MAPPING-CLASS-GROUP ACTION VERSUS GALOIS ACTION ON PROFINITE FUNDAMENTAL GROUPS

MAKOTO MATSUMOTO AND AKIO TAMAGAWA

Abstract. Let X be an algebraic curve of genus g, n-punctured, defined over a number field K. Then, the profinite or the pro-l completion of the topological fundamental group of X admits two actions; the action of the profinite completion of the mapping class group of the orientable surface of topological type (g, n) and the action of the absolute Galois group of K. This paper compares these two. In the profinite case, it is shown that the intersection of the images of these two actions is trivial if X is affine and its fundamental group is non-abelian. On the contrary, in the pro-l case, there are many curves such that the image of the Galois action contains the image of the mapping-class-group action. It is proved that the set of points corresponding to such curves is dense in the moduli space of (g, n)-curves.

1. Mapping-class-group action and Galois action

Let X be a smooth geometrically connected curve of genus g, n-punctured, defined over a number field K ⊂ ℂ (simply called a (g, n)-curve over K). We assume that each puncture is a K-rational point in the smooth compactification X^{∗} of X.

Let Π denote the fundamental group of X(ℂ) in the classical topological sense. The mapping class group of type (g, n), denoted by Γ_{g,n}, is defined to be the group of isotopy classes of orientation preserving self-diffeomorphisms of X^{∗}(ℂ) which fix the n points pointwise. This definition induces a natural homomorphism ρ_{Γ} : Γ_{g,n} → Out(Π) := Aut(Π)/Inn(Π), which is known to be injective.

Let Π̂ denote the profinite completion of Π. Since Π is finitely generated, Out(Π̂) is profinite, and hence ρ_{Γ} induces

ρΓ̂ : Γ̂_{g,n} → Out(Π̂).

On the other hand, there is an (outer) action of the absolute Galois group G_{K} of K

ρ_{G_{K},X} : G_{K} → Out(Π̂).

This is deduced from the following diagram:

1 →  Π̂  → π_{1}^{alg}(X) → G_{K} → 1
↓  ↓  ↓
1 → Inn(Π̂) → Aut(Π̂) → Out(Π̂) → 1,

where the middle arrow is the conjugation on the normal subgroup Π̂ (see [7]). Here Aut denotes the group of continuous automorphisms. Note that ρ_{G_{K},X} has enough information to distinguish the algebraic structure of X [30][16][17], while ρ_{Γ}, and hence ρΓ̂, depend only on the topological type of X.

Composing with Out(Π̂) → Out(Π^{l}), where Π^{l} is the pro-l completion of Π, we obtain ρ_{G_{K},X}^{l} : G_{K} → Out(Π^{l}) and ρl ; Γ̂ : Γ̂_{g,n} → Out(Π^{l}).

The main results of this paper are the following two theorems, which show that opposite phenomena occur in the profinite case and in the pro-l case.

Theorem 1.1. The intersection of the image of ρ_{G_{K},X} with the image of ρΓ̂ is trivial, if X is a (g, n)-curve with 2g − 2 + n ≥ 1 and n ≥ 1.

We conjecture that the condition n ≥ 1 is superfluous.

On the contrary, in the pro-l case, we prove

Theorem 1.2. Fix any (g, n) with 2g −2+n ≥ 1, and consider the following property on a curve X: X is a (g, n)-curve over K and the image of ρ_{G_{K},X}^{l} contains the image of ρl ; Γ̂. Then, the set of points on the moduli space of (g, n)-curves corresponding to such curves, with K not fixed, is dense with respect to the complex topology.

If the moduli space of (g, n)-curves is a rational variety, then most (g, n)-curves over any fixed number field K satisfy this property, i.e., it holds outside a thin set in the K-rational points of the moduli in the sense of Hilbert’s irreducibility theorem.

2. Profinite case

A key to prove Theorem 1.1 is the following (Theorem 2.2 in [15], the case of n = 1). Let us denote by ℙ_{01∞}^{1} the projective line minus three points 0, 1, ∞.

Theorem 2.1. Let L be any subfield of ℂ. Let X be a smooth geometrically connected (g, n)-curve over L with 2g−2+n ≥ 1, with at least one L-rational puncture. Then, the kernel of ρ_{G_{L},X} is contained in the kernel of ρ_{G_{L},ℙ_{01∞}^{1}}.

Note that the statement is still true for pro-l case.

The other essential ingredient, which is not true for pro-l case, is the following Theorem by Belyǐ[3].

Theorem 2.2. ρ_{G_{ℚ},ℙ_{01∞}^{1}} is injective.

Note that from Theorems 2.1 and 2.2, it follows G_{K} ⊂ Out(Π̂).

The following is due to Takayuki Oda.

Lemma 2.1. Let M_{g,n} denote the moduli stack of (g, n)-curves. Then, we have the following exact sequence:

1 → π_{1}^{alg}(X̄) → π_{1}^{alg}(M_{g,n+1}) → π_{1}^{alg}(M_{g,n}) → 1.

(See [22] for the fundamental group of stacks.)

Proof. It is enough to prove the exactness of its geometric part

1 → π_{1}^{alg}(X̄) → π_{1}^{alg}(\overline{M_{g,n+1}}) → π_{1}^{alg}(\overline{M_{g,n}}) → 1.

This is the profinite completion of the well-known exact sequence

1 → Π → Γ_{g,n+1} → Γ_{g,n} → 1,

and since Π̂ is center free, the exactness is preserved by the profinite completion (see [1]).

Proof of Theorem 1.1.

Proof. Let B be a geometrically connected variety over K, and C → B be a smooth family of (g, n)-curves. This means that there is a family of proper smooth (g, 0)- curves C^{∗} → B over K, there are n sections s_{1}, s_{2}, …, s_{n} : B → C^{∗} such that the union of their images is etale over B, and C is the complement of the sections in C^{∗}. Assume that there is an x ∈ B(K), whose fiber is X. Then, we have a short exact sequence

1 → π_{1}^{alg}(B̄, x̄) → π_{1}^{alg}(B, x̄) → G_{K} → 1,

where B̄ := B ⊗ K̄. This sequence splits by the section s_{x} : G_{K} → π_{1}^{alg}(B, x̄). Let

ρ : π_{1}^{alg}(B, x̄) → Out(Π̂)

be the monodromy of C → B with base point x̄. This is given by the short exact sequence

1 → π_{1}^{alg}(X̄) → π_{1}^{alg}(C) → π_{1}^{alg}(B) → 1.  (2.1)

Here, the exactness on the left follows from Lemma 2.1, by considering the map C → M_{g,n+1} induced by the classifying map for C → B.

We define ρ̄ to be the restriction of ρ to its geometric part π_{1}^{alg}(B̄, x̄), and ρ_{x} := ρ ∘ s_{x}. Note that ρ_{x} = ρ_{G_{K},X}.

Let L be the function field of B, and fix an embedding L ↪ ℂ. Then, the absolute Galois group G_{L} of L acts on π_{1}(Cη̄), where η is the generic point of B and η̄ is its geometric point with respect to L ⊂ ℂ. It factors through G_{L} ↠ π_{1}(B, η̄) ≅ π_{1}(B, x̄) → OutΠ̂, by the exactness of (2.1). Let us take an element in the intersection of the image of ρ̄ and ρ_{x}, and let it be ρ̄(z) = ρ_{x}(σ). Let z̃ be an element of G_{L} mapped to z and σ̃ mapped to s_{x}(σ) in π_{1}^{alg}(B, x̄). Then ρ̄(z)ρ_{x}(σ^{−1}) = 1 says that z̃σ̃^{−1} is in the kernel of ρ_{G_{L},X_{L}}. By Theorem 2.1, it lies in the kernel of ρ_{G_{L},ℙ_{01∞}^{1}_{L}}. By the following commutativity and by Belyi’s Theorem 2.2, we have σ = 1.

z̃σ̃^{−1} ∈ G_{L} → Out(π_{1}^{alg}ℙ_{01∞}^{1}L̄)
\overline{↓} ↓ ||
σ^{−1} ∈ G_{K} → Out(π_{1}^{alg}ℙ_{01∞}^{1}K̄).

Thus, the intersection of the image of ρ̄ and the image of ρ_{G_{K},X} is trivial. Theorem 1.1 follows from this. Take a sufficiently large n′ such that the moduli stack M_{g,n′} of (g, n′)-curves lies in the category of scheme. Such n′ exists by Knudsen [13]. Take B to be this moduli space. Then, π_{1}^{alg}(B̄) surjects to the fundamental group of the moduli stack π_{1}^{alg}(\overline{M_{g,n}}). But the image of ρ̄, i.e., the image of ρΓ̂ intersects trivially with the image of ρ_{G_{K},X}, as proved above.

Remark 2.1. Let Out^{∗}(Π) be the subgroup of Out(Π) consisting of the elements stabilizing the conjugacy class of the inertia group at each puncture. Then Γ_{g,n} ⊂ Out^{∗}(Π) is of index two (c.f. [32, Theorem 5.7.1]), and any orientation-reversing diffeomorphism together with Γ_{g,n} generates Out^{∗}(Π). Since ρ_{G_{K},X} is injective, the above theorem can be group theoretically stated as follows. If σ ∈ G_{K} ⊂ Out(Π̂) can be approximated by elements in Out^{∗}(Π), then it is the unit or a conjugate of the complex conjugation. This is because σ^{2} must lie in the image of ρΓ̂, hence must be trivial, and then σ is trivial or a conjugate to the complex conjugation [2].

Remark 2.2. The above proof says that the kernel of ρ coincides with the kernel of ρ̄, which is given in a purely group theoretic way. We do not know whether it is injective or not. This problem can be considered as an analogue of the congruence subgroup problem: does each finite index subgroup of the mapping class group contain the kernel of Γ_{g,n} → Out(Π) → Out(Π/N), for some finite index characteristic subgroup N of Π? M. Asada proved the injectivity for g = 0, 1 and general n (an article is in preparation).

3. Pro-l case

We begin with group theoretical preliminaries.

Definition 3.1. A profinite group G having a finite-index pro-l subgroup P is said to be almost pro-l.

A subgroup and a quotient group of almost pro-l groups are again almost pro-l.

Lemma 3.1. If an almost pro-l group G is finitely generated, then it has a finite index normal subgroup H with the following property:

For any profinite group Γ and any homomorphism Γ → G, if Γ → G → G/H is surjective then Γ → G is surjective.

Proof. Take a pro-l group P ⊂ G in Definition 3.1. We may assume P to be normal in G by replacing it with the intersection of all the conjugates in G if necessary. Being a finite index subgroup of finitely generated group, it is finitely generated. Take the Frattini subgroup of P, i.e., put H := [P, P]P^{l}. (C.f. Proposition in §10.6 in [29].)

Lemma 3.2. For any finitely generated pro-l group G, Aut(G) and Out(G) is almost pro-l.

Proof. It is enough to show this for Aut(G), and then it follows from Corollary 7 in [1].

Consider

ρ^{l} : π_{1}^{alg}(B) → Out(Π̂) → Out(Π^{l})

in the same situation as in the proof of Theorem 1.1, i.e., C → B is a smooth family of (g, n)-curves, where B is a geometrically connected variety over K.

Lemma 3.3. The image of ρ^{l} in Out(Π^{l}) is finitely generated.

Proof. Take the section s_{x} : G_{K} → π_{1}^{alg}(B). Then ρ^{l}(π_{1}^{alg}(B)) is generated by ρ^{l}(π_{1}^{alg}(B̄)) and ρ^{l}s_{x}(G_{K}). It is sufficient to prove that each of these is finitely generated. The former is finitely generated, since so is π_{1}^{alg}(B̄). For the latter we proceed as follows. Take a finite extension L/K, so that ρ^{l}(G_{L}) is pro-l. By smooth base change theorem [7, Expose XIII], ρ^{l} ∘ s_{x} : G_{L} → Out(Π^{l}) ramifies only at the primes 𝔭 of O_{L} such that 𝔭 divides l or X has bad reduction at 𝔭. Thus, only finitely many primes may ramify. Let S be this finite set of primes. Then, ρ^{l}s_{x} factors through G_{L} → G_{L,S} := π_{1}^{alg}(SpecO_{L} − S). By class field theory, l-Frattini quotient of G_{L,S}, namely G_{L,S}/([G_{L,S}, G_{L,S}]G_{L,S}^{l}), is finite, and so is the l-Frattini quotient of ρ^{l}s_{x}(G_{L}). Thus, ρ^{l}s_{x}(G_{L}) is finitely generated, and so is ρ^{l}s_{x}(G_{K}).

Proposition 3.1. There is a finite etale connected cover

B′ → B

with the following property. Let L be a finite extension of K and y be an L-rational point of B. If y is unramified in B′ → B and the fiber above y is connected, then

ρ^{l}(π_{1}^{alg}(B)) = ρ^{l}s_{y}(G_{L}).

Proof. Let G be ρ^{l}(π_{1}^{alg}(B)) ⊂ Out(Π^{l}). By Lemma 3.3, one may use Lemma 3.1 for G to obtain H ⊂ G. Take the inverse image of H by the surjection ρ^{l} : π_{1}^{alg}(B) → G, and let B′ be the cover of B corresponding to H. Thus B′ is connected. By the connectedness of the fiber above y, we know that

G_{L} →^{s_{y}} π_{1}^{alg}(B_{L}) →^{ρ^{l}} G → G/H

is surjective. This and Lemma 3.1 showed that ρ^{l}s_{y}(G_{L}) ⊂ G = ρ^{l}(π_{1}^{alg}(B)) is an isomorphism.

Proof of Theorem 1.2.

Proof. Let n′ ≥ n be a sufficiently large integer such that M_{g,n′} is a scheme. Let B be an open affine subvariety of M_{g,n′}, and let C → B be the family of (g, n)- curves induced from B → M_{g,n}. Then, the dimension of B is N := 3g − 3 + n′. Apply Proposition 3.1 to obtain B′ → B. Take any quasi-finite dominant morphism B → ℙ^{N}. By Hilbert’s irreducibility theorem, there is a thin set ⊂ ℙ^{N}(K), outside which every fiber in B′ → B → ℙ^{N} is connected and unramified. We denote by S ⊂ ℙ^{N}(K) the complement of the thin set. Take any point p outside this thin set, and take a (unique) point y in B above p. Let L be the field k(y). Then, the curve C_{y}/L obtained as the fiber of C → B at y satisfies the unramified-connectedness condition of Proposition 3.1, thus (C_{y}, L) satisfies the condition of Theorem 1.2, because π_{1}^{alg}(B̄) surjects to π_{1}^{alg}(\overline{M_{g,n}}). Now S is dense in ℙ^{N}(ℂ) with respect to the complex topology. Then, since B(ℂ) → ℙ^{N}(ℂ) is locally isomorphic outside a closed subscheme, the set of geometric points lying in the inverse image of S is dense in B(ℂ), which completes the proof of the first statement in the theorem.

If M_{g,n} is rational, we take a rational open subvariety B ⊂ M_{g,n} which is an open subvariety of ℙ^{N}. In this case, the above argument shows that every y ∈ B(K) outside a thin set gives (C_{y}, K) with the desired property.

4. Problems and Motivations

4.1. Problems. In Theorem 1.2, there are curves which do not satisfy the conclusion. Suppose that the automorphism group Aut_{K}(X) is nontrivial. Then, the image ρ_{G_{K},X}^{l}(G_{K}) must commute with the image of Aut_{K}(X). The centralizer of nontrivial element in Γ_{g,n} is much smaller than Γ_{g,n}, (except for (g, n) = (2, 0) with the hyperelliptic involution and (1, 1) with the elliptic involultion, c.f. [10]). Sometimes it is even trivial (see [4]). Thus the conclusion of the theorem can never hold for such curves. What can be said on the locus of the curves satisfying this property? For example, we can consider the following problem.

Problem 4.1. Is the locus in the moduli space consisting of the curves X such that ρ_{G_{K},X}^{l}(G_{K}) contains an open subgroup of ρl ; Γ̂(Γ̂_{g,n}) independent of the choice

of prime l? Moreover, for any such curve X, does ρ_{G_{K},X}^{l}(G_{K}) contain ρl ; Γ̂(Γ̂_{g,n}) for all primes l but finite?

This locus is not Zariski open. For example, assume (g, n) = (1, 1). If an elliptic curve E has a potential complex multiplication (CM), then the Galois action on the abelianization of π_{1}^{alg}(Ē) does not contain the action of SL_{2}(ℤ_{l}) (which is the image of the action of Γ̂_{1,1} on the abelianization). Since CM points are dense, the locus is not open. If (g, n) = (1, 0), then by Serre’s Theorem [28], the Galois action coincides with the action of an open subgroup of GL_{2}(ℤ_{l}) if and only if the elliptic curve is not CM. Thus, the locus is independent of l.

Problem 4.2. Is it possible that the centralizer of Aut_{K}(X) in the image of ρΓ̂ (or ρl ; Γ̂) becomes trivial? If this occurs, it would be interesting to consider the centralizer as an approximation of the image of G_{K}.

4.2. Motivations. Motivations of this study came from two works by Takayuki Oda.

Monodromy of the inertia for bad reduction. Oda proved that the action of the inertia can be described by Dehn twists (see (2.7) in [25], c.f. [23]).

Theorem 4.1. Let l be a prime, A be a mixed characteristic strictly henselian discrete valuation ring with residue characteristic prime to l with maximal ideal 𝔭. Let X → A be a family of stable curves, with generic fiber smooth and having stable bad reduction at 𝔭. Let L be the quotient field of A. Then, the monodromy

π_{1}^{alg}(SpecL) → Out(π_{1}^{l}(Xη̄))

factors through π_{1}^{l}(SpecL) ≅ ℤ_{l}(1) and the image is generated by a (nontrivial) product of Dehn twists along the vanishing cycles modulo 𝔭.

This gives an explicit element in the intersection of the mapping class group action and the Galois action. Stimulated from this result, we considered which mapping class can be realized by the Galois action. Our Theorem 1.2 says that in many cases any mapping class can be realized.

Oda’s conjecture. It is natural to consider how much larger ρ_{G_{K},X}^{l}(G_{K}) is than ρl ; Γ̂(Γ̂_{g,n}), in the case where the conclusion of Theorem 1.2 holds. Takayuki Oda conjectured the following (c.f. [26], and several seminars including those at RIMS around 1992), which motivated this study. For the definitions and properties of the filtrations, refer to [24].

Assume that B is the moduli stack of (g, n)-curves and C is the universal (g, n)- curves over B in §3 (c.f. [13][22]). Let Π^{l} be the pro-l fundamental group of a (g, n)-curve. Then, Π^{l} has a natural central filtration called weight filtration. For a group Γ, a group homomorphism ρ : Γ → OutΠ^{l} induces induced filtration on Γ, Γ ⊃ Γ[0] ⊃ Γ[1] ⊃ Γ[2] ⊃ · · ·. We denote

grΓ := ⨁_{m≥1} gr_{m}Γ := ⨁_{m≥1} Γ[m]/Γ[m + 1],

which has a natural structure of a Lie algebra. By ρ^{l} : π_{1}^{alg}(M_{g,n}) → OutΠ^{l}, we obtain induced filtrations π_{1}^{alg}(M_{g,n})[m], π_{1}^{alg}(\overline{M_{g,n}})[m]. We define G_{ℚ}[g, n, l; m] to be the image of π_{1}^{alg}(M_{g,n})[m] by π_{1}^{alg}(M_{g,n}) → G_{ℚ}. Its graded quotients also yield a Lie algebra, which we denote grG_{ℚ}[g, n, l].

Conjecture 4.1. Assume 2g − 2 + n ≥ 1. Then, the filtration G_{ℚ}[g, n, l; m] does not depend on the choice of g, n.

¿From this, Oda also conjectured the existence of obstructions to the surjectivity of Johnson homomorphisms (introduced by Johnson[11][12], and named in [18]) other than the Morita’s obstructions[19]. Conjecture 4.1 is nearly solved [15][20][9]:

(1) ∩_{m}G_{ℚ}[g, n, l; m] is independent of g, n, (2) G_{ℚ}[g, n, l; m] and G_{ℚ}[g′, n′, l; m] are commensurable; i.e., their intersection is open in each group. The existence of the obstruction to the surjectivity of Johnson morphism was also proved in [20].

Thus, the graded Lie algebra grG_{ℚ}[g, n, l] ⊗_{ℤ_{l}} ℚ_{l} is independent of (g, n). Note that grG_{ℚ}[0, 3, l] ⊗ ℚ_{l} is the Galois realization of the “motivic” graded Lie algebra (c.f. §5 in [8], and [5]), which is conjecturally isomorphic to a free Lie algebra generated by a countably many generators, one for each gr_{4k+2}G_{ℚ}[0, 3, l], k = 1, 2, 3, … (see calculative results in [14][31]). Oda’s conjecture gives a possibility to study this Lie algebra by using general (g, n).

There are researches on a Hodge-realization of this Lie algebra, e.g. [6]. It seems an interesting problem to consider a Hodge analogue of Oda’s conjecture.

Note that ρ_{G_{K},X} gives an induced filtration G_{K}[m] on G_{K}. If G_{K}[0]/G_{K}[1] is GSp(g, ℤ_{l}) and G_{K}[m]/G_{K}[m + 1] is nontrivial for some m ≥ 1, then Galois centralizer in Out(Π^{l}) is finite (see [21, Theorem 3.3] and its proof). Theorem 1.2 shows that for infinitely many curves the graded quotient of the induced filtration of G_{K} is bounded below, because so is the filtration of Γ_{g,n} [27]. Hence this theorem implies the finiteness of the Galois centralizer for infinitely many curves, but now this finiteness is a consequence of more general pro-l Grothendieck conjecture for curves [16][17].

The profinite version of Oda’s conjecture seems to have no sense (holds trivially without filtration, and no way to give a suitable filtration) in the presence of Theorem 1.2.

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Max Planck Institut Für Mathematik, Gottfried-Claren-Strasse 26 D-53225 BONN, GERMANY. (Until Sept. 1997). Department of Mathematics, Keio University, 3-14-1 Hiyoshi Kohoku-ku Yokohama 223 Japan

   E-mail address: matumoto@math.keio.ac.jp

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-01, Japan

   E-mail address: tamagawa@kurims.kyoto-u.ac.jp