Curves over Kummer-faithful Fields

Curves over Kummer-faithful Fields

Yuichiro Hoshi July 2014

———————————–

Abstract. — In the present paper, we discuss the Grothendieck conjecture for hyperbolic curves over Kummer-faithful fields. In particular, we prove that every point-theoretic and Galois-preserving outer isomorphism between the ´etale/tame fundamental groups of affine hyperbolic curves over Kummer-faithful fields arises from a uniquely determined isomor- phism between the original hyperbolic curves.

Contents

Introduction . . . 1

§0. Notations and Conventions . . . .3

§1. Fundamental Groups of Hyperbolic Curves over Kummer-faithful Fields . 4

§2. Maximal Cuspidally Abelian Quotients . . . 13

§3. The Grothendieck Conjecture over Kummer-faithful Fields . . . .21 References . . . 27

Introduction

In the present paper, we discuss the [semi-absolute version of the] Grothendieck con- jecture for hyperbolic curves over Kummer-faithful fields. In Introduction, let the symbol

“” stand for either “◦” or “•”. Let k be a perfect field and X an affine hyperbolic curve over k. Write X^cpt for the smooth compactification of X, D_X ⊆ X^cpt for the divisor at infinity of X, and

Π_X ^def= π₁^tame(X^cpt, D_X) for the tame fundamental group of (X^cpt, D_X).

Then it follows immediately from the functoriality of the tame fundamental group, together with the elementary theory of algebraic curves, that an isomorphism X◦

→∼ X•

of schemes gives rise to a(n) [continuous] outer isomorphism ΠX◦

→∼ ΠX• of profinite groups. In this situation, one of the main questions in the anabelian geometry may be stated as follows.

2010 Mathematics Subject Classification. — 14H30.

Key words and phrases. — Grothendieck conjecture, hyperbolic curve, Kummer-faithful field.

This research was supported by Grant-in-Aid for Scientific Research (C), No. 24540016, Japan Society for the Promotion of Science.

1

What is a necessary and sufficient condition for an outer isomorphism Π_X◦ →^∼ Π_X• of profinite groups to arise [in the above sense] from an isomorphism X◦

→∼ X• of schemes?

Let

α: Π_X◦ −→^∼ Π_X•

be an outer isomorphism of profinite groups. In [4],§2, S. Mochizuki proved the following assertion [cf. [4], Corollary 2.2], which may be regarded as an answer to the above question in the case where k◦ (respectively,k•) is either anMLF [i.e., a finite extension of Q^p for some prime number p] or an FF [i.e., a finite field].

Suppose that k◦ (respectively,k•) is either anMLForFF. Then it holds that α arises from an isomorphism X_◦ →^∼ X_• of schemes if and only if α is point-theoretic [i.e., satisfies the condition that

(∗)_P: α induces a bijection between the set of decomposition subgroups of Π_X◦associated to closed points ofX_◦^cptand the set of decomposition subgroups of Π_X• associated to closed points of X_•^cpt — cf. Definition 3.1, (i); also [4], Definition 1.5, (ii)].

Moreover, such an isomorphism X_◦ →^∼ X_• of schemes is uniquely deter- mined.

In the present paper, by refining various arguments given in [4], §1, §2; [6], §1, we generalize the above result of Mochizuki to the case of affine hyperbolic curves over arbitrary Kummer-faithful fields.

We shall say that a perfect fieldk isKummer-faithfulif, for every finite extensionK of k and every semi-abelian varietyAoverK, the Kummer map associated toAis injective, or, equivalently, it holds that

\

N

N ·A(K) = {0}

— whereN ranges over the positive integers [cf. Definition 1.2; also [6], Definition 1.5]. A typical example of a Kummer-faithful field of characteristic zero is a sub-p-adic field for some prime number p[i.e., a field which is isomorphic to a subfield of a finitely generated extension of an MLF — cf. [3], Definition 15.4, (i)] [cf. [6], Remark 1.5.4]; a typical example of a Kummer-faithful field of positive characteristic is an FF. In particular, a field that appears in the statement of the above result of Mochizuki [i.e., a field which is either an MLF or FF] isKummer-faithful. Here, we note that, in [6], §1, Mochizuki also proved a result on the [semi-absolute version of the] birational Grothendieck conjecture for curves over Kummer-faithful fields [cf. [6], Corollary 1.11].

Let us observe that one verifies easily that if the outer isomorphism α arises from an isomorphism of schemes, thenαsatisfies the above condition (∗)_P. Next, let us introduce another necessary condition to arise from an isomorphism of schemes. We shall say that [under the assumption that k◦ and k• are Kummer-faithful] the outer isomorphism α is Galois-preserving if

(∗)G: α is compatible with the natural quotients ΠX◦ π1(Spec(k◦)), ΠX• π1(Spec(k•)) [cf. Definition 3.1, (ii)].

Here, we note that it follows from [5], Corollary 2.8, (ii), that if k◦ (respectively, k•) is either a finite extension of Q, an MLF, or an FF, then every outer isomorphism Π_X◦ →^∼ Π_X• [hence also every outer isomorphism discussed in the above result of Mochizuki] is

Galois-preserving. Now let us observe that it follows immediately from the elementary theory of algebraic curves that ifαarises from an isomorphism of schemes, thenαsatisfies the above condition (∗)_G.

The main result of the present paper may be summarized as follows [cf. Theorem 3.4, (ii), in the case where we take “Π_X” to be “π₁^tame”].

THEOREMA. — Let the symbol “” stand for either “◦” or “•”. Letk be aKummer- faithful field [cf. Definition 1.2] and X an affine hyperbolic curve over k. Write X^cpt for the smooth compactification of X, D_X ⊆ X^cpt for the divisor at infinity of X, and Π_X ^def= π₁^tame(X^cpt, D_X) for the tame fundamental group of (X^cpt, D_X). Let

α: Π_X◦ −→^∼ Π_X•

be an outer isomorphism of profinite groups. Then it holds thatα arises from an isomor- phism of schemes

X◦

−→∼ X•

if and only if α is point-theoretic [cf. Definition 3.1, (i)] and Galois-preserving [cf.

Definition 3.1, (ii)]. Moreover, such an isomorphism X◦

→∼ X• of schemes is uniquely determined.

The present paper is organized as follows. In §1, we discuss various objects related to the ´etale/tame fundamental group of a hyperbolic curve over aKummer-faithfulfield. In particular, we consider the image, via the Kummer theory, of the multiplicative group of the function field of the hyperbolic curve in a certain injective limit of cohomology modules [cf. Lemma 1.10]. In§2, we discuss themaximal cuspidally abelian quotientof the

´

etale fundamental group of the second configuration space of a proper hyperbolic curve.

In particular, in order to study the maximal cuspidally abelian quotient, we consider a fundamental extension of an open subgroup of the fundamental group of the second configuration space by means of a certain projective system of cohomology modules [cf.

Lemma 2.6]. In §3, we prove the main result of the present paper [cf. Theorem 3.4].

0. Notations and Conventions

Numbers. — We shall write Primesfor the set of all prime numbers. Let Σ ⊆Primes be a subset of Primes. Then we shall say that a positive integer is a Σ-integer if every prime divisor of the integer is contained in Σ. We shall refer to a finite extension of Q^p (respectively, F^p) for some prime number p as an MLF [i.e., a mixed-characteristic local field] (respectively, FF[i.e., a finite field]).

Profinite Groups. — We shall say that a profinite group isslimif every open subgroup of the profinite group is center-free. One verifies immediately that an extension ofcenter- free (respectively,slim) profinite groups is center-free (respectively,slim).

Let G be a profinite group and H ⊆ G a closed subgroup. We shall say that H is characteristic if every [continuous] isomorphism of G preserves H. We shall write Z_G(H) for the centralizer of H in G, Z(G) ^def= Z_G(G) for the center of G, G^ab for the

abelianization of G [i.e., the quotient of G by the closure of the commutator subgroup of G], Aut(G) for the group of [continuous] automorphisms of G, Inn(G) ⊆ Aut(G) for the group of inner automorphisms ofG, and Out(G)^def= Aut(G)/Inn(G) for the group of outer automorphisms ofG. Note that ifGistopologically finitely generated, then it follows immediately that the topology ofGadmits a basis ofcharacteristic open subgroups, which thus induces a profinite topologyon Aut(G), hence also Out(G).

Let G be a center-free and topologically finitely generated profinite group and ρ: J → Out(G) a homomorphism of profinite groups. Thus, we have a naturalexact sequence of profinite groups

1 −→ G −→ Aut(G) −→ Out(G) −→ 1.

Then, by pulling back this exact sequence by the homomorphismρ, we obtain aprofinite group G^outo J, which fits into an exact sequence of profinite groups

1 −→ G −→ G^outo J −→ J −→ 1.

Curves. — Let S be a scheme and X a scheme over S. Then we shall say that X is a smooth curveover S if there exist a scheme X^cpt which is smooth, proper, geometrically connected, and of relative dimension one over S and a closed subscheme D ⊆ X^cpt of X^cpt which is finite and ´etale over S such that the complement X^cpt\D of D in X^cpt is isomorphic to X over S. Note that if S is the spectrum of a field k, then it follows immediately from elementary algebraic geometry that the pair “(X^cpt, D)” is uniquely determined up to canonical isomorphism over k; we shall refer to X^cpt as the smooth compactification of X and toD as the divisor at infinity of X.

Let S be a scheme. Then we shall say that a smooth curve X over S is hyperbolic if there exist a pair (X^cpt, D) satisfying the condition in the above definition of the term

“smooth curve” and a pair (g, r) of nonnegative integers such that 2g−2 +r > 0, the [necessarily locally free]O_S-module (X^cpt→S)∗(Ω¹_Xcpt/S) is of rankg, and the finite ´etale covering D ,→X^cpt→S is of degree r.

1. Fundamental Groups of Hyperbolic Curves over Kummer-faithful Fields

In the present §1, we discuss various objects related to the ´etale/tame fundamental group of a hyperbolic curve over a Kummer-faithful field [cf. Definition 1.2 below]. In the present §1, let k be a perfect field, k an algebraic closure of k, and X a hyperbolic curveoverk. WriteGk

def= Gal(k/k). Note that one verifies immediately from the various definitions involved that since k isperfect, every connected finite ´etale covering of X is a hyperbolic curve over the finite extension of k obtained by forming the algebraic closure of k in the function field of the covering.

DEFINITION1.1. — We shall write

• Primes^×/k ⊆Primes for the set of prime numbers which are invertible in k,

• Zb×/k for the maximal pro-Primes^×/k quotient of Zb,

• X^cpt for the smooth compactification of X,

• D_X ⊆X^cpt for the divisor at infinity of X,

• g_X for the genus of X^cpt,

• r_X ^def= ]D_X(k),

• K_X for the function field of X,

• X^cl+ for the set of closed points of X^cpt, and

• Div(X) for the group of divisors on X^cpt.

If x∈X^cpt(k) is a k-rational point of X^cpt, then we shall write

• ord_x: K_X^× Z for the [uniquely determined] surjective valuation associated to x ∈ X^cpt(k).

In the following, let

Π_X

be either the ´etale fundamental group π₁(X) of X or the tame fundamental group π₁^tame(X^cpt, D_X) of (X^cpt, D_X). Write

∆X ⊆ ΠX

for the quotient of the ´etale fundamental groupπ₁(X⊗_kk)⊆π₁(X) ofX⊗_kkdetermined by Π_X. Thus, we have an exact sequence of profinite groups

1 −→ ∆_X −→ Π_X −→ G_k −→ 1.

Now let us recall [cf., e.g., [7], Corollary 1.4; [7], Proposition 1.11] that ∆_X isslim.

DEFINITION1.2. — We shall say that k isKummer-faithful if, for every finite extension K of k and every semi-abelian varietyA over K, it holds that

\

N

N ·A(K) = {0}

— where N ranges over the positive integers [cf. Remark 1.2.1 below].

REMARK1.2.1. — If k is of characteristic zero, then it is immediate that k is Kummer- faithful in the sense of Definition 1.2 if and only if k is Kummer-faithful in the sense of [6], Definition 1.5.

REMARK1.2.2. — If k is Kummer-faithful, then the following assertion holds:

If K is a finite extension ofk, then it holds that

\

N

(K^×)^N = {1}

— where N ranges over thePrimes^×/k-integers.

Indeed, this follows immediately, by considering the semi-abelian variety “Gm”, from the definition of the term “Kummer-faithful”, together with our assumption that k, hence also K, is perfect.

REMARK1.2.3. — A typical example of a Kummer-faithful field of characteristic zero is a sub-p-adic field for some prime numberp [i.e., a field which is isomorphic to a subfield of a finitely generated extension of an MLF — cf. [3], Definition 15.4, (i)] [cf. [6], Remark 1.5.4]. A typical example of a Kummer-faithful field of positive characteristic is an FF.

DEFINITION1.3.

(i) LetGbe a profinite group. Then we shall writeC(G) for the set of closed subgroups of G. Note that G acts onC(G) by conjugation.

(ii) We shall write

S_X: X^cl+ −→ C(Π_X)/Π_X

[cf. (i)] for the map given by mapping a closed point of X^cpt to the Π_X-conjugacy class of a decomposition subgroup associated to the closed point.

DEFINITION1.4. — We shall write

Λ_X

for the cyclotomeassociated to the semi-graph of anabelioids of pro-Primes^×/k PSC-type [with no nodes] arising from the hyperbolic curve X⊗_kk [cf. [2], Definition 3.8, (i)].

REMARK1.4.1. — In the notation of Definition 1.4:

(i) The cyclotome ΛX is isomorphic, as a Gk-module, to lim←−^N µN(k) — where the projective limit is taken over the Primes^×/k-integers N, and we write µ_N(k) for the group of N-th roots of unity in k.

(ii) If X isproper over k [i.e., r_X = 0], then Λ_X ^def= Hom

Zb×/k H²(∆_X,Zb^×/k),Zb^×/k .

PROPOSITION1.5. — Suppose that k isKummer-faithful. Then the following hold:

(i) Write

χ^×/k_cyc : G_k −→ Aut(lim←−

N

µ_N(k)) = (bZ^×/k)^×

— where the projective limit is taken over the Primes^×/k-integers N, and we write µ_N(k) for the group of N-th roots of unity in k — for the Primes^×/k-adic cyclotomic character of G_k. Then it holds that Z(G_k)∩Ker(χ^×/kcyc ) ={1}.

(ii) The profinite group Π_X is slim.

Proof. — First, we verify assertion (i). Assume that there exists a nontrivial element γ ∈Z(G_k)∩Ker(χ^×/kcyc ). Let K be a finite Galois extension ofk contained ink such that the corresponding normal open subgroup G_K ⊆G_k doesnot containγ ∈G_k. Then since γ ∈Z(G_k)∩Ker(χ^×/kcyc), the natural action of γ onH¹(G_K,lim

←−^N µ_N(k)) istrivial. On the other hand, it follows from Remark 1.2.2, together with the Kummer theory, that this triviality implies the triviality of the action of γ onK. Thus, since γ 6∈GK, we obtain a contradiction. This completes the proof of assertion (i).

Next, we verify assertion (ii). Let us first observe that since a connected finite ´etale covering ofX is ahyperbolic curve over a Kummer-faithful field [cf. our assumption that k isperfect], to verify assertion (ii), it suffices to verify that Π_X iscenter-free. Next, let us observe that since ∆_X is center-free, the composite Z(Π_X),→Π_X G_k is an injection, whose image is contained in the center Z(G_k) of G_k. On the other hand, it follows immediately from the various definitions involved that the natural action of Z(Π_X) on Λ_X is trivial, i.e., that the image of the composite Z(Π_X) ,→ Π_X G_k is contained in Ker(χ^×/kcyc ) [cf. Remark 1.4.1, (i)]. Thus, it follows from assertion (i) that Z(Π_X) ={1}.

This completes the proof of assertion (ii), hence also of Proposition 1.5.

REMARK 1.5.1. — In the situation of Proposition 1.5, (i), in general, it does not hold that Z(G_k) ={1}. Indeed, although [one verifies easily that] an FF is Kummer-faithful, the absolute Galois group of an FF is abelian and nontrivial.

LEMMA1.6. — The following hold:

(i) The quotient of Π_X by the normal closed subgroup generated by the intersections

∆_X∩H — where H ranges over the closed subgroups of Π_X whose images in C(Π_X)/Π_X are contained in the image of the map S_X of Definition 1.3, (ii) — coincides with the quotient Π_X π₁(X^cpt).

(ii) The subset Primes^×/k ⊆ Primes is the [uniquely determined] maximal infinite subset on which the mapPrimes→Zgiven by mappingp∈Primesto dim_Qp π₁(X^cpt⊗_k k)^ab⊗_b

ZQp

(<∞ — cf. [7], Proposition 1.1) is constant.

(iii) For everyp∈Primes^×/k, it holds thatdim_Qp(∆^ab_X ⊗_b

ZQp) = 2g_X+ max{0, r_X−1}, dim_Qp π₁(X^cpt⊗_kk)^ab ⊗

bZQp

= 2g_X.

(iv) It holds that X is proper over k [i.e., r_X = 0] if and only if, for every p ∈ Primes^×/k, the maximal pro-p quotient of ∆X is not free pro-p.

(v) It holds that∆_X isnot topologically finitely generatedif and only ifchar(k)6=

0, r_X 6= 0, and Π_X = π₁(X). In particular, it holds that ∆_X is topologically finitely generated if and only if Π_X =π₁^tame(X^cpt, D_X).

Proof. — Assertion (i) follows immediately from the various definitions involved. As- sertions (ii) and (iii) follow immediately from [7], Corollary 1.2. Assertions (iv) and (v) follow immediately from [7], Proposition 1.1. This completes the proof of Lemma 1.6.

LEMMA1.7. — Suppose that k is Kummer-faithful. Then the following hold:

(i) The map S_X:X^cl+ →C(Π_X)/Π_X of Definition 1.3, (ii), is injective.

(ii) Suppose that X is proper over k [i.e., r_X = 0]. Let m be a positive integer and x₁, . . . , x_m ∈ X(k) distinct k-rational points. Thus, for each i ∈ {1, . . . , m}, the k-rational point x_i ∈X(k) determines a splitting s_i [well-defined up to ∆^ab_X-conjugation]

of the exact sequence of profinite groups

1 −→ ∆^ab_X −→ Π_X/Ker(∆_X ∆^ab_X) −→ G_k −→ 1.

Let (n₁, . . . , n_m) ∈ Z^⊕m be such that Pm

i=1 n_i = 0. Thus, by considering the linear combination “Pm

i=1 n_i ·s_i” of the s_i’s, we obtain a cohomology class [Pm

i=1 n_i · s_i] ∈ H¹(G_k,∆^ab_X). Then the divisorPm

i=1 n_i·x_i [of degree zero]on X isprincipal if and only if [Pm

i=1 n_i·s_i] = 0 in H¹(G_k,∆^ab_X).

Proof. — Assertion (i) follows immediately from a similar argument to the argument applied in the proof of [7], Proposition 2.8, (i). Assertion (ii) follows immediately from a similar argument to the argument applied in the proof of [4], Proposition 2.2, (i). This

completes the proof of Lemma 1.7.

LEMMA 1.8. — Suppose that k is Kummer-faithful, and that D_X(k) = D_X(k). For x∈D_X(k), letI_x ⊆∆_X be an inertia subgroup of∆_X associated tox. Then the following hold:

(i) The inclusions I_x ,→ Π_X — where x ranges over the elements of D_X(k) — and the surjection Π_X G_k determine an exact sequence

0 −→ H¹(G_k,Λ_X) −→ H¹(Π_X,Λ_X) −→ M

x∈DX(k)

HombZ(I_x,Λ_X).

Thus, by considering the isomorphism [well-defined up to a (bZ×/k)^×-multiple]

(k^×)^{×/k def}= lim←−

N

k^×/(k^×)^N −→^∼ H¹(G_k,Λ_X)

— where the projective limit is taken over the Primes^×/k-integers N — obtained by the Kummer theory [cf. Remark 1.4.1, (i)] and the identification

Zb×/k = Hom

bZ×/k(Λ_X,Λ_X) −→^∼ Hom

bZ×/k(I^×/k_x ,Λ_X) −→^∼ Hom

Zb(I_x,Λ_X)

— where we write I^×/kx for the maximal pro-Primes^×/k quotient ofIx — obtained by the synchronization of cyclotomes I^×/kx

→∼ Λ_X discussed in [2], Corollary 3.9, (v), we obtain an exact sequence

0 −→ (k^×)^×/k −→ H¹(Π_X,Λ_X) −→ M

x∈D_X(k)

Zb^×/k.

(ii) The exact sequence of the final display of (i) fits into the following commutative diagram

0 −−−→ k^× −−−→ O^×_X(X)

L

x∈DX(k) ordx

−−−−−−−−−→ L

x∈DX(k) Z



 y



 y



 y 0 −−−→ (k^×)^×/k −−−→ H¹(Π_X,Λ_X) −−−→ L

x∈DX(k) Zb^×/k

— where the horizontal sequences are exact, the vertical arrows are injective, the left- hand and middle vertical arrows are the homomorphisms obtained by the Kummer theory, and the right-hand vertical arrow is the homomorphism determined by the natural inclu- sion Z,→Zb^×/k.

(iii) Let y∈X(k) be a k-rational point. Then the composite

O_X^×(X) −→ H¹(Π_X,Λ_X) −→ H¹(G_k,Λ_X) ←−^∼ (k^×)^×/k

— where the first arrow is the middle vertical arrow of the diagram of (ii), and the second arrow is the homomorphism determined by the splitting [well-defined up to ∆X- conjugation] of ΠX Gk induced by y ∈ X(k), i.e., “SX(y)” — coincides [up to a (bZ^×/k)^×-multiple — cf. the isomorphism (k^×)^×/k →^∼ H¹(G_k,Λ_X) discussed in (i)] with the composite

O_X^×(X) −→ k^× −→ (k^×)^×/k f 7→ f(y)

— where the second arrow is the natural homomorphism.

Proof. — Since the G_k-invariant of the Primes^×/k-adic Tate module of the Jacobian variety of X^cpt is trivial [by our assumption that k is Kummer-faithful], assertion (i) follows immediately from a similar argument to the argument applied in the proof of [4], Proposition 2.1, (ii). Assertions (ii) and (iii) follow immediately — in light of Re- mark 1.2.2 — from the functoriality of the Kummer class, together with the various definitions involved. This completes the proof of Lemma 1.8.

DEFINITION1.9. — Suppose thatk is Kummer-faithful, and thatX isproper[i.e., r_X = 0]. Let S ⊆ X(k) be a finite subset and x∈X(k)\S. Thus, sinceX\S is a hyperbolic curve overk, it follows from Lemma 1.8, (i), that we have an exact sequence

0 −→ (k^×)^×/k −→ H¹(Π_X\S,Λ_X) −→ M

s∈S

Zb^×/k

— where we write ΠX\S

def= π₁^tame(X, S). We shall write P(X, S) ^def= n

(n_s)s∈S ∈M

s∈S

Z

The divisor X

s∈S

n_s·s is principal.o

⊆ M

s∈S

Zb^×/k; O^×(Π_X, S) ⊆ H¹(ΠX\S,Λ_X)

for the submodule obtained by forming the inverse image of the submodule P(X, S) ⊆ L

s∈S Zb×/k via the third arrow of the above exact sequence;

ev_x(Π_X, S) : O^×(Π_X, S) −→ H¹(G_k,Λ_X)

for the restriction toO^×(Π_X, S)⊆H¹(Π_X\S,Λ_X) of the homomorphismH¹(Π_X\S,Λ_X)→ H¹(G_k,Λ_X) determined by the splitting [well-defined up to ∆X\S-conjugation] of Π_X\S G_k induced by x∈X(k)\S, i.e., “SX\S(x)”;

K^×(Π_X) ^def= lim−→

K,T

O^×(ΠX⊗_kK, T)

— where the injective limit is taken over the finite extensionsKofkcontained inkand the finite subsets T ⊆(X⊗_kK)(K). Here, we note that the natural injection Π_X⊗_kK ,→Π_X [well-defined up to ∆X-conjugation] and the natural surjection ΠX⊗_kK GK determine anisomorphism Π_X⊗kK →^∼ Π_X ×_GkG_K [well-defined up to (∆_X × {1})-conjugation].

LEMMA 1.10. — Suppose that k is Kummer-faithful, and that X is proper [i.e., r_X = 0]. Then the following hold:

(i) The middle vertical arrows of the diagram of Lemma 1.8, (ii), in the case where we take “X” of Lemma 1.8, (ii), to be (X ⊗kK)\T — where K ranges over the finite extensions of k contained in k and T ranges over the finite subsets of (X⊗kK)(K) — determine an injective homomorphism

K_X⊗^×

kk ,→ K^×(Π_X) [cf. Definition 1.1].

(ii) LetS ⊆X(k)be a finite subset. Then the diagram of Lemma 1.8, (ii), determines a commutative diagram

0 −−−→ k^× −−−→ O_X^×(X\S)

L

x∈Sordx

−−−−−−→ P(X, S) −−−→ 0



 y



 y

0 −−−→ H¹(G_k,Λ_X) −−−→ O^×(Π_X, S) −−−→ P(X, S) −−−→ 0.

— where the horizontal sequences are exact, and the vertical arrows are injective.

(iii) Let S ⊆ X(k) be a finite subset and x ∈ X(k)\S. Then the kernel of the homomorphism

ev_x(Π_X, S) : O^×(Π_X, S) −→ H¹(G_k,Λ_X)

coincides, relative to the middle vertical injection of the diagram of (ii), with the sub- group

{f ∈ O^×_X(X\S)|f(x) = 1} of O^×_X(X\S), i.e.,

Ker ev_x(Π_X, S)

= {f ∈ O_X^×(X\S)|f(x) = 1} ⊆ O^×_X(X\S).

In particular, for every y ∈ X(k)\S, relative to the left-hand vertical injection of the diagram of (ii), it holds that

ev_y(Π_X, S)(Ker ev_x(Π_X, S)

) ⊆ k^×. (iv) Let x₁, x₂ ∈X(k) be such that x₁ 6=x₂. Then

(a) the subgroups

Ker ev_(xi)K(ΠX⊗_kK, S_i)

⊆ O^×(ΠX⊗_kK, S_i) ⊆ K^×(Π_X)

— where i ranges over the elements of {1,2}, K ranges over the finite extensions of k contained in k, S_i ranges over the finite subsets of (X ⊗_kK)(K) which do not contain (xi)K, and we write (xi)K ∈ (X⊗kK)(K) for the K-rational point determined by xi — and

(b) the subgroups

ev_(x2)K(ΠX⊗_kK, S)(Ker ev_(x1)K(ΠX⊗_kK, S)

)⊆ H¹(G_K,Λ_X) ⊆ K^×(Π_X)

— where K ranges over the finite extensions of k contained in k, S ranges over the finite subsets of (X ⊗_k K)(K) which do not contain (x₁)_K and (x₂)_K, and we write (x_i)_K ∈(X⊗_kK)(K) for the K-rational point determined by x_i —

generate the image of the injection of (i).

Proof. — Assertions (i), (ii) follow immediately from the various definitions involved, together with our assumption that k is Kummer-faithful. Next, we verify assertion (iii).

Let us first observe that one verifies immediately from Lemma 1.8, (iii), together with the various definitions involved, that, to complete the verification of assertion (iii), it suffices to verify that Ker(evx(ΠX, S)) ⊆ O_X^×(X \S). Let f ∈ Ker(evx(ΠX, S)). Next, let us observe that it follows immediately from assertion (ii) that there exist g ∈ O^×_X(X \S) and a∈H¹(G_k,Λ_X) such that f =a·g. Thus, it holds that

1 = ev_x(Π_X, S)(f) = a·ev_x(Π_X, S)(g),

which thus implies that a = ev_x(Π_X, S)(g)⁻¹ ∈ k^× [cf. Lemma 1.8, (iii)]. In particular, we conclude that f =a·g ∈ O^×_X(X\S). This completes the proof of assertion (iii).

Finally, we verify assertion (iv). Write F ⊆ K^×(Π_X) for the subgroup generated by the various subgroups (a), (b) appearing in the statement of assertion (iv) and regard K_X⊗^×

kk as a subgroup of K^×(Π_X) by means of the injection of assertion (i). Then let us observe that it follows from assertion (iii) that F ⊆ K_X⊗^×

kk. Moreover, by considering the subgroups (b), one verifies immediately — in light of Lemma 1.11, (i), below — from assertion (iii), together with Lemma 1.8, (iii), that

k^× ⊆ F ⊆ K^×

X⊗kk.

In particular, by considering the subgroups (a), we conclude from assertion (iii), together with Lemma 1.8, (iii), that, for a rational function f ∈ K_X⊗^×

kk, if f((xi)_k) 6∈ {0,∞} for somei∈ {1,2}[where we write (x_i)_k∈(X⊗_kk)(k) for thek-valued point determined by x_i], then f ∈ F. Thus, the equality F = K_X⊗^×

kk follows immediately from Lemma 1.11, (ii), below. This completes the proof of assertion (iv).

LEMMA1.11. — Let Ω be an algebraically closed field; C a proper hyperbolic curve over Ω; x, y∈C(Ω) distinct Ω-valued points of C. Then the following hold:

(i) For every λ∈Ω\ {0,1}, there exists a rational functionf ∈K_C [cf. Definition1.1]

such that f(x) = 1 and f(y) = λ.

(ii) The multiplicative group K_C^× is generated by rational functions f ∈ K_C^× such that f({x, y})6⊆ {0,∞}.

Proof. — Assertion (i) follows immediately by considering, for instance, a suitable linear fractional transformation (ag +b)/(cg +d) [where a, b, c, d ∈ Ω] of a rational function g ∈ K_C^× such that g(x) 6=g(y) . Next, we verify assertion (ii). Write F ⊆K_C^× for the subgroup of K_C^× generated by rational functions f ∈ K_C^× such that f({x, y}) 6⊆

{0,∞}. To complete the verification of the equality F = K_C^×, let us take a rational function g ∈ K_C^× such that g({x, y}) ⊆ {0,∞}. Now, to verify g ∈ F, we may assume without loss of generality, by replacing g by g⁻¹ if necessary, that g(x) = ∞, i.e., that ordx(g)<0. Then one verifies immediately from the Riemann-Roch theorem that there exists a rational function h ∈ K_C^× such that ord_x(g) = ord_x(h) (= ord_x(h + 1)) and h(y) = 0 [i.e., (h+ 1)(y) = 1]. Thus, sinceg/(h+ 1), h+ 1∈ F, we conclude thatg ∈ F.

This completes the proof of assertion (ii).

DEFINITION1.12. — We shall write

∆^c-ab_X (respectively, ∆^c-cn_X )

for the maximal quotient of ∆_X such that the natural surjection ∆_X π₁(X^cpt⊗_kk) factors through the surjection ∆_X ∆^c-ab_X (respectively, ∆^c-cn_X ), and, moreover, the kernel of the resulting surjection ∆^c-ab_X (respectively, ∆^c-cn_X ) π₁(X^cpt⊗_kk) is pro-Primes^×/k and abelian (respectively, pro-Primes^×/k and contained in the center of ∆^c-cn_X ). We shall write

Π^c-ab_X (respectively, Π^c-cn_X )

for the quotient of ΠX by the kernel of ∆X ∆^c-ab_X (respectively, ∆^c-cn_X ). Thus, we have a commutative diagram of profinite groups

1 −−−→ ∆_X −−−→ Π_X −−−→ G_k −−−→ 1



 y



 y

1 −−−→ ∆^c-ab_X −−−→ Π^c-ab_X −−−→ G_k −−−→ 1



 y



 y

1 −−−→ ∆^c-cn_X −−−→ Π^c-cn_X −−−→ G_k −−−→ 1

— where the horizontal sequences are exact, and the vertical arrows are surjective.

LEMMA1.13. — The following hold:

(i) The natural surjections

Π_X Π^c-ab_X Π^c-cn_X determine isomorphisms

H¹(Π^c-cn_X ,Λ_X) −→^∼ H¹(Π^c-ab_X ,Λ_X) −→^∼ H¹(Π_X,Λ_X).

(ii) Suppose that X is proper [i.e., r_X = 0]. Let x₁, . . . , x_n ∈ X(k) be distinct k- rational points. For i∈ {1, . . . , n}, write U_i ^def= X\ {x_i}andU ^def= X\ {x₁, . . . , x_n}. Then

the natural open immersionsU ,→U_i — whereiranges over the elements of{1, . . . , n}— determine an isomorphism of profinite groups[well-defined up to∆^c-cn_U

1 ×_∆X· · · ×_∆X∆^c-cn_U

n - conjugation]

Π^c-cn_U −→^∼ Π^c-cn_U

1 ×_ΠX · · · ×_ΠX Π^c-cn_Un .

Proof. — Assertion (i) follows immediately from the various definitions involved. As- sertion (ii) follows immediately from a similar argument to the argument applied in the proof of the final portion of [4], Proposition 1.6, (iii).

2. Maximal Cuspidally Abelian Quotients

In the present §2, we discuss the maximal cuspidally abelian quotients “Π^c-ab_UX×

k X” [cf.

Definition 2.1, (ii), below] of the ´etale fundamental groups of the second configuration spaces of proper hyperbolic curves. In the present §2, we maintain the notation of the preceding §1. Suppose, moreover, that X isproper over k [i.e., r_X = 0]. Write

ΠX×_kX

def= π₁(X×_kX) ⊇ ∆_X×_kX

def= π₁((X×_kX)⊗_kk)

for the respective ´etale fundamental groups ofX×_kX, (X×_kX)⊗_kk. Then let us recall that the two projections X×_kX →X determine an isomorphism of profinite groups

Π_X×_kX

−→∼ Π_X ×_Gk Π_X, which restricts to an isomorphism of profinite groups

∆_X×_kX

−→∼ ∆_X ×∆_X. LetN be a Primes^×/k-integer. Write

Λ_X,N ^def= Λ_X/NΛ_X = Hom

bZ×/k H²(∆_X,Zb×/k),Z/NZ [cf. Definition 1.4; Remark 1.4.1, (ii)].

DEFINITION2.1.

(i) We shall writeU_X×_kX ⊆X×_kX for the second configuration space of X, i.e., the open subscheme of X×kX obtained by forming the complement of the diagonal divisor X ⊆X×kX. Thus, the natural inclusion UX×_kX ,→X×kX determines a commutative diagram of profinite groups

1 −−−→ π₁((UX×_kX)⊗_kk) −−−→ π₁(UX×_kX) −−−→ G_k −−−→ 1



 y



 y

1 −−−→ ∆X×_kX −−−→ ΠX×_kX −−−→ G_k −−−→ 1

— where the horizontal sequences are exact, and the vertical arrows are surjective.

(ii) We shall write

∆^c-(×/k)_UX×

k X (respectively, ∆^c-ab_UX×

k X; ∆^c-cn_UX×

k X)

for the maximal quotient ofπ₁((U_X×_kX)⊗_kk) such that the left-hand vertical arrow of the diagram of (i) factors through the surjection π₁((U_X×_kX)⊗_kk)∆^c-(×/k)_U

X×k X

(respectively,

∆^c-ab_UX×

k X; ∆^c-cn_UX×

k X), and, moreover, the kernel of the resulting surjection ∆^c-(×/k)_UX×

k X (respectively,

∆^c-ab_UX×

k X; ∆^c-cn_UX×

k X)∆X×_kX is pro-Primes^×/k (respectively, pro-Primes^×/k and abelian;

pro-Primes^×/k and contained in the center of ∆^c-cn_UX×

k X). We shall write Π^c-(×/k)_UX×

k X (respectively, Π^c-ab_UX×

k X; Π^c-cn_UX×

k X)

for the quotient ofπ₁(UX×_kX) by the kernel ofπ₁((U_X×_kX)⊗_kk)∆^c-(×/k)_U

X×k X (respectively,

∆^c-ab_UX×

k X; ∆^c-cn_UX×

k X). Thus, the diagram of (i) determines a commutative diagram of profi- nite groups

1 −−−→ ∆^c-(×/k)_U

X×k X

−−−→ Π^c-(×/k)_U

X×k X

−−−→ G_k −−−→ 1



 y



 y

1 −−−→ ∆^c-ab_U

X×k X −−−→ Π^c-ab_U

X×k X −−−→ G_k −−−→ 1



 y



 y

1 −−−→ ∆^c-cn_UX×

k X −−−→ Π^c-cn_UX×

k X −−−→ G_k −−−→ 1



 y



 y

1 −−−→ ∆_X×_kX −−−→ ΠX×_kX −−−→ G_k −−−→ 1

— where the horizontal sequences are exact, and the vertical arrows are surjective.

LEMMA 2.2. — Let x ∈ X(k) be a k-rational point. Write U ^def= X \ {x}. Then the splitting [well-defined up to ∆_X-conjugation] s: G_k → Π_X induced by x, i.e., “S_X(x)”, determines an isomorphism of profinite groups over G_k

Π^c-cn_U −→^∼ Π^c-cn_UX×

k X ×ΠX×k X (s(Gk)×G_kΠX) [cf. Definition 1.12].

Proof. — This follows immediately from a similar argument to the argument applied

in the proof of [4], Proposition 1.6, (iii), (iv).

LEMMA2.3. — Suppose that Π_X is slim, and that k isp-cyclotomically fullfor every p ∈ Primes^×/k [i.e., the image of the p-adic cyclotomic character of Gk is open in Z^×p

for every p∈Primes^×/k]. Then an extension of an open subgroup of eitherΠ_X or ΠX×_kX

by a [possibly empty] finite product of copies of Λ_X is slim.

Proof. — Let us first observe that since the profinite group Π_X×_kX

→∼ Π_X×_GkΠ_X has a structure of extension of ΠX by ∆X, and ∆X is slim, it follows that ΠX×_kX is slim. In particular, sincek isp-cyclotomically fullfor everyp∈Primes^×/k, it follows immediately from Remark 1.4.1, (i), that an extension of an open subgroup of either Π_X or Π_X×kX by a finite product of copies of Λ_X is slim. This completes the proof of Lemma 2.3.

REMARK 2.3.1. — It follows immediately from Proposition 1.5, (ii), together with a similar argument to the argument given in [6], Remark 1.5.1, that ifk isKummer-faithful, then the two assumptions in the statement of Lemma 2.3 are satisfied. In particular, in this situation, it follows from Lemma 2.3 that an extension of an open subgroup of either Π_X or ΠX×_kX by a [possibly empty] finite product of copies of Λ_X is slim.

DEFINITION2.4. — Let ∆⊆∆_X be acharacteristicopen subgroup of ∆_X and Π_Y ⊆Π_X an open subgroup of Π_X such that ∆_Y = Π_Y ∩∆_X = ∆. Write G_kY ⊆ G_k for the image of the composite Π_Y ,→ Π_X G_k. [Thus, the connected finite ´etale covering Y →X [corresponding to Π_Y ⊆Π_X] is a hyperbolic curve over the finite extension k_Y of k [corresponding to G_kY ⊆G_k].]

(i) By conjugation, we obtain an action Π_X → Aut(∆), hence also a semi-direct product ∆oΠ_X, which fits into an exact sequence of profinite groups

1 −→ ∆o∆_X −→ ∆oΠ_X −→ G_k −→ 1.

Observe that since ∆X is slim, it follows that ∆o∆X is slim.

(ii) By restricting the action Π_X →Aut(∆) of (i) to Π_Y ⊆Π_X, we obtain a semi-direct product ∆oΠ_Y. Then one verifies easily from the fact that ∆ is center-free that the centralizerZ_∆o∆Y(∆) ⊆∆o∆_Y determines a splittingof the exact sequence of profinite groups

1 −→ ∆ −→ ∆o∆_Y −→ ∆_Y −→ 1.

Thus, the natural surjections ∆o∆_Y (∆o∆_Y)/Z_∆o∆Y(∆), ∆_Y determine isomor- phisms of profinite groups

∆o∆_Y −→^∼ (∆o∆_Y)/Z_∆o∆Y(∆)

×∆_Y ←−^∼ ∆×∆_Y,

which arecompatible with the natural outer actions ofG_kY. In particular, by considering

“(−)^outo G_kY” [cf. the slimnessof ∆o∆_X discussed in (i)], we obtain an isomorphism of profinite groups

∆oΠ_Y −→^∼ Π_Y ×_GkY Π_Y (←−^∼ Π_Y×_kYY).

Observe that one verifies immediately from the various definitions involved that the natu- ral splitting of the surjection Π_Y×_kYY

→∼ ∆oΠ_Y Π_Y arises from thediagonal morphism Y ,→Y ×_kY Y.

(iii) We shall writeZ_∆ →X×_kX for the connected finite ´etale covering corresponding to the open subgroup

Π_Z∆ ^def= ∆oΠ_X ⊆ ∆_X oΠ_X →^∼ Π_X×_kX

— where the “→” is the isomorphism obtained in (ii). Thus, the exact sequence of (i)^∼ determines an exact sequence of profinite groups

1 −→ ∆_Z∆ ^def= Ker(Π_Z∆ G_k) −→ Π_Z∆ −→ G_k −→ 1.

Observe that one verifies immediately from the various definitions involved that the sur- jection Π_Z∆ = ∆oΠ_X Π_X arises from an “isotrivial” [cf. (ii)]hyperbolic curveZ_∆→X over X, and the natural splitting of the surjection Π_Z∆ = ∆oΠ_X Π_X arises from a

section ι_∆: X ,→ Z_∆ — that lies over the diagonal morphism X ,→X×_kX [cf. (ii)] — of this hyperbolic curve Z_∆→X.

REMARK2.4.1. — One verifies easily from the various definitions involved that, in the notation of Definition 2.4, if k is either an MLF or FF, then the finite ´etale covering Z_∆ →X×_kX of Definition 2.4, (iii), is the diagonal covering associated to the covering Y →X in the sense of [4], Definition 1.2, (i).

LEMMA2.5. — In the notation of Definition 2.4, the following hold:

(i) Let i6= 0 be an integer, G∈ {∆,Π_Z∆}, and A a finite G-module annihilated by a Primes^×/k-integer. Then

lim−→

H

Hⁱ(H, A) = {0}

— where the injective limit is taken over the open subgroups H ⊆G, and the transition morphisms in the limit are given by the restriction maps.

(ii) Let i 6= 2 be an integer and A a finite module equipped with the trivial action of

∆ that is annihilated by a Primes^×/k-integer. Then lim←−

H

Hⁱ(H, A) = {0}

— where the projective limit is taken over the open subgroups H⊆∆, and the transition morphisms in the limit are given by the corestriction maps.

(iii) Leti be an integer andA a finiteΠ_Z∆-module annihilated by aPrimes^×/k-integer.

Then the natural homomorphism

Hⁱ(Π_Z∆, A) −→ Hⁱ(Z_∆, A) is an isomorphism.

Proof. — Assertion (i) follows immediately from a similar argument to the argument applied in the proof of [1], Lemma 4.2, (iii). Next, we verify assertion (ii). Let us recall [cf., e.g., [4], Proposition 1.3, (ii)] that the homomorphism

Hⁱ(H, A) −→ Hom_b

Z×/k H²⁻ⁱ(H,ΛX), A

determined by the cup product in group cohomology and the natural isomorphism of Λ_X with “Λ_X” with respect to H [cf., e.g., [4], Remark 1] is an isomorphism. Thus, assertion (ii) follows immediately from assertion (i). This completes the proof of assertion (ii).

Assertion (iii) is a formal consequence of assertion (i) [cf., e.g., the proof of [1], Lemma

4.2, (iii)]. This completes the proof of Lemma 2.5.

LEMMA2.6. — In the notation of Definition 2.4, write

E₂^i,j(∆) = Hⁱ(ΠX, H^j(∆,ΛX,N)) =⇒ E^i+j(∆) = H^i+j(ΠZ_∆,ΛX,N) for the spectral sequence associated to the exact sequence of profinite groups

1 −→ ∆ −→ Π_Z∆(= ∆oΠ_X) −→ Π_X −→ 1.

Then the following hold:

(i) The natural homomorphism lim←−

∆^†

E₂^0,2(∆^†) −→ E₂^0,2(∆) (= H⁰(Π_X, H²(∆,Λ_X,N)) = Z/NZ)

— where the projective limit is taken over the characteristic open subgroups ∆^† ⊆ ∆_X contained in ∆, and the transition morphisms in the limit are given by the corestriction maps — is an isomorphism.

(ii) The natural homomorphism lim←−

∆^†

E²(∆^†) −→ lim

←−∆^†

E₂^0,2(∆^†)

— where the projective limits are taken over the characteristic open subgroups ∆^†⊆∆_X contained in ∆, and the transition morphisms in the limits are given by the corestriction maps — is an isomorphism.

(iii) The image of 1∈Z/NZ via the composite

Z/NZ = H⁰(Π_X, H²(∆,Λ_X,N)) = E₂^0,2(∆) ←−^∼ lim←−

∆^†

E₂^0,2(∆^†)

←−∼ lim←−

∆^†

E²(∆^†) −→ E²(∆) = H²(Π_Z∆,Λ_X,N) −→^∼ H²(Z_∆,Λ_X,N)

— where the first “←” is the isomorphism of^∼ (i), the second “←” is the isomorphism of^∼ (ii), and the “→” is the isomorphism of Lemma^∼ 2.5,(iii)—coincideswith the first Chern classc₁(ι_∆(X))of the divisorι_∆(X)⊆Z_∆obtained by forming the scheme-theoretic image of the section ι_∆: X ,→Z_∆ of the hyperbolic curve Z_∆→X [cf. Definition 2.4, (iii)].

Proof. — First, we verify assertion (i). Let us recall [cf., e.g., [4], Proposition 1.3, (ii)]

that the homomorphism

(E₂^0,2(∆^†) =) H²(∆^†,ΛX,N) −→ Hom_b

Z×/k H⁰(∆^†,ΛX),ΛX,N

(= Z/NZ) determined by the cup product in group cohomology and the natural isomorphism of Λ_X with “Λ_X” with respect to ∆^† [cf., e.g., [4], Remark 1] is an isomorphism. Thus, assertion (i) follows immediately from the various definitions involved. Assertion (ii) follows immediately from Lemma 2.5, (ii). Assertion (iii) follows immediately from the [easily verified] fact that the image of the compatible system

(c₁(ι_∆^†(X)))_∆^† ∈ lim←−

∆^†

E²(∆^†)

[cf. Lemma 2.5, (iii)] via the composite of natural homomorphisms lim←−

∆^†

E²(∆^†) −→ lim

←−∆^†

E₂^0,2(∆^†) −→ E₂^0,2(∆) = Z/NZ

coincides with 1∈Z/NZ. This completes the proof of Lemma 2.6.