On the fundamental group of a smooth arithmetic surface

Alexander Schmidt and Kay Wingberg Dedicated to Helmut Koch on his 60th birthday

Let X_{K} be a geometrically connected smooth and proper curve over a local or global field K. Following Grothendieck [3] there is a canonical exact sequence for the (etale) fundamental group π_{1}(X_{K}) of X_{K}.(Here and in the following we will omit the base points.)

1⟶π_{1}(XK̄)⟶π_{1}(X_{K})⟶G_{K}⟶1

where K̄ denotes an algebraic closure of K and G_{K} = G(K̄/K) is the absolute Galois group of K. The structure of the geometric fundamental group π_{1}(XK̄) is completely known for the part prime to the characteristic of K (see [3] X Cor. 3.10 ). In particular for a prime l ≠ char(K) the maximal pro-l factor group π_{1}(XK̄)(l) of π_{1}(XK̄) is a l-Poincaré group of dimension 2, i.e. the cohomology groups of this group have the well known duality properties. In this paper we are interested in duality properties of the maximal pro-l factor of the arithmetic fundamental group π_{1}(X_{K}). It is easy to see that the sequence

1⟶π_{1}(XK̄)(l)⟶π_{1}(X_{K})(l)⟶G_{K}(l)⟶1

remains exact for l ≠ char(K) if the l-torsion points of the Jacobian of X_{K} are K-rational. As a consequence we obtain

Theorem 1 Let K be a local field and let l be a prime number different to char(K). Assume that X_{K} has genus g ≥ 1 and that the l-torsion points of the Jacobian of X_{K} are K-rational. Then π_{1}(X_{K})(l) is a Poincaré group of dimension 4 with dualizing module ℚ_{l}/ℤ_{l}(2) = lim ; ⟶_{n}μ_{l^{n}}^{⊗2}, i.e. for all i ≥ 0 and all

finite l-primary π_{1}(X_{K})(l)-modules M the cup product

H^{i}(π_{1}(X_{K})(l), M) × H^{4−i}(π_{1}(X_{K})(l), Hom(M, ℚ_{l}/ℤ_{l}(2)))

⟶^{∪}H^{4}(π_{1}(X_{K})(l), ℚ_{l}/ℤ_{l}(2))⟶^{∼}ℚ_{l}/ℤ_{l}

induces a perfect pairing of finite groups.

This follows from the fact, that π_{1}(XK̄)(l) and G_{K}(l) are Poincaré groups of dimension 2 with dualizing module ℚ_{l}/ℤ_{l}(1) = μ_{l^{∞}} and the general properties of Poincaré groups in group extensions (see [10],[17],[12]). We remark that the last result extends in a natural way to the case where K is a higher dimensional local field (see [12],[17]). Now let K be a global field. Then G_{K}(l) is no longer a Poincaré group not even a duality group (resp. strict Cohen-Macaulay) [15].(For the definition of a duality group see below or [15] [16] [10]. Roughly speaking, such groups satisfy an analogous duality property but the dualizing module is not necessarily isomorphic to ℚ_{l}/ℤ_{l} as an abelian group. In general it has an infinite ℤ_{l}-Corank.) As G_{K}(l) itself is not strict Cohen-Macaulay it is natural to look for factor groups which are of this type. Let S be a finite nonempty set of primes of K. Assume that in the number field case S includes the set of all primes over l (denoted by S_{l}) and the set of all infinite primes (denoted by S_{∞}) and that l ≠ char(K) for function fields. Then the Galois group G_{S}(l) = Gal(K_{S}(l)/K) of the maximal l-extension of K unramified outside S is a duality group of dimension 2 if a few weak additional conditions on K and S are fulfilled. In particular this is true if S is big enough. (See [18] [19] and below for statements and proofs.) We will prove an analogous result as in Theorem 1 for smooth proper curves over

O_{S} = ⋂_{℘∉S}O_{℘}.

Theorem 2 Let K be a global field, let l be a prime (l ≠ char(K)) and let X_{K} be a geometrically connected smooth and proper curve over K having a K-rational point. Assume that the l-torsion points of the Jacobian of X_{K} are K-rational. Let S be a finite set of primes of K including at least two primes, S ⊃ S_{l} ∪ S_{∞} if K is a number field and including all primes where X_{K} has bad reduction. Let X_{O_{S}} be a smooth and proper model of X_{K} over Spec(O_{S}) (i.e. X_{K} is the general fiber of X_{O_{S}}). Then the following holds

(i) There is a canonical exact sequence

1⟶π_{1}(XK̄)(l)⟶π_{1}(X_{O_{S}})(l)⟶G_{S}(l)⟶1 .

Moreover the sequence splits.

(ii) Assume that the genus of X_{K} is not zero and that the following condition holds:

- if K is a number field then K is Galois over ℚ , #(S_{l}(K)) > 1 and K is not real if l = 2. Then π_{1}(X_{O_{S}})(l) is a duality group of dimension 4 with dualizing module

I = T or_{l}C_{S}(K_{S}(l))(1) ,

i.e. I is the first Tate twist of the l-torsion part of the S-idele class group of K_{S}(l). Explicitly this means that for all i ≥ 0 and all finite l-primary π_{1}(X_{O_{S}})(l)-modules M the cup product

H^{i}(π_{1}(X_{O_{S}})(l), M) × H^{4−i}(π_{1}(X_{O_{S}})(l), Hom(M, I))

⟶^{∪}H^{4}(π_{1}(X_{O_{S}})(l), I)⟶ℚ_{l}/ℤ_{l}

induces a perfect pairing of finite groups.

Furthermore we obtain results about the structure of the arithmetic fundamental groups in the local case in terms of generators and relations.

1

For the definition of etale fundamental groups see [3] V or [6] I§5 . In the sequel denoting these groups we will always omit the base point. Let l be a prime and let X be any connected scheme. Using the Hochschild-Serre spectral sequence [6] III 2.20 we get

Lemma 1.1 The canonical map

H^{i}(π_{1}(X), ℤ/lℤ) ⟶ H_{et}^{i}(X, ℤ/lℤ)

is bijective for i = 0, 1 and injective for i = 2.

For a profinite group G and a prime number l we denote the maximal pro-l-factor group by G(l).

Proposition 1.2 Let K be a local or global field and X_{K} a geometrically connected smooth and proper curve over K. Assume that the l-torsion points of the Jacobian of X_{K} are K-rational, where l is a prime number different to char(K). Then there is a canonical exact sequence

1⟶π_{1}(XK̄)(l)⟶π_{1}(X_{K})(l)⟶G_{K}(l)⟶1 .

Proof: Observe that if X_{K} has genus 0 the result follows trivially as π_{1}(XK̄) = 0. Let the genus of X_{K} be nonzero. We denote the Jacobian of X_{K} by J. Using the autoduality of J and the Weil-pairing

J_{l} × J_{l}⟶μ_{l}

the condition on the l-torsion points implies μ_{l} ⊂ K. By [3] X Cor.3.4. we have an exact sequence

π_{1}(XK̄)(l)⟶π_{1}(X_{K})(l)⟶G_{K}(l)⟶1

and hence a surjection

π_{1}(XK̄)(l) −_{↠} π_{1}(X_{K(l)})(l) = ker(π_{1}(X_{K})(l)⟶G_{K}(l))

where K(l) denotes the maximal l-extension of K. Let C be its kernel. We will show that in the Hochschild-Serre sequence (with coefficients ℤ/lℤ)

0⟶H^{1}(π_{1}(X_{K(l)})(l))⟶^{α}H^{1}(π_{1}(XK̄)(l))

⟶H^{1}(C)π_{1}(XK̄)(l)⟶H^{2}(π_{1}(X_{K(l)})(l))⟶^{β}H^{2}(π_{1}(XK̄)(l))

α is an isomorphism and β is injective.Then it follows that

H^{1}(C)π_{1}(XK̄)(l) = 0

hence C = 0 as C is a pro-l-group. The groups on the left are isomorphic to P ic^{∘}(X_{K(l)})_{l} and P ic^{∘}(XK̄)_{l} respectively. As the Brauer group of K(l) has no l-torsion ([13] II Prop.9) the exact sequence ([7] Remark 1.6)

0⟶P ic^{∘}(X_{K(l)})⟶J(K(l))⟶Br(K(l))

shows that by our assumption both groups are just the l-torsion points of the Jacobian

J(K(l))_{l} = J(K)_{l} = J(K̄)_{l} .

We know by [13]II Prop.9 that cd_{l}(Gal(K̄/K(l)) ≤ 1. So we obtain the commutative and exact diagram

0  0
↓  ↓
H^{2}(π_{1}(X_{K(l)})(l)) → H^{2}(π_{1}(XK̄)(l))
↓  ↓
H^{1}(Gal(K̄/K(l)), H^{1}(π_{1}(XK̄))) → H^{2}(π_{1}(X_{K(l)})) → H^{2}(π_{1}(XK̄))

But H^{1}(π_{1}(XK̄)) is a trivial G_{K}-module showing that

H^{1}(Gal(K̄/K(l)), H^{1}(π_{1}(XK̄))) = Hom(Gal(K̄/K(l)), H^{1}(π_{1}(XK̄))) = 0

by the defining property of K(l). This completes the proof. □

2

Definition Let l be a prime number. A profinite group G is called a l-duality group of dimension n (D_{l}^{n}-group for short) if

(i) cd_{l}(G) = n

(ii) For every finite l-torsion G-module M the cup product

H^{i}(G, M) × H^{n−i}(G, Hom(M, I_{l}))⟶^{∪}H^{n}(G, I_{l})⟶^{tr}ℚ_{l}/ℤ_{l}

induces a perfect pairing of finite groups for every i where I_{l} is the ldualizing module of G and tr : H^{n}(G, I_{l})⟶ℚ_{l}/ℤ_{l} is the canonical trace map. G is called a Poincaré group at l if in addition I_{l} is isomorphic to ℚ_{l}/ℤ_{l} as an abelian group.

Remark: The following holds

I_{l} = lim ; ⟶_{{U,cor^{∗} \atop m}}H^{n}(U, ℤ/l^{m}ℤ)^{∗} ,

where the limit is taken with respect to the l-multiplication and the dual of the corestriction map and U runs through all open subgroups of G.

Fundamental for our purposes is the following proposition

Proposition 2.1 Let

1⟶N⟶G⟶Q⟶1

be a short exact sequence of profinite groups. If N is a D_{l}^{n}-group and Q is a D_{l}^{m}-group then G is a D_{l}^{n+m}-group and we have

I_{l,G}^{∗} ≅ I_{l,N}^{∗} ⊗_{ℤ_{l}} I_{l,Q}^{∗} .

For a proof see [10] Theorem 3.9. and [17] Theorem 1.4. □

The groups we are concerned with are of the above type:

For l ≠ char(K) the group π_{1}(XK̄)(l) is as a pro-l-group generated by 2g (g the genus of X_{K}) generators x_{1}, ⋯, x_{2g} with the one relation [x_{1}, x_{2}] ⋯ [x_{2g−1}, x_{2g}]. If g ≥ 1 it is a 2-dimensional Poincaré group, where I_{l} ≅ ℚ_{l}/ℤ_{l} has trivial action.

For a proof see [3] X Cor.3.10 or [20] for a purely algebraic approach. Furthermore we have the classical result:

Let K be a local field and let l be prime to char(K). Assume that μ_{l} ⊂ K. Then G_{K}(l) is a Poincaré group of dimension 2 having ℚ_{l}/ℤ_{l}(1) as it’s dualizing module.

For a proof see e.g. [13]II§2 Th.4. and Ex.2.

Now we are ready to proof theorem 1. Using proposition 1.2 and proposition 2.1 the only thing left to do is determining the dualizing module. Let I_{X} denote the dualizing module of π_{1}(XK̄)(l). As it is a trivial π_{1}(XK̄)(l)-module we only have to control the action of G_{K}(l) on it. To every open subgroup U of π_{1}(XK̄)(l) corresponds a smooth complete curve Y over K̄ lying finite and etale over XK̄ so that U ≅ π_{1}(Y)(l). Consequently we have for all n

H^{1}(U, ℤ/l^{n}ℤ) ≅ H^{1}(Y, ℤ/l^{n}ℤ) ≅ H^{1}(Y, μ_{l^{n}})(−1)

Using the Kummer sequence and the fact that H^{2}(Y, 𝔾_{m}) = 0 (see [6] III §2 2.22 d ), we obtain

P ic(Y)/P ic(Y)^{l^{n}} ≅ H^{2}(Y, μ_{l^{n}}) .

But via the degree map the left term is just ℤ/l^{n}ℤ as a trivial G_{K}-module. By the explicit description of I (see the remark after the definition of a duality group) we see that I_{X} is isomorphic to ℚ_{l}/ℤ_{l}(1) as π_{1}(X_{T})(l)-module. Using the formula given in proposition 2.1 we get the desired result. Thus theorem 1 is proved.□

Let T be a one dimensional connected and regular scheme, K the function field of T, X_{T} a connected proper smooth curve over T, where the general fiber X_{K} of X_{T} is a geometrically connected (smooth,proper) curve over K, XK̄ the base change of X_{K} to the algebraic closure K̄ of K, K(X_{T}) = K(X_{K}) be the function field of X_{T}, K_{T} (l) the composite of all finite Galois l-extensions of K where the

normalization of T lies etale over T, T̃(l) the normalization of T in K_{T} (l), X_{T̃(l)} the base change of X_{T} to T̃(l).

We know that the finite coverings of a connected normal scheme are just the normalizations in certain extensions of the function field (see [6] I Th. 3.21) Going to the inverse limit we conclude

Lemma 3.1 We have the exact sequence

1⟶π_{1}(X_{T̃(l)})(l)⟶π_{1}(X_{T})(l)⟶π_{1}(T)(l)⟶1

If X_{K} has a K-rational point the above sequence splits.

Proof: The only remaining point is the splitting of the sequence. But if X_{K} has a K-rational point then X_{T} has a section over T by the valuative criteria of properness.□

Now let K be a global field,X_{K} a geometrically connected smooth and proper curve over K, l prime to char(K) and S a finite set of places of K including all primes where X_{K} has bad reduction and containing S_{l} ∪ S_{∞} in the number field case and containing at least two elements in the function field case ( we only need that K_{S}(l)/K is not a constant field extension see proposition 4.2 below). Let T be the affine resp. complete curve corresponding to the number field resp. function field K with the points of S removed. So T = Spec(O_{S}), with O_{S} = ⋂_{℘∉S} O_{℘}. Let X_{T} be a smooth (and proper) model of X_{K} over T.

Observe that π_{1}(T)(l) is nothing else but the Galois group G_{S}(l) of the maximal l-extension of K unramified outside S. Now we will identify the left group in the exact sequence of lemma 3.1 with the maximal pro-l factor of the geometric fundamental group of X_{K}.

Proposition 3.2 Assume that X_{K} has a K-rational point and that the l-torsion points of the Jacobian of X_{K} are K-rational. Then the canonical morphism XK̄ → X_{T̃(l)} induces an isomorphism π_{1}(XK̄)(l) → π_{1}(X_{T̃(l)})(l) .

For the proof we need the following lemma.

Lemma 3.3 Let f : X ⟶ Y be a morphism of connected schemes over ℤ[1/l][μ_{l}]. Assume Y to be regular and quasicompact and that the following conditions hold

(i) the map Γ(Y, O_{Y})^{∗}/l ⟶ Γ(X, O_{X})^{∗}/l is an isomorphism,

(ii) the map P ic(Y)_{l} ⟶ P ic(X)_{l} is an isomorphism,

(iii) the map P ic(Y)/l ⟶ P ic(X)/l is injective,

(iv) H^{2}(K(Y), 𝔾_{m})_{l} = 0

then the induced map π_{1}(X)(l)⟶^{f_{∗}(l)}π_{1}(Y)(l) is an isomorphism.

Proof: From [6]III Prop.4.11. we get the following diagram with exact rows

0 → Γ(Y, O_{Y})^{∗}/l → H_{et}^{1}(Y, ℤ/lℤ) → P ic(Y)_{l} → 0
↓  ↓  ↓
0 → Γ(X, O_{X})^{∗}/l → H_{et}^{1}(X, ℤ/lℤ) → P ic(X)_{l} → 0

and by our assumptions the middle vertical arrow is an isomorphism. Now let C be the cokernel of f_{∗}(l). Using the Hochschild-Serre spectral sequence and lemma 1.1 we get the exact sequence (cohomology with values in ℤ/lℤ)

1 → H^{1}(C) → H_{et}^{1}(Y) →^{∼} H_{et}^{1}(X)

and hence C has to be trivial. Let now D be the kernel of f_{∗}(l). Again using the Hochschild-Serre spectral sequence we get

0 → H^{1}(D)^{π_{1}(Y)(l)} → H^{2}(π_{1}(Y)(l)) → H^{2}(π_{1}(X)(l))
[figure: ∩; ∩]
H^{2}(π_{1}(Y)) → H^{2}(π_{1}(X))
[figure: ∩; ∩]
H_{et}^{2}(Y)  →^{α}  H_{et}^{2}(X)

Thus if α is injective we are done. But using the Kummer sequence we get the following diagram

0 → P ic(Y)/l → H_{et}^{2}(Y, ℤ/lℤ) → H_{et}^{2}(Y, 𝔾_{m})_{l} → 0
↓  ↓^{α}  ↓
0 → P ic(X)/l → H_{et}^{2}(X, ℤ/lℤ) → H_{et}^{2}(X, 𝔾_{m})_{l} → 0

By assumption Y is regular,connected and quasicompact so we have the inclusion H_{et}^{2}(Y, 𝔾_{m}) ⊂ H^{2}(K(Y), 𝔾_{m}) (see [6]III 2.22 ) and by conditions (iii) and (iv) the lemma is proved.□

Proof of the proposition: By the same argument as in the proof of proposition 1.2 we get μ_{l} ⊂ K. So we only have to check the conditions of lemma 3.3.

(i) Let p : X_{T̃(l)} → T̃(l) be the canonical projection. By similar argument as in [8] Remark 20.9 we get p_{∗}(O_{X_{T̃(l)}}) = O_{T̃(l)}. One easily checks that Γ(T̃(l), O_{T̃(l)}) is l-divisible. Hence both groups under consideration are zero. (iv) Clearly K(X_{T̃(l)}) = K_{S}(l)K(X_{K}) Looking at the field tower

\overline{K(X_{K})}
|
K̄K(X_{K}) − K̄
|^{I_{S}}  |^{I_{S}}
K_{S}(l)K(X_{K}) − K_{S}(l)

(Observe that K is algebraically closed in K(X_{K}) as X_{K} is geometrically irreducible.) We obtain the exact sequences

H^{2}(I_{S}, μ_{l})

↓

0 → H^{2}(I_{S}, (K̄K(X_{K}))^{∗})_{l} → Br(K_{S}(l)K(X_{K}))_{l} → Br(K̄K(X_{K}))_{l}

↓

0

Since cd_{l}(I_{S}) ≤ 1 ( see [13] II 3.3. Prop. 9 ) the left term is zero and by Tsen’s theorem the right term is zero too. Hence Br(K(X_{T̃(l)}))_{l} = 0. (ii) Claim: P ic(T̃(l))/l = P ic(T̃(l))_{l} = 0.

Proof of the claim: In the number field case this is proved in [1] appendix . The same proof shows P ic^{∘}(T̃(l))/l = P ic^{∘}(T̃(l))_{l} = 0 in the function field case. Looking at the exact sequence of the degree map(see [14])

1⟶P ic^{∘}(T̃(l))⟶P ic(T̃(l))⟶ℚ

in our situation the image in ℚ is equal to ℤ[1/l] hence uniquely l-divisible and so the claim is proved. Now we have

P ic(X_{T̃(l)})_{l} = P ic^{∘}(X_{T̃(l)})_{l} = (P ic^{∘}(X_{T̃(l)})/P ic(T̃(l)))_{l} = P_{X_{T}}^{∘} (T̃(l))_{l} ,

where P_{X_{T}}^{∘} denotes the relative Picard functor. Let J be the generalized Jacobian of X_{T̃(l)} (see [7] §8 for the definition). Observe that by the valuative criteria of properness X_{T̃(l)} has a section over T̃(l). So we get by [7]

P_{X_{T}}^{∘} (T̃(l)) ⟶^{∼} J (T̃(l))
↓  ↓^{b}
P ic^{∘}(X_{K_{S}(l)}) ⟶^{∼} J(K_{S}(l))

But J is just the general fiber of J and by the valuative criteria of properness the map b is an isomorphism. As the l-torsion points of J are already K-rational we have

P_{X_{T}}^{∘} (T̃(l))_{l}⟶^{∼}P ic^{∘}(X_{K_{S}(l)})_{l}⟶^{∼}P ic^{∘}(XK̄)_{l} = P ic(XK̄)_{l} .

(iii) By an argument in [7] §1 we have a degree map on P ic(X_{T̃(l)}) which commutes with the degree map on the general fiber. Taking the isomorphism b and P ic(T̃(l))/l = P ic(T̃(l))_{l} = 0 into consideration we get

P ic(X_{T̃(l)})/l⟶^{∼}P ic(X_{K_{S}(l)})/l

Hence it remains to show that P ic(X_{K_{S}(l)})/l ≅ P ic(XK̄)/l . Observe that the degree map P ic(X_{K_{S}(l)})⟶^{deg}ℤ is surjective as X_{K} has a K-rational point. So it is enough to show that P ic^{∘}(X_{K_{S}(l)}) is l-divisible. By [9] §6 lemma 6.1. we know that P ic^{∘}(X_{K_{S}}) is l-divisible. Now let I_{S}^{l} denote the Galois group Gal(K_{S}/K_{S}(l)). By definition I_{S}^{l} has no nontrivial homomorphisms to l-groups. Taking I_{S}^{l}-invariants of the exact sequence

0⟶P ic(X_{K})_{l}⟶P ic^{∘}(X_{K_{S}})⟶^{·l}P ic^{∘}(X_{K_{S}})⟶0

we get

P ic^{∘}(X_{K_{S}(l)})⟶^{·l}P ic^{∘}(X_{K_{S}(l)})⟶Hom(I_{S}^{l}, P ic(X_{K})_{l}) = 0 □

Summarizing we have proved the first part of theorem 2.

4

To make the proof of theorem 2 complete we need the results on the maximal l-extension unramified outside a finite set S of primes of a global field that we already mentioned in the introduction. As before S = S(K) includes S_{l} ∪ S_{∞} for a number field K.

Proposition 4.1 Let K be a finite Galois extension of ℚ containing the group μ_{2l}.Suppose that #S_{l}(K) ≥ 2. Then G_{S}(l) is a D_{l}^{2}-group with dualizing module T or_{l}C_{S}(K_{S}(l)). Moreover G_{S}(l) is a Poincaré group of dimension 2 with dualizing module μ_{l^{∞}} if and only if there are exactly two finite primes in K_{S}(l) lying above S.

Proof: The second assertion is just the corollary in [18] and proposition 3 in [18] shows that T or_{l}C_{S}(K_{S}(l)) is isomorphic to μ_{l^{∞}} in this case. Suppose that G_{S}(l) is not a D_{l}^{2}-group ( hence not of Poincaré-type ). Then there exists a prime ν_{0} ∈ S_{l}(K) such that

G(K_{S}(L)_{ν_{0}}/K_{ν_{0}}) ≅ G_{S}(l) ≅ ( ⋆_{{ν∈S_{f} (K) \atop ν≠ν_{0}}} G_{ν})⋆F

where G_{ν} = G(K_{ν}(l)/K_{ν}) and F is a free pro-l group, [18] Theorem A. Since K is assumed to be Galois over ℚ all decomposition groups for all ν/l are isomorphic. Therefore, because #S_{l}(K) ≥ 2, the free group F has to be trivial and

S\S_{∞}(K) = S_{l}(K) = {ν_{0}, ν_{1}} .

Hence

G_{S}(l) ≅ G_{ν_{0}} ≅ G_{ν_{1}}

is of Poincaré type in contradiction to the assumption. The determination of the dualizing module has been carried out in [19]. □

Proposition 4.2 Let K be a function field in one variable over the finite field k of characteristic p. Suppose that l ≠ p and that μ_{l} ⊂ k. Then

(i) G_{S}(l) ≅ ℤ_{l} if S consists exactly of one point which is undecomposed in the maximal constant field l-extension and (l, h_{K}) = 1 where h_{K} is the class number of K

(ii) G_{S}(l) is a D_{l}^{2}-group with dualizing module T or_{l}C_{S}(K_{S}(l)) in all other cases.

Proof: Let Ω be the maximal (everywhere) unramified l-extension of K. It is easy to see that Ω/K is a constant field extension if and only if (l, h_{K}) = 1.

(i) Assume (l, h_{K}) = 1. Let S = {P_{0}} where P_{0} is a point remaining undecomposed in the maximal constant field l-extension (e.g. P_{0} is a k-rational point). For a point P of K we denote the completion of K at P by K_{P} and the unit group of K_{P} by U_{P}. Let

\begin{matrix}V^{S}(K) =^{def} \{α ∈ K^{×}| & α_{P} ∈ U_{P} · K_{P}^{×l} for all P \\ & α_{P_{0}} ∈ K_{P_{0}}^{×l} \end{matrix} \} .

For α ∈ V^{S}(K) the extension K( ^{l} \sqrt{α})/K is unramified and decomposed at P_{0}. But by our assumption it has to be constant field extension hence not a proper extension i.e. V^{S}(K) = K^{×l}. Following Koch [5] Satz 11.5, 11.8 we have in our situation

\begin{aligned}dim_{ℤ/lℤ}H^{1}(G_{S}(l), ℤ/lℤ) & = & dim_{ℤ/lℤ}V^{S}(K)/K^{×l} + 1 & = & 1 \\ dim_{ℤ/lℤ}H^{2}(G_{S}(l), ℤ/lℤ) & ≤ & dim_{ℤ/lℤ}V^{S}(K)/K^{×l} & = & 0 \end{aligned}

this proves part (i). (ii) It is easy to see that K_{S}(l)/K is not a constant field extension in the remaining cases. Denoting the maximal constant field l-extension of K by K_{∞} we get the canonical exact sequence

1⟶Gal(K_{S}(l)/K_{∞})⟶G_{S}(l)⟶ℤ_{l}⟶1 .

The left group is free and of finite rank (see [20] Satz 1.7.) and nonzero in our situation, hence a D_{l}^{1}-group and by Prop. 2.1 G_{S}(l) is a D_{l}^{2} group. It remains to determine the dualizing module. Since scd_{l}(G_{S}(l)) = 2 (see [20] Satz 1.6.) the dualizing module is T or_{l}D_{2}(ℤ) (see [15], [16]), where D_{2}(ℤ) =^{def} lim ; ⟶_{{U⊂G_{S} (l) \atop U open}}H^{2}(U, ℤ)^{∗}.

Using the exact sequence

1⟶C_{S}(K)⟶G_{S}^{ab}⟶_{ℤ̂}/ℤ⟶1

(see [9] page 66) we get

T or_{l}D_{2}(ℤ) =^{def} T or_{l} lim ; ⟶_{{U⊂G_{S} (l) \atop U open}}H^{2}(U, ℤ)^{∗}
= T or_{L} lim ; ⟶_{{U⊂G_{S} (l) \atop U open}}H^{1}(U, ℚ/ℤ)^{∗}
= lim ; ⟶_{{U⊂G_{S} (l) \atop U open}}T or_{l} U^{ab}
= T or_{l}C_{S}(K_{S}(l))

This finishes the proof of proposition 4.2. □

Now the second part of theorem 2 is a simple consequence of the above propositions and the first part, which is already proved. The only point open is the action of G_{S}(l) on the dualizing module of π_{1}(XK̄) but here we are in full analogy to the local case already handled within the proof of theorem 1. Hence the proof of theorem 2 is complete.□

5

Now we want to give a description of the fundamental groups in terms of generators and relations. Doing this in full generality seems to be very hard, so we will only treat some special cases. First we need a technical lemma which will guarantee the minimality of the given representations. For a pro-l group G we denote the minimal number of generators by e(G) and the minimal number of relations by r(G). It is well known that we have

e(G) = dim_{ℤ/lℤ}H^{1}(G, ℤ/lℤ) , r(G) = dim_{ℤ/lℤ}H^{2}(G, ℤ/lℤ) .

Lemma 5.1 Let

1⟶U⟶G⟶G/U⟶1

be an exact sequence of pro-l-groups. Assume that the following conditions hold (unless otherwise specified all cohomology groups have coefficients in ℤ/lℤ)

(i) the sequence splits

(ii) H^{1}(U) is a trivial G/U-module (iii) cd_{l}(G/U) ≤ 2

(iv) the cup product

∪ : H^{1}(U) × H^{1}(U)⟶H^{2}(U)

is surjective.

Then we can calculate e(G) and r(G) as follows

\begin{aligned}e(G) & = & e(U) & + & e(G/U) \\ r(G) & = & r(U) & + & r(G/U) & + & e(U)e(G/U) \end{aligned}

Proof: In the Hochschild-Serre spectral sequence

E_{2}^{pq} = H^{p}(G/U, H^{q}(U)) ⟹ H^{p+q}(G)

the E_{2}^{pq} terms are zero for p ∉ {0, 1, 2} by condition (iii). By [4] Ch.3. Th.4 the differential

\begin{aligned}d_{2} : & E_{2}^{p−1,1} & ⟶ & E_{2}^{p+1,0} \\ & H^{p−1}(G/U, Hom(U, ℤ/lℤ)) & ⟶ & H^{p+1}(G/U, ℤ/lℤ) \end{aligned}

is just −c∪? , i.e. the cup product pairing with -c, where c is the cohomology class in H_{cont.}^{2}(G/U, U/[U, U]) which corresponds to the group extension

1⟶U/[U, U]⟶G/[U, U]⟶G/U⟶1 .

Hence by condition (i) these diffentials are zero. The only remaining differential that is interesting to us is now d_{2} : E_{2}^{0,2} → E_{2}^{2,1}. Using the identity

d(u ∪ v) = d(u) ∪ v − u ∪ d(v) (see[4])

condition (iv) and the fact that the differential on E_{2}^{0,1} vanishes yield that this differential is zero too. So we have E_{2}^{pq} = E_{∞}^{p,q} for all p + q ≤ 2. If we now compute the ℤ/lℤ-vector space dimensions with respect to the filtrations given by the Hochschild-Serre spectral sequence on the cohomology groups of G the desired results fall out.(Observe that H^{2}(U) is a trivial G/U-module by conditions (ii) and (iv).)□

Let K be a local field, (O_{K}, ℘) it’s ring of integers and k it’s residue field. Let X_{K} be a geometrically connected smooth curve of genus g over K, having a Krational point and so that that l-torsion points of the Jacobian of X_{K} are rational where l ≠ p = char(k).

Theorem 3 Assume that X_{K} has good reduction at ℘ (denoted by X_{k}) and that H^{1}(X_{k}, ℤ_{l}) (l-adic cohomology) decomposes into 1-dimensional eigenspaces for the Frobenius action. Then for l odd π_{1}(X_{K})(l) is generated by 2g + 2 generators σ, τ, x_{1}, … x_{2g} with 4g + 2 defining relations

[x_{1}, x_{2}] ⋯ [x_{2g−1}, x_{2g}] = 1
[x_{i}, τ] = 1  all i
τ^{q−1}[τ, σ] = 1
x_{i}^{a_{i}−1}[x_{i}, σ] = 1 mod[π̄, π̄]^{l^{m}}[π̄, π̄, π̄] i even
x_{i}^{a_{i}−1}[σ, x_{i}] = 1 mod[π̄, π̄]^{l^{m}}[π̄, π̄, π̄] i odd

where q = #k, {a_{1}, … a_{2g}} are the eigenvalues of the Frobenius action on H^{1}(X_{k}, ℤ_{l}),m = min_{i} ν_{l}(a_{i} − 1) ( we always have m ≥ 1 ) , σ is a Frobenius lift, τ is a lift of a generator of the cyclic inertia group of ℘ and π̄ = π_{1}(XK̄)(l).

Remark: For g = 0, 1 the above relations are obviously exact and not only given modulo commutators.

For the proof we need the following propositions.

Proposition 5.2 For g ≥ 2 the group π_{1}(XK̄)(l) has trivial center.

Proof: See [2] §3 corollary.

Proposition 5.3 Let l be an odd prime number and let

1⟶D⟶G⟶Γ⟶1

be an exact sequence of pro-l groups where Γ is isomorphic to ℤ_{l} and D is a Poincaré group of dimension 2 and rank 2g. Suppose that D^{ab} is torsion free and that

D_{Γ}^{ab} ≅ ⨁_{i=1}^{2g} ℤ/b_{i}ℤ , ν_{l}(b_{i}) ≥ 1 .

Let m = min_{i} ν_{l}(b_{i}). Then G is a Poincaré group of dimension 3 and there are 2g + 1 generators x_{0}, … x_{2g} of G with 2g + 1 defining relations

\begin{aligned}r_{o} & = & ∏_{i=1,3,...,2g−1}[x_{i}, x_{i+1}] \\ r_{i} & = & \{ \begin{matrix}x_{i}^{b_{i}}[x_{i}, x_{0}] \\ x_{i}^{b_{i}}[x_{0}, x_{i}] \end{matrix} mod [D, D]^{l^{m}}[D, D, D], & \begin{matrix}i > 0 even \\ i odd \end{matrix} \end{aligned}

Proof: From Proposition 2.1 we know that G is of Poincaré type, hence

dimH^{2}(G, ℤ/lℤ) = dimH^{1}(G, ℤ/lℤ)
= dimH^{1}(D, ℤ/lℤ)^{Γ} + dimH^{1}(Γ, ℤ/lℤ)
= 2g + 1 .

For shortness we put H^{i}(−) = H^{i}(−, ℤ/l^{m}ℤ). As H^{i}(D) for i = 1, 2 are trivial Γ-modules we obtain the following exact diagram of perfect pairings

H^{1}(D)  H^{1}(D)  H^{2}(D)
∥  ∥  ∥
H^{1}(D)_{Γ} × H^{1}(D)^{Γ} ⟶^{∪} H^{2}(D)_{Γ}
[figure: ∩; φ; res; ↓^{≀}]
H^{2}(G) × H^{1}(G) ⟶^{∪} H^{3}(G)
[figure: res; ∪; inf; ∥]
H^{2}(D) × H^{1}(Γ) ⟶^{∪} H^{3}(G)

where the exact sequence on the left is part of the Hochschild-Serre spectral sequence. Observe that inf H^{1}(Γ) = (φH^{1}(D))^{⊥} as both ℤ/l^{m}ℤ-modules are free of rank 1. Let {χ_{i} , i = 1, … 2g} ⊂ H^{1}(G) be lifts of a symplectic basis of H^{1}(D) such that

χ_{i} ∪ χ_{i+1} = 1 = −χ_{i+1} ∪ χ_{i} , i = 1, 3, … 2g − 1

and all other pairs are zero.

Furthermore it is possible to choose these characters in the following way: {resχ_{i}} is the dual basis of {x_{i} mod D^{l^{m}}[D, D] , i = 1, …, 2g} of D/D^{l^{m}}[D, D] and Γ operates on the generators x_{i} of D by

x_{i}^{x_{0}} ≡ x_{i}^{b_{i}} mod [D, D]

where x_{0} is a generator of Γ. ( Decompose H^{1}(D) into eigenspaces for the action of Γ and observe that

x_{0} (χ ∪ χ′) = q (χ ∪ χ′)

where the principal unit q is given by the action of Γ on the dualizing module of D.) This is possible for arbitrary m ≥ 1. Let χ_{0} ∈ H^{1}(G) be such that χ_{0}(x_{0}) = 1 , χ_{0}(x_{i}) = 0 , i ≥ 1 . Let ϱ_{0} ∈ H^{2}(G) be such that ϱ_{0} ∪ χ_{0} = 1 and ϱ_{0} ∈ < χ_{1}, …, χ_{2g} >^{⊥}, then resϱ_{0} is a generator of H^{2}(D). Let

φ(resχ_{i}) = \{ \begin{matrix}ϱ_{i+1} & , & i odd \\ ϱ_{i−1} & , & i even \end{matrix} ; then
H^{2}(G) =< ϱ_{0}, ϱ_{1}, …, ϱ_{2g} >

and

ϱ_{i} ∪ χ_{i} = \{ \begin{matrix}1 & , & i = 0, 2, …, 2g \\ −1 & , & i = 1, 3, …, 2g − 1 \end{matrix}

and all other pairs are zero.

Let

1⟶R⟶F ⟶G⟶1

be a minimal presentation of G by a free pro-l group F of rank 2g + 1 and a normal subgroup R of F. Obviously R ⊂ F^{l^{m}}[F, F]. Now consider the two commutative diagrams

H^{1}(R)^{F} × R/R^{l^{m}}[R, F] ⟶^{∪} ℤ/l^{m}ℤ
↓^{≀ tg}  ↑^{≀ tg^{∗}}  ∥
H^{2}(G) × H^{2}(G)^{∗}  ⟶^{∪} ℤ/l^{m}ℤ
∥  ↑^{≀ φ}  ∥
H^{2}(G) ×  H^{1}(G)  ⟶^{∪} ℤ/l^{m}ℤ

and

∏_{k}φ_{k} : H^{1}(R)^{F} ⟶^{∼} ∏_{k}ℤ/l^{m}ℤ , α ↦ (α ∪ (−1)^{k}r_{k})_{k}
↑^{≀ tg^{−1}}  ↑^{≀ tg∘φ}
H^{2}(G) ⟶^{∼} ∏_{k}ℤ/l^{m}ℤ , ϱ ↦  (ϱ ∪ χ_{k})_{k}

where tg denotes the transgression map,φ is the duality isomorphism and {r_{k}}, r_{k} ∈ R/R^{l^{m}}[R, F], is a dual basis of {tg^{−1}ϱ_{k}}. It follows that

tg^{−1} (χ_{i} ∪ χ_{j})(r_{k}) = φ_{k} (χ_{i} ∪ χ_{j}) = χ_{i} ∪ χ_{j} ∪ χ_{k} =: c_{ijk}

If

δ : H^{1}(G)⟶H^{2}(G)

is the Bockstein operator induced by the exact sequence

1⟶ℤ/l^{m}ℤ⟶ℤ/l^{2m}ℤ⟶ℤ/l^{m}ℤ⟶1

then

φ_{k} (δχ_{i}) = δχ_{i} ∪ χ_{k} =: b_{ik}

According to Koch [5] 7.23, 7.24 we have

r_{k} = ∏_{i=0}^{2g} x_{i}^{l^{m}b_{ik}} ∏_{i<j} [x_{i}, x_{j}]^{c_{ijk}} mod G^{(3,l^{m})}

where G^{(3,l^{m})} = G^{l^{2m}}[G, G]^{l^{m}}[G, G, G]. By our special choice of the basis {χ_{0}, …, χ_{2g}} we obtain the following relations:

r_{o} ≡ ∏_{i=1,3,...,2g−1}[x_{i}, x_{i+1}] mod G^{(3,l^{m})} ∩ [D, D, D]
r_{i} ≡ x_{i+1}^{b_{i+1}}[x_{i+1}, x_{0}] mod G^{(3,l^{m})} ∩ [D, D, D] , i odd
r_{i} ≡ x_{i−1}^{b_{i−1}}[x_{0}, x_{i−1}] mod G^{(3,l^{m})} ∩ [D, D, D] , i even i ≥ 2

By the well known procedure of changing generators we can make the relation r_{0} exact without changing the structure of the relations r_{i} i ≥ 1. This concludes the proof of the proposition. □

Proof of Theorem 3 : Keeping the notations of section 3 let T be Spec(O_{K}) and X_{T} be a model of X_{K} over T. Then as already proved we have exact sequences

1 → π_{1}(XK̄)(l) → π_{1}(X_{K})(l) → G_{K}(l) → 1
∥  ↓  ↓
1 → π_{1}(XK̄)(l) → π_{1}(X_{T})(l) → π_{1}(T)(l) → 1
↓^{≀}
G_{k}(l)

Furthermore we know that G_{K}(l) is generated by σ, τ where σ is any lift of the Frobenius and τ is any generator of the inertia group and the one relation τ^{q−1}[τ, σ] (q = #k, for a proof see e.g. [5]). Claim: We find a section S : G_{K}(l) → π_{1}(X_{K})(l) so that [S(τ), α] = 1 for all α ∈ π_{1}(XK̄)(l). By the existence of a K-rational point on X_{K} we have at least one section, i.e. we find lifts s, t of σ, τ satisfying the one relation t^{s} = t^{q}. By the above diagram we know that t acts on π̄ = π_{1}(XK̄)(l) as an inner automorphism. In the case g = 1 we are ready so let g ≥ 2. Then by proposition 5.2 there is exactly one θ ∈ π̄ inducing the same inner automorphism like t. Define t′ = tθ^{−1}. Then [t′, α] = 1 for all α ∈ π̄. Hence [t, θ] = 1. Now θ^{s} acts as an inner automorphism of π̄ like t^{s} = t^{q}. Hence θ^{s} = θ^{q} and so the relation between s and t is not destroyed when we change t to t′. Furthermore we now can change to any lift of σ again without destroying the relation. At this point we are exactly in the situation of proposition 5.3 and so the proof of the theorem is complete. □

6

Now we will handle a special case of a complete surface over a finite field. Keeping the notations of section 3 we state:

Theorem 4 Let T be a complete smooth curve of nonzero genus over the finite field k of characteristic different to l and let X_{T} be a geometrically connected smooth and proper curve over T with general fiber X_{K}. Assume that the l-torsion points of the Jacobian of T are k-rational and that the l-torsion points of the Jacobian of X_{K} are K-rational. Furthermore assume that X_{K} has a K-rational point. Then the following holds:

(i) There is a canonical exact sequence

1⟶π_{1}(XK̄)(l)⟶π_{1}(X_{T})(l)⟶π_{1}(T)(l)⟶1 .

Moreover the sequence splits.

(ii) If the genus of X_{K} is nonzero then π_{1}(X_{T})(l) is a Poincaré group of dimension 5 having ℚ_{l}/ℤ_{l}(2) as dualizing module, i.e. for all i ≥ 0 and all finite l-primary π_{1}(X_{T})(l)-modules M the cup product

H^{i}(π_{1}(X_{T})(l), M) × H^{5−i}(π_{1}(X_{T})(l), M^{∗}(2))

⟶H^{5}(π_{1}(X_{T})(l), ℚ_{l}/ℤ_{l}(2))⟶^{∼}ℚ_{l}/ℤ_{l}

induces perfect pairings of finite groups.(M^{∗} denotes Hom(M, ℚ/ℤ) carrying the canonical module structure.)

Remark: One can generalize this theorem in two directions. For once one can built higher towers of curves proper and smooth one above the other, on the other side one can change to other ground fields having good duality properties e.g. higher dimensional local fields.

Proof of theorem 4: The proof of (i) is completely the same as the proof of the first part of theorem 2 given in section 3. The only point is that P ic(T̃(l)) is uniquely l-divisible ( this is needed in the proof of proposition 3.2). This is proved in [14] 1. Just in the same way as in the proof of theorem 1 one can show that π_{1}(T)(l) is a Poincaré group of dimension 3 having ℚ_{l}/ℤ_{l}(1) as dualizing module. One can also cite this from [14] Th.1. The determination of the π_{1}(T̃(l))-module structure of the dualizing module of π_{1}(XK̄)(l) goes the same way as in the other cases already handled and so also part (ii) is proved.□

One can again ask for a representation in terms of generators and relations. For simplicity we work over an algebraically closed field, leaving the difficult Frobenius action (see section 5) aside. Then we can state

Proposition 6.1 Let T be a complete smooth curve of nonzero genus g over an algebraically closed field k (char(k) ≠ l) and let X_{T} be a geometrically connected smooth and proper curve over T with general fiber X_{K}. Assume that X_{K} is an elliptic curve with K-rational l-torsion points. The the following holds

(i) π_{1}(X_{T})(l) is a Poincaré group of dimension 4 having the trivial module ℚ_{l}/ℤ_{l} as it’s dualizing module.

(ii) For l ≠ 2, 3 the fundamental group π_{1}(X_{T})(l) is as a pro-l group generated

by the 2g + 2 generators

x_{1}, x_{2}, y_{1}, ⋯, y_{2g}

and the 4g + 2 relations

[x_{1}, x_{2}] = 1
[x_{i}, y_{j}] = 1 for all i, j
[y_{1}, y_{2}] ⋯ [y_{2g−1}, y_{2g}] = 1

Proof: (i) Analogous to the proof of the first part of Theorem 2. (ii) In view of the exact sequence

1 → π_{1}(XK̄)(l) → π_{1}(X_{T})(l) → π_{1}(T)(l) → 1

and the information we already have it is enough to show that π_{1}(T)(l) acts trivially on π_{1}(XK̄)(l). Following Shafarevich [11] Theorem 6 there is an elliptic curve E over k, a finite group G ⊂ Aut(E) and a finite Galois covering S/T with Galois group G so that

X_{S} = X_{T} ×_{T} S ≅ E ×_{k} S

By [3] X.Cor.1.7. we have π_{1}(S ×_{k} E) = π_{1}(S) × π_{1}(E) and by [3] X Cor.1.8. π_{1}(E) and π_{1}(XK̄) are isomorphic. So we get the following diagram

1 ⟶ π_{1}(E) ⟶ π_{1}(E) × π_{1}(S) ⟶ π_{1}(S) ⟶ 1
↓^{≀}  ↓  ^{∩}
1 ⟶ π_{1}(XK̄) ⟶  π_{1}(X_{T})  ⟶ π_{1}(T) ⟶ 1
[figure]
G

Hence the action of π_{1}(T) factors through G and we get the following commutative diagram

π_{1}(T) → G → Autπ_{1}(XK̄)
[figure: ↓]
π_{1}(T)(l) ⟶ Autπ_{1}(XK̄)(l)

But by it’s definition G has order divisible at most by 2 and 3. Hence for l ≠ 2, 3 the image of π_{1}(T) in Autπ_{1}(XK̄)(l) must be trivial. □

References

[1] A.Brumer Galois groups of extensions of algebraic number fields with given ramification Mich.Math.J. 13(1966) 33-40

[2] Mamoru Asada and Masanobu Kaneko On the automorphism groups of some pro-l-fundamental groups Advances Studies in Pure Mathematics 12(1987)137-159

[3] A.Grothendieck Revètements Etales et Groupe Fondamental (SGA 1) LNM 224 , Springer Berlin 1971

[4] G.Hochschild,J.P.Serre Cohomology of group extensions Trans.Am.Math.Soc.74(1953) 110-134

[5] H.Koch Galoissche Theorie der p-Erweiterungen Dt. Verl. d. Wiss. Berlin 1970 [6] J.S. Milne Etale Cohomology

Princeton University Press 1980

[7] J.S.Milne Jacobian Varieties in Arithmetic Geometry edited by G.Cornell and J.H.Silverman Springer New York 1986

[8] J.S.Milne Abelian Varieties in Arithmetic Geometry edited by G.Cornell and J.H.Silverman Springer New York 1986

[9] J.S.Milne Arithmetic Duality Theorems Persp. in Math. 1 Academic Press Boston 1986

[10] A.Pletch Profinite duality groups I Journ. o. Pure a. Applied Algebra 16(1980) 55-74

[11] I.R.Shafarevich Principal homogeneous spaces defined over function fields (russ.) Trudy.M.Inst. ANSSSR 64 (1961) 316-346; engl. transl. in AMS Transl. series 2,37 85-114

[12] A.Schmidt Maximale p-Erweiterungen mehrdimensionaler lokaler Körper Diplomarbeit Humboldt-Universität Berlin 1991

[13] J.P.Serre Cohomologie Galoisienne Lecture Notes in Math. 5 Springer 1964

[14] T.Takahashi Galois cohomology in unramified extensions of algebraic function fields Tohoku Math.J. 24(1972),33-39

[15] J.Tate Letter to Serre in [13]

[16] J.L.Verdier Dualité dans la cohomologie des groupes profinies in [13]

[17] K.Wingberg On Poincaré Groups J.of the London Math. Soc. (2) 33(1986) 271-278

[18] K.Wingberg On Galois groups of p-closed algebraic number fields with restricted ramification J. reine u. ang. Math. 400(1989) 185-202

[19] K.Wingberg On Galois groups of p-closed algebraic number fields with restricted ramification II J. reine u. ang. Math. 416(1991), 187-194

[20] K.Wingberg Ein Analogon zur Fundamentalgruppe einer Riemann’schen Fläche im Zahlkörperfall Invent. Math. 77(1987), 557-584

Alexander Schmidt  Kay Wingberg
Universität Heidelberg  Universität Heidelberg
Mathematisches Institut  Mathematisches Institut
Im Neuenheimer Feld 288  Im Neuenheimer Feld 288
D-6900 Heidelberg  D-6900 Heidelberg
Germany  Germany