyze L r G. First of all, let us consider Lr. For each node ν Y (), write Y ν for the completion of Y at ν, and let γ ν and δ ν be the two irreducible components of Y ν. Let D Y (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 γ ν, δ ν (respectively, ), as ν (respectively, ) ranges over all the nodes of Y ν (respectively, irreducible components of Y ). Let D Y D Y be the submodule generated by (γ ν δ ν ) F l ( 1) (where ν ranges over all the nodes of Y ν ). Note that we have a natural morphism D Y E Y given by assigning to the symbol γ ν (respectively, δ ν ) the unique irreducible component in which γ ν (respectively,δ ν ) is contained. n particular, restricting to D Y, we obtain a morphism D Y E Y Let Y D Y be the ernel of this morphism. Now let us note that we have a natural morphism λ : L r D Y given by restricting an admissible covering of Y to the various γ ν and δ ν. t follows immediately from the definition of an admissible covering that m(λ) D Y. Moreover, by considering the Leray-Serre spectral sequence in étale cohomology for the morphism Ĭ Ĩ (where Ĩ Y is the normalization of an irreducible component of Y, and Ĭ is the complement of the points that map to nodes), plus the definition of an admissible covering, one

24 594 Shinichi Mochizui sees easily that, in fact, λ( L r )= Y. Finally, by counting dimensions, we see that λ is injective. Thus, we see that λ defines a natural isomorphism of L r with Y. n the following, we shall identify L r and Y by means of λ. Now let us consider the subset L r G Lr. Let µ () be a node. For each such µ, let us fix a node ν Y () over µ. f σ G n (regarded as the Galois group of Y ), we shall write a σ F l for σ regarded as an element of F l. Fix a generator ω F l( 1). Let def ω µ = (a 1 σ ω)(σ(γ ν ) σ(δ ν )) D Y σ G n One checs easily that ω µ is, in fact, an element of Y = L r. Moreover, by calculating τ(ω µ ) (for τ G n ), one sees that ω µ is manifestly an element of L r G Lr. Finally, it is routine to chec that, in fact, L r G is freely generated by the ω µ (as µ ranges over the nodes of () but ω is fixed). This completes our analysis of L r G. Suppose that α L. Let Z α Y be the corresponding covering. Let ɛ : L Z be the morphism that maps α to N Zα (i.e., the number of nodes of Z α ). Let M L be the subset of elements α on which ɛ attains its maximum. Let us define a pre-equivalence relation onm as follows: f α, β M, then we write α β if, for every λ, µ F l λ α + µ β L, we have λ α + µ β M. for which Now we have the following result: Proposition 5.1. Suppose that is stable and sturdy. Then is, in fact, an equivalence relation, and moreover, M/ is naturally isomorphic to the set of nodes of. Proof. Suppose that α L maps to a linear combination (with nonzero coefficients) of precisely c 1 of the elements ω µ L r G. Then one calculates easily that Z α has precisely ɛ(α) = l(n Y cn) +cn = l N Y + cn(1 l) nodes. Thus, ɛ(α) attains its maximum precisely when c = 1. Thus, M L consists of those α which map to a nonzero multiple of one of the ω µ s. t is thus easy to see (as in the proof of Proposition 1.3) that M/ is naturally isomorphic to the set of nodes µ ().

25 Profinite Grothendiec Conjecture 595 Remar. Note that at first glance the set M/ appears to depend on the choice of n, l, and Y. However, it is not difficult to see that in fact, if one chooses different data n n, l l, and Y, and hence obtains a resulting M /, then there is a natural isomorphism (M/ ) = (M / ) (compatible with the isomorphisms just obtained of M/ and M / to the set of nodes of ) as follows: f α M and α M, let us consider the product Z αα = Z α Z α. Thus, we have an admissible covering Z αα of degree (ln)(l n ). Then one checs easily that α and α correspond to the same node if and only if (Z αα ) has precisely nn {l l (N 1) + 1} nodes. Proposition 5.2. Suppose that is stable and sturdy. Then the set of nodes of (together with its natural Π S -action) can be recovered entirely from Π adm Π Slog. Moreover, (relative to Proposition 1.4) for each node µ of, the set of irreducible components of containing µ can also be recovered entirely from Π adm Π S log. Proof. ndeed, (after possibly replacing by a finite extension of ) one can always choose l, n, and Y as above. Then one can recover L e and L a from Π adm Π Slog and the subgroup of Πadm defined by Y. Thus, one can also recover L r. We saw in Section 1 that for any Z α, N Zα Zα, as well as Zα, may be recovered group-theoretically. n particular, N Zα can also be recovered group-theoretically. Thus, M and can also be recovered group-theoretically. Moreover, by the above Remar, M/ is independent of the choice of n, l, and Y. (That is to say, the isomorphism (M/ ) = (M / ) of the above Remar 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 (relative to the reconstruction of the set of irreducible components of given in Proposition 1.4) µ lies on. To this end, note first that (by Corollary 1.6) the genus g of each irreducible component of can also be recovered group-theoretically. (ndeed, J l () has precisely l 2g elements.) Thus, if α M corresponds to the node µ, the irreducible components of containing µ are precisely those that are the image of irreducible components J of Z α such that (g J 1) >l n (g 1). Corollary 5.3. Suppose that is stable and sturdy. Let H Π adm be an open orderly subgroup. Let Y log H log be the corresponding covering.