45(2009), 135–186
A Prime-to-p Version of Grothendieck’s Anabelian Conjecture for Hyperbolic Curves
over Finite Fields of Characteristic p>0
By
MohamedSa¨ıdi∗and AkioTamagawa∗∗
Contents
§0. Introduction
§1. Characterization of Decomposition Groups
§2. Cuspidalizations of Proper Hyperbolic Curves
§3. Kummer Theory and Anabelian Geometry
§4. Recovering the Additive Structure References
Abstract
In this paper, we prove a prime-to-p version of Grothendieck’s anabelian con- jecture for hyperbolic curves over finite fields of characteristicp >0, whose original (full profinite) version was proved by Tamagawa in the affine case and by Mochizuki in the proper case.
§0. Introduction
Letkbe a finite field of characteristicp >0 andU a hyperbolic curve over k. Namely,U =XS, whereX is a proper, smooth, geometrically connected
Communicated by S. Mukai. Received July 2, 2007. Revised February 10, 2008.
2000 Mathematics Subject Classification(s): 11G20, 14G15, 14H30, 14H25.
∗School of Engineering, Computer Science and Mathematics, University of Exeter, Harri- son Building, North Park Road, Exeter EX4 4QF, United Kingdom.
e-mail: M.Saidi@exeter.ac.uk
∗∗Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.
e-mail: tamagawa@kurims.kyoto-u.ac.jp
c 2009 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
curve of genusg overk and S ⊂X is a divisor which is finite ´etale of degree r overk, such that 2−2g−r < 0. We have the following exact sequence of profinite groups:
1→π1(U ×k¯k,∗)→π1(U,∗)→Gk →1.
Here, ¯kis an algebraic closure ofk, Gk is the absolute Galois group Gal(¯k/k),
∗ means a suitable geometric point, and π1 stands for the ´etale fundamen- tal group. The following result is fundamental in the anabelian geometry of hyperbolic curves over finite fields.
Theorem (Tamagawa, Mochizuki). Let U,V be hyperbolic curves over finite fieldskU,kV, respectively. Let
α:π1(U,∗)→∼ π1(V,∗)
be an isomorphism of profinite groups. Thenα arises from a uniquely deter- mined commutative diagram of schemes:
U˜ −−−−→∼ V˜
⏐⏐
⏐⏐ U −−−−→∼ V
in which the horizontal arrows are isomorphisms, and the vertical arrows are the profinite ´etale universal coverings determined by the profinite groups π1(U,∗), π1(V,∗), respectively.
This result was proved by Tamagawa (cf. [Tamagawa1], Theorem (4.3)) in the affine case (together with a certain tame version), and more recently by Mochizuki (cf. [Mochizuki2], Theorem 3.2) in the proper case. It implies, in particular, that one can embed a suitable category of hyperbolic curves over finite fields into the category of profinite groups. It is essential in the anabelian philosophy of Grothendieck, as was formulated in [Grothendieck], to be able to determine the image of this functor. Recall that the full structure of the profinite group π1(U×k¯k,∗) is unknown (for any single example of U which is hyperbolic). Hence, a fortiori, the structure of π1(U,∗) is unknown.
(Even if we replace the fundamental groupsπ1(U×k¯k,∗),π1(U,∗) by the tame fundamental groups π1t(U ×kk,¯ ∗), π1t(U,∗), respectively, the situation is just the same.) Thus, the problem of determining the image of the above functor seems to be quite difficult, at least for the moment. In this paper we investigate the following question:
Question 1. Is it possible to prove any result analogous to the above Theorem whereπ1(U,∗) is replaced by some (continuous) quotient of π1(U,∗) whose structure is better understood?
The first quotients that come into mind are the following. Let C (re- spectively, Cl) be the class of finite groups of order prime to p(respectively, finite l-groups, where l = p is a fixed prime number). Let ΔU be the max- imal pro-prime-to-p (i.e., pro-C) quotient of π1(U ×k ¯k,∗). For a profinite group Γ, Γl stands for the maximal pro-l (i.e., pro-Cl) quotient of Γ. Thus, in particular, ΔlU coincides with π1(U ×kk,¯ ∗)l. Here, the structures of ΔU and ΔlU are well understood — ΔU (respectively, ΔlU) is isomorphic to the pro-prime-to-p(respectively, pro-l) completion of a certain well-known finitely generated discrete group (i.e., either a free group or a surface group). Let ΠU def= π1(U,∗)/Ker(π1(U×kk,¯ ∗)ΔU), Π(Ul)def= π1(U,∗)/Ker(π1(U×k¯k,∗)ΔlU be the corresponding quotients of π1(U ×k ¯k,∗), respectively. We shall re- fer to ΠU as the geometrically pro-ΣU ´etale fundamental group of U, where ΣU def= Primes{char(k)}, andPrimesstands for the set of all prime numbers.
Question 2. Is it possible to prove any result analogous to the above Theorem whereπ1(U,∗) is replaced by ΠU, Π(Ul), respectively?
Our main result in this paper is the following (cf. Corollary 3.10):
Theorem 1 (A Prime-to-p Version of Grothendieck’s Anabelian Con- jecture for Hyperbolic Curves over Finite Fields). Let U, V be hyperbolic curves over finite fieldskU,kV, respectively. Let ΣU def
= Primes{char(kU)}, ΣV def= Primes{char(kV)}, and write ΠU, ΠV for the geometrically pro-ΣU
´etale fundamental group ofU, and the geometrically pro-ΣV ´etale fundamental group ofV, respectively. Let
α: ΠU →∼ ΠV
be an isomorphism of profinite groups. Thenα arises from a uniquely deter- mined commutative diagram of schemes:
U˜ −−−−→∼ V˜
⏐⏐
⏐⏐ U −−−−→∼ V
in which the horizontal arrows are isomorphisms and the vertical arrows are the profinite ´etale coverings corresponding to the groupsΠU,ΠV, respectively.
As an important consequence of Theorem 1, we deduce in Corollary 3.11 the following prime-to-pversion of Uchida’s Theorem on isomorphisms between absolute Galois groups of function fields (cf. [Uchida]).
Theorem 2. Let X, Y be proper, smooth, geometrically connected curves over finite fields kX, kY, respectively. Let KX, KY be the function fields of X, Y, respectively. Let GKX, GKY be the absolute Galois groups of KX,KY, respectively, and let ΓKX, ΓKY be their geometrically pro-prime-to- characteristic quotients (cf. the discussion before Corollary 3.11). Let
α: ΓKX →∼ ΓKY
be an isomorphism of profinite groups. Thenα arises from a uniquely deter- mined commutative diagram of field extensions:
(KX)∼ −−−−→∼ (KY)∼ ⏐
⏐ ⏐⏐ KX −−−−→∼ KY
in which the horizontal arrows are isomorphisms and the vertical arrows are the extensions corresponding to the Galois groupsΓKX,ΓKY, respectively.
Our proof of Theorem 1 relies substantially on the methods of Tamagawa and Mochizuki. We shall explain this briefly in the case where U is proper (hence, U = X). (cf. Theorem 3.9. The general case can be reduced to this special case.) Starting from ΠX, Tamagawa’s method characterizes group- theoretically the decomposition groups at points of X in ΠX. The problem of recovering the points ofX from the corresponding decomposition groups is related to the question of whether the natural map
Xcl→Sub(ΠX)ΠX
from the set of closed points ofX to the set of conjugacy classes of closed sub- groups of ΠX, which maps a pointxto the conjugacy class of its decomposition groupDx, is injective. This map is known to be injective in the full profinite case, i.e., when one starts fromπ1(X,∗) instead of ΠX. In our case we are only able to prove that the above map is almost injective, i.e., injective outside a finite setEX ⊂Xcl. Thus, we can only recover from ΠX the set of points in a nonempty open subsetXEX.
In [Mochizuki2], Mochizuki developed a theory of cuspidalizations of ´etale fundamental groups of proper hyperbolic curves. One of the consequences of
the main results of this theory is that, starting from ΠX, one can recover in a functorial way the Kummer theory of open affine subsetsUS def
= XS, whereS is a finite set of closed points contained inXEX. Using Kummer theory, one then recovers (up to Frobenius twist) the multiplicative groupOE×X of rational functions on X whose divisor has support disjoint from EX. Thus, starting from an isomorphism
ΠX→∼ ΠY
as in Theorem 1 we can recover, up to Frobenius twist, an injective embedding between multiplicative groups
O×EX → O×EY.
The issue is then to show that this embedding arises from a uniquely determined embedding of fieldsKX→KY, between the corresponding function fields. This kind of problem of recovering the additive structure of function fields has been treated in [Uchida] and [Tamagawa1], using certain auxiliary functions called the minimal elements, i.e., functions with a minimal pole. The arguments of loc. cit. work well in the case of a bijection between multiplicative groups, but fail in our case where we only have an embeddingO×EX → O×EY between multiplicative groups. In our case, instead of using minimal elements, we can recover the additivity by using functions whose divisor has a unique pole. Also, the fact that we can only evaluate functions at all but finitely many points ofX (or, more precisely, all points ofXEX) presents an additional difficulty, which we overcome, roughly speaking, by passing to an infinite algebraic extension of the base field, and using “infinitely many” auxiliary functions.
In§1, we review some (mostly known but partly new) results which show that various invariants of the curveX (among other things, the set of decom- position groups at closed points of X) can be recovered group-theoretically, starting from ΠX. We also prove the almost injectivity of the above map from the set of closed points ofX to the set of conjugacy classes of decomposition groups. In§2, we review the main results of Mochizuki’s theory of cuspidal- izations of ´etale fundamental groups of proper hyperbolic curves, which plays an essential role in this paper. In §3, we prove our main results, assuming the results of§4. Finally, In§4, we investigate the problem of recovering the additive structure of functions mentioned above. Using the above “one pole argument”, we prove Proposition 4.4, which is used in§3.
§1. Characterization of Decomposition Groups
Let X be a proper, smooth, geometrically connected curve over a finite fieldk=kX of characteristicp=pX>0. WriteK=KX for the function field ofX.
LetS be a (possibly empty) finite set of closed points ofX, and set U = US def= XS. We assume thatU is hyperbolic.
Fix a separable closure Ksep = KXsep of K, and write k = kX for the algebraic closure ofkinKsep. Write
GK def
= Gal(Ksep/K), Gkdef= Gal(k/k)
for the absolute Galois groups ofK andk, respectively.
The tame fundamental groupπ1t(U) with respect to the base point defined byKsep(where “tame” is with respect to the complement of U in X) can be naturally identified with a quotient ofGK. Write Gal(KUt/K) for this quotient.
(In caseS =∅, we also write KUur forKUt.) It is easy to see thatKUt contains Kk.
Let Σ = ΣX be a set of prime numbers that contains at least one prime number different fromp. Write
Σ† def= Σ{p}.
Thus, Σ† =∅by our assumption. Denote by ˆZΣ† the maximal pro-Σ† quotient of ˆZ. Set Σ = ΣX =PrimesΣX. We say that Σ is cofinite if (Σ) <∞. Note that, if Σ is cofinite, then Σ is of (Dirichlet) density 1.
We define ˜KU to be the maximal pro-Σ subextension ofKkinKUt. Now, set
ΠU = Gal( ˜KU/K),
which is a quotient ofπ1t(U) = Gal(KUt/K). This fits into the exact sequence 1→ΔU →ΠU
prU
→ Gk→1.
Here, ΔU is the maximal pro-Σ quotient ofπ1t(U), where, for ak-schemeZ, we setZdef= Z×kk.
Define ˜XU to be the integral closure of X in ˜KU. Define ˜U to be the integral closure ofU in ˜KU, which can be naturally identified with the inverse image (as an open subscheme) ofU in ˜XU. Define ˜SU to be the inverse image (as a set) ofS in ˜XU.
For a scheme Z, writeZclfor the set of closed points ofZ. Then we have Xcl=Ucl
S, ( ˜XU)cl= ˜Ucl
S˜U.
Moreover, ( ˜XU)cl admits a natural action of ΠU, and the corresponding quo- tient can be naturally identified withXcl.
For each ˜x ∈ ( ˜XU)cl, we define the decomposition group Dx˜ ⊂ ΠU (re- spectively, the inertia groupIx˜⊂Dx˜) to be the stabilizer at ˜xof the natural action of ΠU on ( ˜XU)cl(respectively, the kernel of the natural action ofD˜xon k(˜x) =k(x) =k). These groups fit into the following commutative diagram in which both rows are exact:
1→ I˜x →Dx˜→Gk(x)→1
∩ ∩ ∩
1→ΔU→ΠU→ Gk →1
Moreover,Ix˜ ={1} (respectively, Ix˜ is (non-canonically) isomorphic to ˆZΣ†), if ˜x∈U˜cl (respectively, ˜x∈S˜U). SinceIx˜ is normal inDx˜, Dx˜ acts onIx˜ by conjugation. Since Ix˜ is abelian, this action factors through Dx˜ →Gk(x) and induces a natural action ofGk(x) onIx˜.
Lemma 1.1. Assume x˜∈S˜U. Then:
(i)The subgroupI˜xGk(x) ofI˜xthat consists of elements fixed by theGk(x)-action is trivial.
(ii)Assume moreover that Σis of density 1. Then the map Gk(x) →Aut(Ix˜) is injective.
Proof. By assumption, Ix˜ ZˆΣ†, and it is well-known that the map Gk(x)→Aut(I˜x) = (ˆZΣ†)× coincides with the cyclotomic character and sends the(k(x))-th power Frobenius elementϕk(x)∈Gk(x), which is a (topological) generator ofGk(x), to (k(x))∈(ˆZΣ†)×. The assertion of (i) follows from this, since(k(x))−1 is not a zero divisor of the ring ˆZΣ†. The assertion of (ii) also follows from this, together with a theorem of Chevalley ([Chevalley], Th´eor`eme 1, see also [GS]). More precisely, by applying Chevalley’s theorem to the finitely generated subgroup(k(x))of Q×, we see that the map ˆZ →(ˆZΣ†)×, α→ (k(x))αis injective, as desired.
LetGbe a profinite group. Then, define Sub(G) (respectively, OSub(G)) to be the set of closed (respectively, open) subgroups ofG.
By conjugation,Gacts on Sub(G). More generally, letH andKbe closed subgroups of Gsuch thatK normalizes H. Then, by conjugation,K acts on Sub(H). We denote by Sub(H)K the quotient Sub(H)/K by this action. In particular, Sub(G)G is the set of conjugacy classes of closed subgroups ofG.
For any closed subgroups H, K of G with K ⊂ H, we have a natural inclusion Sub(K) ⊂ Sub(H), as well as a natural map Sub(H) → Sub(K), J →J∩K. By using this latter natural map, we define
Sub(G)def= lim−→
H∈OSub(G)
Sub(H).
Observe that Sub(G) can be identified with the set of commensurate classes of closed subgroups ofG. (Closed subgroupsJ1 andJ2 ofGare called commen- surate (to each other), ifJ1∩J1 is open both inJ1and inJ2.)
With these notations, we obtain natural maps
D=D[U] : ( ˜XU)cl→Sub(ΠU),x˜→Dx˜, I=I[U] : ( ˜XU)cl→Sub(ΔU)⊂Sub(ΠU),x˜→I˜x, which fit into the commutative diagram
( ˜XU)cl −−−−→D Sub(ΠU)
⏐⏐
( ˜XU)cl −−−−→I Sub(ΔU),
where the vertical arrow stands for the natural map Sub(ΠU) → Sub(ΔU), J →J∩ΔU. By composition with the natural map Sub(ΠU)→Sub(ΠU),D, I yield
D=D[U] : ( ˜XU)cl→Sub(ΠU), I=I[U] : ( ˜XU)cl→Sub(ΔU)⊂Sub(ΠU).
Remark 1.2. Unlike the case of D, I, the maps D, I are essentially un- changed if we replaceU by any covering corresponding to an open subgroup of ΠU.
Since the mapsD, I are ΠU-equivariant, they induce natural maps DΠU =D[U]ΠU :Xcl→Sub(ΠU)ΠU,
IΠU =I[U]ΠU :Xcl→Sub(ΔU)ΠU ⊂Sub(ΠU)ΠU, respectively.
Definition 1.3. Letf :A→B be a map of sets.
(i) We define μf :B →Z∪ {∞} byμf(b) =(f−1(b)). (Thus, f is injective (respectively, surjective) ifμf(b)≤1 (respectively, μf(b)≥1) for any b∈ B.
We also havef(A) ={b∈B|μf(b)≥1}.)
(ii) We say thatf is quasi-finite, ifμf(b)<∞for anyb∈B.
(iii) We say that an element a of A is an exceptional element off (in A), if μf(f(a))>1. We refer to the set of exceptional elements as the exceptional set.
(iv) We say that a pair (a1, a2) of elements ofAis an exceptional pair of f (in A), ifa1=a2 andf(a1) =f(a2) hold.
(v) We say thatf is almost injective (in the strong sense), if the exceptional set off is finite. (Observe that almost injectivity implies quasi-finiteness.)
Lemma 1.4. Let f :A→B and g:B →C be maps of sets. Then we have:
Bothf andg are quasi-finite (respectively, almost injective).
⇓
g◦f is quasi-finite (respectively, almost injective).
⇓
f is quasi-finite (respectively, almost injective).
Proof. Easy.
Definition 1.5. Denote byEU˜ the exceptional set ofD in ( ˜XU)cl Definition 1.6. Let G be a profinite group andH a closed subgroup.
Then we denote byZG(H),NG(H) andCG(H) the centralizer, the normalizer and the commensurator, respectively, ofH in G. Namely,
ZG(H) ={g∈G|ghg−1=hfor anyh∈H},
∩
NG(H) ={g∈G|gHg−1=H} ⊃H,
∩
CG(H) ={g∈G|gHg−1andH are commensurate}.
Lemma 1.7. LetZ be a closed subgroup ofΠU such thatprU(Z)is open inGk and thatprU|Z is injective. Then prU induces an injection CΠU(Z)→ Gk, and we haveCΠU(Z) =NΠU(Z) =ZΠU(Z)⊃Z and(CΠU(Z) :Z)<∞.
Proof. Take anyσ∈CΠU(Z)∩ΔU. Thus,Z0def= Z∩σZσ−1is open both in Z and in σZσ−1. We claim that σ commutes with any element τ of Z0. Indeed, first, observe thatτ ∈Z0 ⊂σZσ−1 and στ σ−1 ∈σZ0σ−1 ⊂σZσ−1 hold. Or, equivalently, σ−1τ σ, τ ∈ Z. Second, observe that prU(σ−1τ σ) = prU(τ) holds, since prU(σ) = 1. Now since prU|Z is injective, the equality prU(σ−1τ σ) = prU(τ) impliesσ−1τ σ=τ, as desired.
Next, we proveσ= 1. To see this, supposeσ= 1 and take any sufficiently small, open characteristic subgroup N of ΔU such that σ ∈ N. Set H def= N , σ ⊂ΔU. Then the image ofσinHabis nontrivial. (Indeed, the image of σinH/N is nontrivial by definition. Since H/N is cyclic, the surjectionH → H/Nfactors through the surjectionH →Hab.) Observe thatZ0normalizesH, sinceNis characteristic in ΔU andσcommutes withZ0. So, the open subgroup H def= H, Z0of ΠU can be regarded as a semidirect-product extension ofZ0 by H and satisfies H ∩ΔU = H. Now, the image of σ in Hab is nontrivial and fixed by the action of Z0. This is impossible, as can be easily seen by observing the Frobenius weights in the action ofZ0, or of prU(Z0), which is an open subgroup ofGk.
Thus, we have provedσ= 1, and the first assertion follows from this. In particular,CΠU(Z) (→Gk) is abelian, hence the second assertion follows. Fi- nally, since prU induces an isomorphism CΠU(Z)→∼ prU(CΠU(Z)) and prU(Z) is open inGk, the third assertion holds.
The first main result in this§is:
Proposition 1.8. (i)I|S˜U : ˜SU →Sub(ΠU)is injective.
(ii)EU˜ is disjoint from S˜U. (Or, equivalently, EU˜ ⊂U˜cl.)
(iii) Let ρ denote the natural morphism X˜U → X. Then, for each x ∈ Xcl, D|ρ−1(x) is injective.
(iv) Let ρ denote the natural morphism X˜U → X. Then, for each x ∈ Xcl, D|ρ−1(x) is quasi-finite. If, moreover, k(x) = k holds (i.e., x is a k-rational point ofX), then D|ρ−1(x) is injective.
(v)EU˜ isΠU-stable.
Assume, moreover, thatΣis cofinite. Then:
(vi)The quotientEU˜/ΠU is finite.
(vii)D: ( ˜XU)cl→Sub(ΠU)is quasi-finite.
Proof. (i) Take any ˜x,x˜ ∈ S˜U, and assume ˜x = ˜x. Then there exists an open subgroupH0 of ΠU, such that the following holds: LetU0 denote the covering ofU corresponding toH0⊂ΠU andX0the integral closure ofXinU0 (i.e.,X0 is the smooth compactification of U0), then the imagesx0, x0 of ˜x,x˜ in X0 are distinct from each other. Moreover, by replacingH0 by a smaller open subgroup if necessary, we may assume that the cardinality of the point setX0U0 is≥3 (see, e.g., [Tamagawa1], Lemma (1.10)).
Now, to show the desired injectivity, it suffices to prove that I˜x∩H1 = Ix˜∩H1 holds for any open subgroupH1 ofH0. LetU1 denote the covering of U corresponding toH1⊂ΠU andX1the integral closure ofX inU1. Then, by the choice ofH1, we see that the images of ˜x,x˜ inS1def= X1U1 are distinct from each other and that the cardinality ofS1 is≥3. Then it is easy to see that the images ofIx˜∩H1, Ix˜ ∩H1 in Hab1 are isomorphic to ˆZΣ† and that the intersection of these images is {0}. (Observe (the pro-Σ† part of) exact sequence (1-5) in [Tamagawa1].) Thus, a fortiori,I˜x∩H1=Ix˜∩H1holds, as desired.
(ii) Take any ˜x ∈ S˜U and ˜x ∈ ( ˜XU)cl, such that ˜x = ˜x holds; then we shall prove that the images of ˜x,x˜ by D are distinct from each other. To see this, it suffices, by definition, to prove that, for any open subgroup H of ΠU, the images Dx˜∩H, Dx˜ ∩H of ˜x,˜x in Sub(H) are distinct from each other. Now, replacing U by the covering of U corresponding to H ⊂ ΠU, it suffices to prove that Dx˜, D˜x are distinct from each other. Now, recall that Dx˜∩ΔU =I˜x, Dx˜ ∩ΔU =Ix˜. Thus, if ˜x ∈ S˜U, the last assertion follows from (i). On the other hand, if ˜x∈U˜cl, the last assertion follows from the fact Ix˜ZˆΣ†,Ix˜ ={1}.
(iii) Ifx∈S def= XU, the assertion follows from (ii). So, we may and shall assumex∈Ucl. Take any ˜x,x˜∈ρ−1(x). Then there exists σ∈ΔU such that
˜
x =σ˜xholds. (Suchσis unique by the assumptionx∈Ucl, though we do not use this fact in the proof.) Now, suppose that the images of ˜x,x˜byDcoincide with each other. Namely, Dx˜ and Dx˜ = Dσ˜x = σDx˜σ−1 are commensurate to each other. Thus,σ∈CΠU(Dx˜)∩ΔU, and it follows from Lemma 1.7 that σ= 1 holds, hence ˜x =σ˜x= ˜x. Namely,D|ρ−1(x)is injective, as desired.
(iv) Letπdenote the natural morphismX →X, so thatρ=π◦ρholds. Since (π−1(x)) = [k(x) :k]<∞, the assertions follow directly from (iii).
(v) This follows from the fact thatD is ΠU-equivariant.
(vi) To prove this (assuming that Σ is cofinite), we may replace U by any covering corresponding to an open subgroup of ΠU. Thus, we may assume that the genus ofX is>1 and thatXis non-hyperelliptic. (See, e.g., [Tamagawa1],
Lemma (1.10) for the former, and either [Tamagawa3], §2 or the proof (in characteristic zero) of [Mochizuki1], Lemma 10.4(4) for the latter.) We shall prove that ρ(EU˜), which can be identified with EU˜/ΠU, is finite, or, more strongly, thatρ(EU˜), which can be identified withEU˜/ΔU, is finite.
Take any pair of elements ˜x,x˜∈U˜cl, and denote byx, xthe images of ˜x,x˜ inUcl, respectively. The images prU(D˜x) and prU(Dx˜) are open inGk, hence so is the intersectionG0def= prU(Dx˜)∩prU(Dx˜). Lets, s be the inverse maps of the isomorphisms prU|D˜x : D˜x → prU(D˜x), prU|Dx˜ : Dx˜ → prU(Dx˜), respectively. Then, it is well-known and easy to see that the mapφ:G0→ΔU, γ→s(γ)s(γ)−1 is a continuous 1-cocycle (with respect to the left, conjugacy action of G0 on ΔU via the section s). Thus, φ defines a cohomology class in H1(G0,ΔU). We denote by φab0,X = φab0,X(˜x,˜x) the image of this class in H1(G0,ΔabX). (Note that theG0-action on ΔabX induced by that on ΔU extends to a canonicalGk-action, hence, in particular, is independent of the choice of
˜
x.) Moreover, we set
HX def
= lim−→
G∈OSub(Gk)
H1(G,ΔabX)
(where the transition maps are the restriction maps) and denote by φabX = φabX(˜x,x˜) the image ofφab0,X inHX.
On the other hand, it is well-known that ΔabX is canonically isomorphic as a Gk-module to the pro-Σ part T(J)Σ of the full Tate module T(J) of the Jacobian variety J (tensored with k) of X. Thus, by Kummer theory (for the abelian varietyJ), we obtain an injective mapJ(kG)/(J(kG){Σ})→ H1(G,ΔabX), where G is an open subgroup of Gk, kG is the finite extension ofk corresponding toG, and, for an abelian group M, M{Σ} stands for the subgroup of torsion elementsaofM such that every prime divisor of the order of a belongs to Σ. (In fact, the above injective map is bijective by Lang’s theorem, though we do not use this fact in the proof.) By taking the inductive limit, we obtain an injective mapJ(k)/(J(k){Σ})→ HX. Now, it is widely known that the image in HX of the class of x−x in Jcl = J(k) coincides with φabX. For this, see [NT], Lemma (4.14). (See also [Nakamura2], 2.2 and [Tamagawa1], Lemma (2.6).)
Suppose moreover that (˜x,x˜) is an exceptional pair ofD. Then it follows from the various definitions thatφabX ∈ HX is trivial. Therefore the class of x−x in J(k)/(J(k){Σ}) is trivial, or, equivalently, the class cl(x−x) in J(k) lies in J(k){Σ}. On the other hand, by (iii), it holds that x= x, or, equivalently (by the assumption that the genus ofX is>1), cl(x−x)= 0.
Consider the morphismδ:X×X →J, (P, Q)→cl(P−Q). We claim:
Claim 1.9. (i) δ|X×X−ι(X) is injective (on k-valued points), where ι : X→X×X is the diagonal morphism.
(ii) The imageW ofδdoes not contain any translate of a positive-dimensional abelian subvariety ofJ.
Indeed, for (i), suppose that (P, Q),(P, Q) ∈ (X×X ι(X))(k) have the same image underδ. Namely, the divisors P−QandP−Q are linearly equivalent: P −Q ∼ P−Q, or, equivalently, P +Q ∼P+Q. Since we have assumed thatX is of genus >1 and non-hyperelliptic, this implies that P +Q and P+Q coincide with each other as divisors. This implies that either P = P, Q = Q or P = Q, P = Q holds. The former implies that (P, Q) = (P, Q), as desired, and the latter implies that (P, Q),(P, Q) ∈ ι(X), which contradicts the assumption. For (ii), suppose that W contains a translate B of some positive-dimensional abelian subvariety B of J. As dim(W)≤ dim(X×X) = 2, we have dim(B) ≤ 2, i.e., either dim(B) = 2 or dim(B) = 1. The former implies thatB =W, sinceW is defined as the image ofX ×X, hence irreducible of dimension ≤2. Since 0∈W =B, we concludeW =B=B. Now, sinceJ is generated byW, we must haveJ =B.
This implies that the genus ofX (i.e., dim(J)) is 2, which implies that X is hyperelliptic. This contradicts the assumption. So, suppose dim(B) = 1. By (i), we see thatδinduces a bijective morphismX×Xι(X)→W{0}. From this, we deduce that there exists a finite radicial coveringBofB that admits a non-constant morphism toX×X. In particular, considering a suitable one of two projections, we see thatB admits a non-constant morphism toX. This is absurd, since the genus ofB(respectively,X) is 1 (respectively,>1). This completes the proof of Claim 1.9.
By Claim 1.9(ii) and [Boxall] (which is the most nontrivial ingredient of the proof of Proposition 1.8), we see thatW(k)∩(J(k){Σ}) is finite. Now, by Claim 1.9(i), we conclude that there exists a finite subsetS of (X×X)(k) that contains any pair (x, x) as above. This implies the desired assertion that ρ(EU˜) is a finite set.
(vii) Note thatρ(EU˜) can be identified withEU˜/ΠU by (v). Thus, the assertion of (vii) directly follows from (vi) and the first part of (iv).
Definition 1.10. We defineEU to be the image ofEU˜ inXcl. (This can be identified withEU˜/ΠU. Thus, if Σ is cofinite, then it is finite by Proposition 1.8(vi).)
Corollary 1.11. (i)DΠU|XclEU :XclEU →Sub(ΠU)ΠU is injective.
(ii)EU is disjoint from S. (Or, equivalently, EU ⊂Ucl.) Assume, moreover, thatΣis cofinite. Then:
(iii)DΠU :Xcl→Sub(ΠU)ΠU is almost injective.
Proof. (i) As D|( ˜XU)clEU˜ : ( ˜XU)clEU˜ → Sub(ΠU) is injective by definition and ΠU-equivariant, its quotient by ΠU, which is naturally identified withDΠU|XclEU :XclEU →Sub(ΠU)ΠU, is also injective. This completes the proof.
(ii) This follows from Proposition 1.8(ii).
(iii) This follows from (i) and the fact thatEUis finite (Proposition 1.8(vi)).
Corollary 1.12. (i)For each x˜∈U˜cl,prU induces an injection CΠU(Dx˜)→Gk,
and we have
CΠU(Dx˜) =NΠU(Dx˜) =ZΠU(Dx˜)⊃Dx˜
and
(CΠU(Dx˜) :D˜x)<∞. If, moreover,x˜∈U˜clEU˜, we have
CΠU(Dx˜) =NΠU(Dx˜) =ZΠU(D˜x) =Dx˜. (ii)For eachx˜∈S˜U, we have
CΠU(Dx˜) =NΠU(D˜x) =Dx˜, ZΠU(Dx˜) =ZD˜x(Dx˜) and
CΠU(I˜x) =NΠU(I˜x) =Dx˜, ZΠU(Ix˜) =ZD˜x(Ix˜).
If, moreover,Σis of density 1, thenZDx˜(Dx˜) ={1} andZDx˜(Ix˜) =I˜x. (iii)Assume, moreover, that Σis cofinite. Then there exists an open subgroup G0 of Gk, such that, for any open subgroupH of pr−1U (G0)and any element x˜ of( ˜XU)cl= ˜UclS˜U, we have
CH(Dx˜∩H) =NH(Dx˜∩H) =Dx˜∩H,
ZH(Dx˜∩H) =
Dx˜∩H, forx˜∈U˜cl, {1}, forx˜∈S˜U.
In other words, if we replaceU by a covering corresponding to suchH, we have, for anyx˜∈( ˜XU)cl,
CΠU(Dx˜) =NΠU(Dx˜) =Dx˜, ZΠU(Dx˜) =
D˜x, forx˜∈U˜cl, {1}, forx˜∈S˜U.
Proof. First, sinceD|( ˜XU)clEU˜ : ( ˜XU)clEU˜ →Sub(ΠU) is injective and ΠU-equivariant, we see thatCΠU(Dx˜) =Dx˜holds for any ˜x∈( ˜XU)clEU˜. (i) The first assertion follows from Lemma 1.7. The second assertion follows from the first assertion and the fact shown at the beginning of the proof.
(ii) Let ˜x∈S˜U. Then ˜x∈EU˜by Proposition 1.8(ii). Thus, we haveCΠU(D˜x) = NΠU(Dx˜) =D˜x. From this, we also haveZΠU(Dx˜) =ZD˜x(Dx˜).
Next, by Proposition 1.8(i), the map I|S˜U : ˜SU → Sub(ΠU) is injec- tive. Since this map is also ΠU-equivariant, we see that CΠU(I˜x) = Dx˜. As CΠU(Ix˜) ⊃NΠU(Ix˜)⊃Dx˜, we have NΠU(Ix˜) =Dx˜. From this, we also have ZΠU(Ix˜) =ZD˜x(Ix˜).
If Σ is of density 1, then this last group coincides withIx˜by Lemma 1.1(ii).
In particular,ZDx˜(D˜x) ⊂Ix˜, which implies ZD˜x(Dx˜) =I˜x∩ZDx˜(Dx˜) ={1} by Lemma 1.1(i).
(iii) DefineG0 to be the intersection (inGk) ofGk(x) forx∈EU. SinceEU is finite by Proposition 1.8(vi),G0is an open subgroup ofGk. By (i) and (ii), it is easy to see that thisG0 satisfies the desired properties.
Next, we shall show that various invariants and structures of U can be re- covered group-theoretically (orϕ-group-theoretically) from ΠU, in the following sense.
Definition 1.13. (i) We say that Π = (Π,Δ, ϕΠ) is aϕ-(profinite )group, if Π is a profinite group, Δ is a closed normal subgroup of Π and ϕΠ is an element of Π/Δ.
(ii) An isomorphism from aϕ-group Π = (Π,Δ, ϕΠ) to another ϕ-group Π = (Π,Δ, ϕΠ) is an isomorphism Π →∼ Π as profinite groups that induces an isomorphism Δ →∼ Δ, hence also an isomorphism Π/Δ →∼ Π/Δ, such that the last isomorphism sendsϕΠto ϕΠ.
From now on, we regard ΠU as aϕ-group by ΠU = (ΠU,ΔU, ϕk), whereϕk
stands for the(k)-th power Frobenius element in Gk = ΠU/ΔU. We shall say that an isomorphismα: ΠU →∼ ΠU of profinite groups is Frobenius-preserving ifαdetermines an isomorphism ofϕ-groups.
Definition 1.14. (i) Given an invariant F(U) that depends on the iso- morphism class (as a scheme) of a hyperbolic curveU over a finite field, we say that F(U) can be recovered group-theoretically (respectively, ϕ-group- theoretically) from ΠU, if any isomorphism (respectively, any Frobenius- preserving isomorphism) ΠU →∼ ΠV impliesF(U) =F(V) for two such curves U, V.
(ii) Given an additional structureF(U) (e.g., a family of subgroups, quotients, elements, etc.) on the profinite group ΠU that depends functorially on a hy- perbolic curve U over a finite field (in the sense that, for any isomorphism (as schemes) between two such curvesU, V, any isomorphism ΠU →∼ ΠV in- duced by this isomorphismU →∼ V (unique up to composition with inner au- tomorphisms) preserves the structures F(U) and F(V)), we say that F(U) can be recovered group-theoretically (respectively,ϕ-group-theoretically) from ΠU, if any isomorphism (respectively, any Frobenius-preserving isomorphism) ΠU →∼ ΠV between two such curves U, V preserves the structures F(U) and F(V).
Proposition 1.15. I. The following invariants and structures can be recovered group-theoretically fromΠU:
(i)The subgroup ΔU of ΠU, hence the quotient Gk= ΠU/ΔU. (ii)The subsetsΣandΣ† of Primes.
II. The following invariants and structures can be recovered ϕ-group- theoretically fromΠU:
(iii)The prime number p.
(iv)The cardinality (k) (or, equivalently, the isomorphism class of the finite fieldk).
(v)The genus g = gX of X and the cardinalityr = rU def
= (S), where S def= XU.
(vi)The kernelIU of the natural surjectionΠU →ΠX(which coincides with the kernel of the natural surjectionΔU →ΔX), hence the quotientsΠX= ΠU/IU, ΔX = ΔU/IU.
(vii)The cardinalities(X(k)),(U(k))and (S(k)).
III. Assume, moreover, that Σ is of density 1. Then the following structure (hence also (iii)–(vii) above)can be recovered group-theoretically fromΠU: (viii)The (k)-th power Frobenius element ϕk∈Gk.
Proof. (i) Similar to [Tamagawa1], Proposition (3.3)(ii). (See also [Mochizuki2], Theorem 1.1(ii).)
