Rational connectedness and Galois covers of the projective line

By Jean-Louis Colliot-Thélène

Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely there exists a finite field extension F/k(t), Galois with group G, such that F is a regular extension of k (i.e. k is algebraically closed in F). Moreover, one may arrange that a given k-place of k(t) be totally split in F. Harbater proved this theorem for k an arbitrary complete valued field. Rather formal arguments ([10], §4.5; §2 hereafter) then imply that the theorem holds over any ‘large’ field k. This in turn is a special case of a result of Pop [15], hence will be referred to as the Harbater/Pop theorem. We refer to [10], [16], [6] for precise references to the literature (work of Dèbes, Deschamps, Fried, Haran, Harbater, Jarden, Liu, Pop, Serre, Völklein).

Most proofs (see [10], [19] 8.4.4 p. 93 and Liu’s contribution to [16] – see however [15]) first use direct arguments to establish the theorem when G is a cyclic group (here the nature of the ground field is irrelevant), then proceed by patching, using either formal or rigid geometry, together with GAGA theorems.

In the present paper, where I take the case of algebraically closed fields for granted, I show how a technique recently developed by Kollár [12] may be used to give a quite different proof of the Harbater/Pop theorem, when the ‘large’ field k has characteristic zero. This proof actually gives more than the original result (see comment after statement of Theorem 1).

Before I formally state the main result, let me recall what a ‘large’ field is. Let k be a field and let k((y)) be the quotient field of the ring k[[y]] of formal power series in one variable. Following F. Pop, we shall say that k is ‘large’ if it satisfies one of the three equivalent properties ([15], Prop. 1.1) :

(i) It is existentially closed in k((y)) : any k-variety with a k((y))-point has a k-point. (ii) On a smooth integral k-variety with a k-point, k-points are Zariski dense. (iii) On a smooth integral k-curve with a k-point, k-points are Zariski dense. (Such a field is clearly infinite. By going over to the completion at a smooth k-point of a

curve, one sees that (i) implies (iii). That (iii) implies (ii) is easy (consider a regular system of parameters). In characteristic zero, one may use resolution of singularities to show that (ii) implies (i).)

Known examples of ‘large’ fields k are fraction fields of a henselian discrete valuation ring, such as a p-adic field or a field of the shape k = F ((x)) for F some field.

Other well-known examples are real closed fields. That these are ‘large’ is a special instance of the following fact, which seems to have escaped the attention of specialists : any field F all finite field extensions of which are of degree a power of a fixed prime p is a ‘large’ field. To see this, one only needs to observe that on a regular, projective, connected curve C over a field F, given any nonempty open set U, any zero-cycle (divisor) z on C is rationally equivalent to a zero-cycle z_{1} whose support is contained in U (a semi-local Dedekind ring is a principal ideal domain); the degree (over F ) of z and z_{1} clearly coincide. Applying this to an F -point of C, one produces a zero-cycle ∑_{i} n_{i}P_{i} (n_{i} ∈ Z, P_{i} closed points) with support in U, such that the degree ∑_{i} n_{i}[F (P_{i}) : F] = 1. For F as above, this forces one of the degrees [F (P_{i}) : F] to be one.

Other known examples are the fields of totally real algebraic numbers and of totally p-adic algebraic numbers (that these fields are ‘large’ is a very special case of a theorem of Moret-Bailly [14], Thm. 1.3). The property trivially holds for so-called pseudo algebraically closed fields, such as infinite algebraic extensions of a finite field.

Theorem 1. Let G be a finite group. Let k be a ‘large’ field of characteristic zero. Let E = Spec(K) be a G-torsor over Spec(k). Then there exist an open set U of the affine line A_{k}^{1} containing a k-point O and a G-torsor V → U such that the following two properties hold :

(i) The fibre of V → U over O is isomorphic to E (as a G-torsor over Spec(k)); (ii) the smooth k-curve V is geometrically connected. The ring K is a finite separable extension of k, it need not be a field. In loose terms :

given a Galois extension K/k with group G, one may realize G as the Galois group of a ‘regular’ extension of k(t), in such a way that over a suitable k-place of k(t), the extension specializes to K/k.

When the G-torsor E/Spec(k) is trivial, i.e. E = ∐_{g∈G} Spec(k), we recover the result of Harbater and Pop. The question whether E may be chosen arbitrary had been investigated for special groups by several authors (see [6]). For arbitrary groups, Dèbes proves a weaker result ([6], Thm. 3.1) when k is ‘large’, and he proves the theorem in the case where k is a pseudo algebraically closed field ([6], Thm. 3.2).

Using general results from [EGA IV_{3}], we immediately obtain a series of concrete corollaries. These will be detailed in §2. In the case of a split E/k, most of them had already been obtained, with somewhat different proofs.

After the paper was submitted, I was asked whether in Theorem 1 one may impose arbitrary G-torsors as fibres of V → U at more than one k-point of U ⊂ A_{k}^{1}. The answer is in general in the negative, as shown in the appendix.

Let us say a few words on the tools used in this article. In a series of papers which appeared in 1992, Kollár, Miyaoka and Mori developed a technique which enables them, under some assumptions, to smooth a tree of rational curves into a single rational curve ([13], Thm (2.1); see also [11], Chap. II. 7, p. 154-158 and [5], §4.2). That work was over an algebraically closed field. In his recent paper [12], Kollár extends the technique over ‘large’ fields (e.g. local fields). Under certain assumptions, he manages to deform a set of conjugate P^{1}’s into a single P^{1} defined over the ground field. From this he gets the finiteness of the set of R-equivalence classes on k-points of a geometrically rationally connected variety defined over a local field k. That the key lemma of [12] precisely holds for ‘large’ fields provided the incentive for the present paper.

The proof I give for Theorem 1 starts from the classical fact that a finite group G is a Galois group over k(t) when k is algebraically closed of characteristic zero. It then uses a natural versal model for a G-torsor, and applies the deformation result of [12] to (a smooth compactification of) the base space of this G-torsor. The proof uses the existence of such a smooth compactification, but it avoids any consideration of the divisor at infinity : there is no discussion of inertia groups at all.

The idea of using a versal model of a G-torsor, originally due to E. Noether, has come up a number of times in the literature, notably in work of E. Fischer, D. Saltman [17], F. A. Bogomolov [1], see [20] and [21] for further references.

I am much indebted to János Kollár for having shown me his work [12] while in progress. I thank Pierre Dèbes, David Harbater and Laurent Moret-Bailly for their interest in my paper. Proposition A.3 was found during a stay at M.S.R.I., Berkeley, in September 1999.

§1. Proof of Theorem 1

In this section, we shall assume that the ground field k (which is of characteristic zero) is uncountable. The proof in the countable case will be given in §2.

Let \overline{k} be an algebraic closure of k. Given a k-scheme Z, let us write \overline{Z} = Z ×_{k} \overline{k}.

(1) Let G be a finite group and E/Spec(k) a G-torsor. Let us fix an embedding of G into some general linear group GL_{n}. Here G is viewed as a constant (split) k-group scheme, GL_{n} is the linear group over k and i : G → GL_{n} is a homomorphism of k-group schemes. Let U = GL_{n}/G the affine k-variety of ‘left classes’. This is the affine k-scheme whose ring is the ring of invariants for G acting on the ring k[GL_{n}]. The projection map GL_{n} → U makes GL_{n} into a right G-torsor V over U. The left action of GL_{n} on itself induces a left action of GL_{n} on U = GL_{n}/G and the projection V → U is equivariant for these (left) actions.

Let us recall basic facts from non-commutative étale cohomology. Given any smooth affine k-group scheme H, and any commutative k-algebra A, we denote by H_{ét}^{1}(A, H) the pointed cohomology set which classifies (étale) (right) H ×_{k} A-torsors over Spec(A) (up to non-unique isomorphism). Such torsors will simply be called H-torsors over A. For any such A, there is an “exact sequence”

V (A) → U(A) → H_{ét}^{1}(A, G) → H_{ét}^{1}(A, GL_{n}).

Let us detail this sequence. The map V (A) → U(A) is the obvious one; it respects the (left) action of GL_{n}(A) on both sets. The right G-torsor V → U defines an element ξ ∈ H_{ét}^{1}(U, G). To an element ρ ∈ U(A) = Hom_{k}(Spec(A), U), the map U(A) → H_{ét}^{1}(A, G) associates the class ρ^{∗}(ξ) ∈ H_{ét}^{1}(A, G) of the pull-back ρ^{∗}(V → U), which is a G-torsor over A. Two points x, y ∈ U(A) have the same image in H_{ét}^{1}(A, G) if and only if there exists α ∈ GL_{n}(A) such that α.x = y. By Grothendieck’s version of Hilbert’s theorem 90, the set H_{ét}^{1}(A, GL_{n}) classifies projective modules of rank n over A. Thus if A is semi-local, or if A is a Dedekind ring with trival class group, then H_{ét}^{1}(A, GL_{n}) is reduced to one element, and for any right G-torsor T over A there exists an element ρ ∈ U(A) such that T and ρ^{∗}(V → U) are isomorphic G-torsors over A. In particular, there exists a k-point P ∈ U(k) such that the fibre V_{P} of V above P is a G-torsor isomorphic to the given E/k. We shall fix such a k-point P.

(2) By classical results (see [19], chap. 6), we know that G is a ‘regular’ Galois group over \overline{k}(t). In other words there exist a nonempty open set W of the affine line A1 ; \overline{k} = Spec(\overline{k}[t]) and a G-torsor over W whose underlying variety is integral. Let A be the semi-local ring of \overline{k}[t] at t = 0 and t = 1, and let S = Spec(A). Let us abuse notation and call 0, resp. 1, the points of S defined by t = 0, resp. t = 1. Changing coordinates and semi-localizing produces a G-torsor T over S such that T is an integral scheme.

By (1), there exists a nonconstant \overline{k}-morphism ρ : S → \overline{U} such that the pull-back of the Gtorsor \overline{V} → \overline{U} under ρ is isomorphic to the G-torsor T /S. Given any α ∈ GL_{n}(A), the G-torsor (α.ρ)^{∗}(\overline{V} → \overline{U}) is G-isomorphic to the G-torsor T . In particular, it is an integral scheme.

(3) The action of GL_{n}(\overline{k}) on \overline{U}(\overline{k}) is transitive, hence the obvious action of GL_{n}(\overline{k})×GL_{n}(\overline{k}) on \overline{U}(\overline{k})×\overline{U}(\overline{k}) is also transitive. Reduction of A modulo t and modulo t−1 induces a surjective homomorphism GL_{n}(A) → GL_{n}(\overline{k}) × GL_{n}(\overline{k}). Thus given two points M, N ∈ \overline{U}(\overline{k}), there exists α ∈ GL_{n}(A) such that α.ρ ∈ \overline{U}(A) sends the point t = 0 to M and the point t = 1 to N.

Remark One should compare the present general position argument with ‘Kuyk’s lemma’ (see [20], Lemma 4.5).

(4) Since char(k)=0, by Hironaka’s theorem, there exist smooth, projective, geometrically integral k-varieties X_{1} and X, with V open in X_{1} and U open in X, together with a k-morphism p : X_{1} → X extending the map V → U and inducing a k-isomorphism V ≃ p^{−1}(U).

(5) According to a theorem of Kollár, Miyaoka and Mori ([13]; [11], Theorem II. 3.11 p. 118), to the point \overline{P} ∈ \overline{U}(\overline{k}) ⊂ \overline{X}(\overline{k}) one may associate countably many proper subvarieties V_{i} (i ∈ I) of the smooth projective variety \overline{X} such that if f : P1 ; \overline{k} → \overline{X} is a nonconstant morphism, f(0) = \overline{P} and the image of f is not contained in the union of the V_{i}’s, then f is free over 0 ∈ P1 ; \overline{k}. By definition (see [11], II. 3.1, p. 113), this means that the coherent cohomology group H^{1}(P1 ; \overline{k}, f^{∗}T\overline{X}(−2)) vanishes (here T\overline{X} denotes the tangent bundle of \overline{X}), which amounts to the hypothesis that in Grothendieck’s decomposition of the vector bundle f^{∗}T\overline{X} over P1 ; \overline{k} as a sum of line bundles O_{P^{1}}(n_{j}), we have n_{j} > 0 for each j (this is the ampleness property for the vector bundle f^{∗}T\overline{X} on P1 ; \overline{k} , see [11], II.3.8 p. 116).

Since k is uncountable, there exists a point Q ∈ \overline{U}(\overline{k}), Q ≠ \overline{P}, which does not lie on any of the V_{i}’s (proof : use a generically finite projection to projective space and induct on dimension). By (3), there exists α ∈ GL_{n}(A) such that α.ρ ∈ \overline{U}(A) sends the point t = 0 to \overline{P} and the point t = 1 to Q. Since X/k is proper, the morphism α.ρ : S → \overline{U} extends to a (nonconstant) morphism f : P1 ; \overline{k} → \overline{X}. The image of f contains \overline{P} and is not contained in the union of the V_{i}’s, since this image contains Q. By the quoted theorem ([11], II.3.11), we conclude :

(5.1) The vector bundle f^{∗}T\overline{X} on P1 ; \overline{k} is ample.

On the other hand, we have :

(5.2) The underlying variety of the G-torsor f^{∗}(\overline{V} → \overline{U}) over f^{−1}(\overline{U}) is integral.

Indeed, this follows from the same statement for the restriction of this G-torsor over S = Spec(A) ⊂ f^{−1}(\overline{U}), which was pointed out at the end of (2).

(6) We have now reached the situation studied in [12]. Starting from f : P1 ; \overline{k} → \overline{X} such that f(0) = \overline{P} and f^{∗}T\overline{X} is ample, Kollár ([12], 3.2, I change notation) produces, over the ground field k, a smooth integral k-curve C with a k-point O, a smooth geometrically integral k-surface Z proper over C, together with a k-morphism h : Z → X, with the following properties :

(6.a) The projection Z → C admits a k-section σ : C → Z which by h is mapped to P ∈ X. (6.b) The geometric fibre Z\overline{O} of Z → C at the point O is a comb D + ∑_{i∈I} C_{i} on \overline{Z} (here

I is a nonempty finite set, the C_{i}’s are the teeth of the comb, see [11], II.7.7 p. 156), each component of which is a nonsingular curve of genus zero; the map \overline{h} : \overline{Z} → \overline{X} sends D to \overline{P} and induces on C_{i} a conjugate of f : P1 ; \overline{k} → \overline{X}.

(6.c) Over any closed point M of C different from O, the fibre Z_{M} of Z → C is k(M)-iso-morphic to the projective line P_{k(M)}^{1} : the fibre is a smooth, geometrically irreducible, projective curve of genus zero over the residue field k(M), and it contains the k(M)-rational point σ(M).

(7) Since the map \overline{h} : Z\overline{O} → \overline{X} is not constant (because its restriction to any C_{i} is not constant), the closed set h^{−1}(P) ⊂ Z is a proper closed set. Thus, after shrinking C, we may assume : for no M ∈ C is h constant on the fibre Z_{M} (note that on any fibre Z_{M}, h assumes the value h(σ(M)) = P ×_{k} k(M)).

Let Ω ⊂ Z be the inverse image of U under h. Note that Ω contains σ(C), hence the composite map Ω ⊂ Z → C is surjective. Let Ω_{1} → Ω be the inverse image of the G-torsor V → U under h : Ω → U. Let M be a closed point in C. We shall show : For all but finitely many M ∈ C, the total space of the induced G-torsor Ω_{1,M} → Ω_{M} ⊂ Z_{M} ≃ P_{k(M)}^{1} is a smooth geometrically integral k(M)-variety.

To prove this, it is enough to prove the corresponding statement over \overline{k}. For the rest of the proof of (7), to simplify notation, let us set k = \overline{k}. Points M will be \overline{k}-rational points on C. For M ≠ O, the (nonempty) variety Ω_{M} is smooth and connected and the variety Ω_{1,M} is a finite étale cover of Ω_{M}, hence is smooth. To prove that a given Ω_{1,M}, M ≠ O, is integral, it is thus enough to show that it is connected.

The inverse image in Ω_{1} of D ∩ Ω is a disjoint union of copies D_{g} (g ∈ G) of D ∩ Ω, each with multiplicity one; by (5.2) and (6.b), for a given i ∈ I the inverse image in Ω_{1} of each C_{i} ∩ Ω is a (smooth) connected curve, which meets each D_{g} (g ∈ G), since C_{i} meets D (see (6.b)). Thus Ω_{1,O}, which is the inverse image of D + ∑_{i∈I} C_{i}, is a reduced connected divisor on Ω_{1}.

That Ω_{1,M} is connected for all but finitely many M ∈ C now follows from the general lemma (where X and Y have nothing to do with the previous Y and X), to be applied to X = Ω_{1} and Y = Ω :

Lemma. Let C be a smooth connected curve over an algebraically closed field k, and let O ∈ C(k). Let X, Y, C be smooth varieties over k, equipped with faithfully flat k-morphisms X → Y and Y → C. Assume that the generic fibre of Y → C is smooth and geometrically integral. Assume that X → Y is finite and étale. Assume moreover that the inverse image of O under the composite map X → Y → C is a connected divisor on X and is not a multiple divisor. Then there exists a finite set S of points of C such that for M ∈ C, M ∉ S, the inverse image X_{M} of M under the composite map X → Y → C is a smooth connected variety.

Proof Note first that X is connected. Indeed if it was not connected, the finite étale cover X → Y would break up into a disjoint union of finite étale (hence faithfully flat) covers X_{i} → Y, and the fibre of X → Y → C over O would not be connected. Thus X is connected; since it is smooth, it is integral. Let D be the normalization of C in the function field of X. This is a smooth integral curve, and the map D → C is flat and finite. Since X is normal, the map X → C factors through D. The finite (étale) map X → Y factors through the scheme Y ×_{C} D. The scheme Y ×_{C} D is integral, because C is its own normalization in Y, since we have assumed that the generic fibre of Y → C is geometrically integral. The finite map of integral varieties X → Y ×_{C} D is dominant, hence surjective as a morphism of schemes (it need not be flat). In particular, it is surjective on k-points (recall k = \overline{k}). The projection map Y ×_{C} D → D is faithfully flat, since it is obtained by base change from the faithfully flat map Y → C. In particular, Y ×_{C} D → D is surjective on k-points. We conclude that X → D is surjective on k-points. But then the scheme-theoretic inverse image of O ∈ C under the map D → C must consist of one reduced point, since the inverse image of O under the composite map X → D → C is a connected divisor which is not multiple. Since D → C is finite and flat, this implies that D → C is an isomorphism. Thus the function field of C is algebraically closed in the function field of X, hence the generic fibre of X → C is a smooth geometrically integral variety. By [EGA IV_{3}] (9.7.7) this implies the same statement for all fibres of X → C away from a proper closed subset of C.

(8) We finally make use of the hypothesis that the field k is ‘large’. Since the curve C has a k-rational point, namely O, this hypothesis implies that there exists a k-point M on C away from the finitely many points excluded in (7), such that the map P_{k}^{1} → X induced by h on the fibre Z_{M} ≃ P_{k}^{1} does what we want : the inverse image of the G-torsor V → U under the map h : h^{−1}(U) ∩ P^{1} → U is a G-torsor over the open set h^{−1}(U) ⊂ P_{k}^{1}, whose fibre at σ(M) ∈ h^{−1}(U)(k) ⊂ P^{1}(k) is isomorphic to the fibre of V → U at P, hence is isomorphic to E (by the very choice of P, see (1)), and whose total space is a geometrically integral k-variety (see (7)).

§2. Corollaries

Theorem 2. Let O be a Q-point of the projective line P_{Q}^{1}. Let G be a finite group and let E = Spec(K) → Spec(Q) be a G-torsor. There exist a smooth, geometrically integral curve Y/Q whose smooth compactification has a Q-point, an open set U ⊂ P^{1} ×_{Q} Y containing O ×_{Q} Y, and a G-torsor V → U (an étale Galois cover with group G), whose restriction to O ×_{Q} Y is the G-torsor E ×_{Q} Y, and such that the fibre of the composite map V → U → Y at any geometric point of Y is nonempty and connected (hence integral).

Proof Let G ↪ GL_{n,Q} be an embedding. The varieties U, V, X, X_{1} which appear in the proof of Theorem 1 may all be defined over Q. We also have P ∈ U(Q) ⊂ X(Q).

For any field F with Q ⊂ F, let us in this proof say that an F -morphism f : P_{F}^{1} → X_{F} is good if f(O) = P_{F} and the inverse image of V_{F} → U_{F} under f (restricted to f^{−1}(U_{F})) is a geometrically integral F -variety. Let Z = Hom_{Q}(P^{1}, X, O ↦ P) (notation as in [11], II.1.4 p. 94). This is a countable union of Q-varieties Z_{d} (d for degree of the image of P^{1}, in a fixed projective embedding of X). An F -point of Z will be called good if the corresponding F -morphism f : P_{F}^{1} → X_{F} is good. Given arbitrary field extensions Q ⊂ E_{1} ⊂ E_{2}, a point in Z(E_{1}) is good if and only if its image in Z(E_{2}) is good.

The field Q((x)) is uncountable. By Theorem 1 over such a field, as proved in §1, there exists a good Q((x))-point on Z, hence on Z_{d} for some d. Let Y ⊂ Z_{d} be the scheme-theoretic closure of the image of the corresponding morphism Spec(Q((x))) → Z_{d}. The Q-variety Y is geometrically integral. We have the field embeddings Q ⊂ Q(Y) ⊂ Q((x)). Thus on the one hand the generic point of Y is a good Q(Y)-point of Z, on the other hand any Q-compactification of Y has a Q-point. Indeed, for any such compactification Y_{c}, the map Spec(Q((x))) → Y extends to a Q-morphism Spec(Q[[x]]) → Y_{c}; the image of x = 0 is a Q-point of Y_{c}.

Replacing Y by a nonempty open set, one may ensure ([EGA IV_{3}] (8.8.2)) that the corresponding good Q(Y)-morphism P_{Q(Y)}^{1} → X_{Q(Y)} extends to a Y -morphism φ : P^{1} ×_{Q} Y → X ×_{Q} Y which sends O ×_{Q} Y to P ×_{Q} Y.

Let Ω = φ^{−1}(U ×_{Q} Y) ⊂ P^{1} ×_{Q} Y and let Ω_{1} → Ω be the G-torsor which is the inverse image of the G-torsor V ×_{Q} Y → U ×_{Q} Y under φ. Upon replacing Y by a nonempty open set (this is actually not necessary), the restriction of this G-torsor over O ×_{Q} Y ⊂ Ω is isomorphic to E ×_{Q} Y (indeed, this is true over the generic point of Y). We have the maps Ω_{1} → Ω → Y. The first map is finite étale of constant rank, the second one is smooth and surjective. Thus the composite map Ω_{1} → Y is smooth. Since the generic point of Y corresponds to a good point of Z, the generic fibre Ω_{1,Q(Y)} is geometrically integral over Q(Y). Upon replacing Y by a nonempty open set ([EGA IV_{3}], (9.7.7)(iv)), we therefore have that all geometric fibres of the map Ω_{1} → Y are smooth and geometrically integral. In particular for any field F with Q ⊂ F and any F -point of Y, the morphism φ_{F} : P_{F}^{1} → X_{F} induced by φ is good.

On a smooth projective model Y_{c} of Y over Q, there exists a Q-point R. By considering a regular system of parameters at R one produces a geometrically integral Q-curve C ⊂ Y_{c}, smooth at R, and which meets Y. One now replaces Y by Y ∩ C. This completes the proof of Theorem 2.

Remarks and Corollaries

(1) Note that Y in Theorem 2 need not have a Q-point. But for any field k containing Q such that Y (k) ≠ ∅, G is a ‘regular’ Galois group over the rational field k(t), with the added information that the fibre at the point t = 0 is isomorphic to the torsor E ×_{Q} k. This applies in particular to any ‘large’ field of characteristic zero, thus completing the proof of Theorem 1 for fields which are countable.

(2) One should compare Theorem 2 with the contribution of Deschamps in [16], and the proof given here with that given in [7], 4.2.

(3) One amusing corollary is that for any finite group G, there exists a finite set of number fields k_{i} such that the g.c.d. of the degrees [k_{i} : Q] is equal to one, and such that G is a ‘regular’ Galois group over each k_{i}(t), hence in particular a Galois group over each k_{i}. The proof is simple : on the smooth compactification Y_{c} of the curve Y, there exists a Q-point, call it M. If we let S ⊂ Y_{c} be the complement of Y in Y_{c}, there exists a zero-cycle ∑_{i∈I} n_{i}P_{i} (here the n_{i} are integers, P_{i} is a closed point and I is finite) on Y_{c} which is rationally equivalent to M, hence of degree one, and whose support is foreign to S, i.e. whose support is contained in Y. Let k_{i} be the residue field at the closed point P_{i}. Then ∑_{i∈I} n_{i}[k_{i} : Q] = 1 and Y (k_{i}) ≠ ∅ for each i, hence the claim.

One could say that, for any group G, the inverse Galois group problem over Q acquires a positive answer when passing from rational points to ‘zero-cycles of degree one’.

This could have been noticed earlier. For any prime p, let K_{p} be the fixed field of a pro-p-Sylow subgroup of the absolute Galois group of Q. As proved in the introduction of this paper, K_{p} is a ‘large’ field. By Theorem 1 (or, for that matter, the Harbater/Pop theorem), G is a regular Galois group over K_{p}(t). There exists a finite subextension L_{p}/Q of K_{p}/Q, such that G is a regular Galois group over L_{p}(t). By Hilbert’s irreducibility theorem, G is a Galois group over the number field L_{p}, whose degree [L_{p} : Q] is prime to p.

(4) Starting from the statement of Theorem 2 and writing a model of the whole situation over an open set of the ring of integers (same references to [EGA IV_{3}] as above), one easily deduces the following result, which is a special case of a theorem of Fried and Völklein : For a given finite group G, for almost all primes p (“almost all” depending on G), G is a ‘regular’ Galois group over F_{p}(t) (see [10] and [7], 3.9 for references; in [7] a model-theoretic argument is given). Simply note that if Y/Z is a smooth integral model of the smooth, geometrically integral curve Y/Q, then by classical estimates (Weil) we have Y(F_{p}) ≠ ∅ for almost all primes p. Here again, the present proof enables us to get more : if we start off with a given G-torsor E over a nonempty open set of Spec(Z), we may satisfy the additional requirement that for almost all primes p the ‘regular’ Galois extension over F_{p}(t) be unramified at t = 0, the fibre being isomorphic to E ×_{Z} F_{p}.

Appendix

In this appendix, where for simplicity I assume all fields to be of characteristic zero, I address the question :

Let k be a field, G a finite group, n ≥ 1 an integer. Let E_{1}, ⋯, E_{n} be G-torsors over k. Can one find an open set U ⊂ A_{k}^{1}, a G-torsor V → U and n points P_{1}, ⋯, P_{n} ∈ U(k) such that for each i, the fibre V_{P_{i}} is isomorphic to E_{i} as a G-torsor over k ?

Here are two cases where the answer is in the affirmative :

(i) G is an abelian group, its 2-primary subgroup is of exponent 2^{r}, the cyclotomic field extension k(μ_{2^{r}})/k is cyclic, and n is arbitrary. This is a special case of [3], Thm. 7.9 (various versions of this statement exist in the literature, see [17], [20]).

(ii) G is arbitrary, k is ‘large’ and n = 1 : this is Theorem 1 of the present paper (with the additional piece of information that V may be chosen geometrically integral).

In this appendix, I show by examples that for n ≥ 2 and k ‘large’ the answer to the above question is in general in the negative.

In the first part of the appendix, written in April 1999, I consider the case left open in (i) above. I give an example with G = Z/8 and k the 2-adic field Q_{2}. As may be expected, this example is closely related to Wang’s counterexample to Grunwald’s theorem.

In the second part of the appendix, written in November 1999, for an arbitrary prime p, I give examples with G a p-group and k a suitable ‘large’ field. That part builds upon work of Saltman [18].

Background and references for the first part of the appendix (algebraic tori, quasi-trivial and flasque tori, groups of multiplicative type, R-equivalence) will be found in [2], [3], [21]. For G a commutative algebraic group over a field k, the étale cohomogy group H_{ét}^{1}(k, G) may be identified with a Galois cohomology group, and will be simply denoted H^{1}(k, G).

Proposition A.1 Let k be a field and A be a finite abelian group. One may embed the constant k-group scheme A into a commutative diagram of exact sequences of k-groups of multiplicative type :

1 → A → P_{1} → T → 1
↓  ↓  ↓ =
1 → F → P_{2} → T → 1

where T is a k-torus, F is a flasque k-torus and P_{1} and P_{2} are quasi-trivial k-tori.

Proof By the well-known duality M ↦ M̂ = Hom_{k−gr}(M, G_{m,k}) between k-groups of multiplicative type and finitely generated Galois modules over k, it is enough to prove the dual result. There exist exact sequences of finitely generated Galois modules

0 → T̂ → P̂_{1} → Â → 0

and

0 → P̂ → F̂ → Â → 0

with P̂_{1} and P̂ permutation modules, and F̂ a flasque module (for the second sequence, see [3] (0.6.2)). The pull-back of the first sequence under the map F̂ → Â is an exact sequence

0 → T̂ → P̂_{2} → F̂ → 0

where the module P̂_{2} is an extension of the permutation module P̂_{1} by the permutation module P̂, hence is itself a permutation module. Taking duals yields the proposition.

For a quasi-trivial k-torus P, Hilbert’s theorem 90 implies H^{1}(k, P) = 0. Passing over to Galois cohomology in the diagram of Prop. A.1, we get the commutative diagram of exact sequences

P_{1}(k) → T (k) → H^{1}(k, A) → 0
↓  ↓ =  ↓
P_{2}(k) → T (k) → H^{1}(k, F) → 0.

From this diagram it immediately follows that the map H^{1}(k, A) → H^{1}(k, F) is onto.

Let us recall the following basic fact from [2] : the map T (k) → H^{1}(k, F) induces an isomorphism T (k)/R ≃ H^{1}(k, F). Here R denotes R-equivalence ([2], §4) on the set of k-points of the k-torus T.

Proposition A.2 With notation as above, assume that there exists ξ ≠ 0 ∈ H^{1}(k, F). Let η ∈ H^{1}(k, A) denote a lift of ξ under the surjective map H^{1}(k, A) → H^{1}(k, F). Then there do not exist an open set U ⊂ A_{k}^{1} and an A-torsor X → U with the following properties : there exist points M, N ∈ U(k) such that the fibre of X → U at M is trivial while the fibre of X → U at N has class η ∈ H^{1}(k, A).

Proof Let us assume there exist such U, M, N. Since P_{1} is a quasi-trivial k-torus, for any kscheme V the étale cohomology group H_{ét}^{1}(V, P_{1}) is isomorphic to a sum of groups Pic(V ×_{k} K_{i}), where the K_{i}/k are finite separable field extensions of k. For U ⊂ A_{k}^{1}, we thus have H_{ét}^{1}(U, P_{1}) = 0. Hence the map T (U) → H_{ét}^{1}(U, A) associated to the upper exact sequence in the diagram of Proposition A.1 is onto. There thus exists a k-morphism φ : U → T such that φ^{∗}(P_{1} → T) is isomorphic to the A-torsor X → U. The map T (k) → H^{1}(k, A) sends φ(M) to 0, and it sends φ(N) to η. Thus the map T (k) → H^{1}(k, F) sends φ(M) to 0, and it sends φ(N) to ξ ≠ 0. Now since U is an open set of A_{k}^{1}, the points φ(M) ∈ T (k) and φ(N) ∈ T (k) are R-equivalent : their images under the map T (k) → H^{1}(k, F) should coincide. This contradiction establishes our contention.

We still need to exhibit one case where the hypotheses of Proposition A.2 are fulfilled. Let k be a field, let A = Z/8 and let T and F be two k-tori as in Proposition A.1. Suppose the cyclotomic field extension k(μ_{8})/k has degree 4. Its Galois group is then Z/2 × Z/2. In that case, we have H^{1}(k, F̂) = Z/2 ([21], §7.4 p. 79). If k is a p-adic field, then the finite abelian groups H^{1}(k, S) and H^{1}(k, Ŝ) are dual (Tate-Nakayama). Let k be the 2-adic field Q_{2}. The field extension Q_{2}(μ_{8})/Q_{2} has degree 4, we thus have H^{1}(Q_{2}, F) ≠ 0.

This completes the construction of the announced example, but one can be more explicit. Let k = Q_{2}. As a class η ≠ 0 ∈ H^{1}(k, Z/8), let us take the class of the degree 8 unramified field extension E of k = Q_{2}. Let us write the commutative diagram in Prop. A.1 over Q. One may then write the ensuing commutative diagram over Q and over Q_{2}, in a compatible manner. Let M ∈ T (k) be any point with image η in H^{1}(k, Z/8). Suppose the image of η in H^{1}(k, F) is trivial. Then M comes from a k-point of P_{2}. But then the point M lies in the closure of T (Q) in T (Q_{2}), since P_{2}/Q is a quasi-trivial torus, hence Q-isomorphic to an open set of some affine space over Q. One can then find a Q-point N of T such that the fibre of P_{1} → T at N is a Galois extension F/Q with group Z/8 and such that F ⊗_{Q} Q_{2} ≃ E (as Galois extensions of Q_{2} with group Z/8). But there is no such extension (Wang’s well-known counterexample to Grunwald’s theorem, see [17] and [20]). Thus the image of η in H^{1}(k, F) is nontrivial.

Let us now turn to other types of examples.

Proposition A.3 Let p be a prime number. There exist a p-group G, a ‘large’ field k, and G-torsors E_{1} and E_{2} over k with the following property : given any G-torsor f : V → U over an open set U of A_{k}^{1}, there do not exist k-points P, Q ∈ U(k) such that the G-torsor V_{P} is isomorphic to E_{1} and the G-torsor V_{Q} is isomorphic to E_{2}.

Proof Saltman’s work [18] (extended by Bogomolov [1], see [21], §7.6 and §7.7) produces finite p-groups G together with faithful (finite dimensional) linear representations W of G over the complex field C, such that the unramified Brauer group Br_{nr}(F) of F = C(W)^{G} is a nontrivial (p-primary) group. Here by C(W) we denote the fraction field of the symmetric algebra on W. The unramified Brauer group of F is the subgroup of the Brauer group Br(F) consisting of classes which are unramified with respect to any (rank one) discrete valuation on F. As is well-known, the group Br_{nr}(C(W)^{G}) does not depend on the particular faithful (finite dimensional) linear representation of G.

Let us fix one such p-group G. As in the beginning of §1, let us fix a homomorphic embedding G → GL_{n} = GL_{n,C}. We may take for W the vector space of C-points of M_{n} (the ring scheme of n by n matrices over C), with the action induced by left multiplication. Let U = GL_{n}/G and V = GL_{n} ⊂ M_{n}. Projection V → U makes V into a G-torsor, whose properties are described at the beginning of §1.

By Hironaka’s theorem, there exists a smooth projective variety X/C containing U as a dense open set. The function field C(X) of X is F. By results of Grothendieck, the natural map from the étale Brauer group Br(X) = H_{ét}^{2}(X, G_{m}) to Br(F) is one-to-one, and it induces an isomorphism Br(X) ≃ Br_{nr}(F) (see [4]). Let A ∈ Br(X) ⊂ Br(F) be a nontrivial element. Let X_{F} be the smooth, projective F -variety X_{F} = X×_{C}F. This contains the open set U_{F} = U ×_{C}F. On the one hand, the natural field embedding C ⊂ F induces an inclusion X(C) ⊂ X_{F} (F) of the set of C-rational points of X into the set of F -rational points of X_{F}, and similarly U(C) ⊂ U_{F} (F). Let P ∈ U_{F} (F) be an arbitrary point in that subset. On the other hand, the generic point Spec(F) → X of X gives rise (via the diagonal map) to an F -rational point Q of Y. Let A_{F} ∈ Br(X_{F}) be the inverse image of A under the projection map X_{F} → X. Let us evaluate A_{F} on the F -rational points P and Q. We have A_{F} (P) = 0 ∈ Br(F) because A_{F} (P) comes from Br(C). We have A_{F} (Q) ≠ 0 ∈ Br(F) because A_{F} (Q) is none other than the image of A ∈ Br(X) under the embedding Br(X) ↪ Br(F). Let k be a field, F ⊂ k, such that the induced map Br(F) → Br(k) is one-to-one. Changing the base field from F to k, we obtain rational points which we still denote P, Q in X_{k}(k), such that A_{k}(P) = 0 and A_{k}(Q) ≠ 0 in Br(k). The points P, Q both lie in U_{k} = U ×_{C} k. Let E_{1} = V_{P}, resp. E_{2} = V_{Q}, be the G-torsors over k defined as the fibre of the G-torsor V → U at P, resp. Q. Suppose there exist a G-torsor Z → Y over an open set Y ⊂ A_{k}^{1} and two k-points p, q ∈ Y (k) such that the fibre Z_{p}, resp Z_{q}, is a G-torsor over k isomorphic to E_{1}, resp. E_{2}. By the general properties of the G-torsor V_{k} → U_{k} (see beginning of §1) and the fact that Pic(Y) = 0, there exists a k-morphism r : Y → U_{k} such that the inverse image of the G-torsor V_{k} → U_{k} under r is isomorphic to the G-torsor Z → Y. Let P_{1} = r(p) ∈ U(k) and Q_{1} = r(q) ∈ U(k). Then V_{P} and V_{P_{1}} are isomorphic as G-torsors over k, and similarly V_{Q} and V_{Q_{1}}. The general properties of the G-torsor V → U then imply that there exist g, h ∈ GL_{n}(k) such that gP_{1} = P and hQ_{1} = Q. Since GL_{n} is an open set of an affine space over k, this implies that the k-points P_{1} and P of U_{k}(k) ⊂ X_{k}(k) are R-equivalent. Similarly, Q_{1} and Q are R-equivalent. Clearly, P_{1} and Q_{1} are R-equivalent. Thus P and Q are R-equivalent on the projective k-variety X_{k}. By Prop. 16 of [2] (p. 213) this implies A_{k}(P) = A_{k}(Q). But then we cannot have A_{k}(P) = 0 and A_{k}(Q) ≠ 0.

To complete the proof of Proposition A.3, it remains to notice that the field k = F ((t)) of formal power series in one variable is a ‘large’ overfield of F for which the map Br(F) → Br(k) is one-to-one.

Whether examples as in Prop. A.3 may be exhibited over a p-adic field remains to be seen.

C.N.R.S., U.M.R. 8628, Université de Paris-Sud, Orsay, France

E-mail address: colliot@math.u-psud.fr

References

[1] F. A. Bogomolov, The Brauer group of quotients spaces by linear group actions, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987) 485-516; engl. transl. Math. USSR Izv. 30 (1988) 455-485.

[2] J.-L. Colliot-Thélène et J.-J. Sansuc, La R-équivalence sur les tores, Ann. scient. Éc. Norm. Sup., 4ème série, 10 (1977) 175-229.

[3] J.-L. Colliot-Thélène and J.-J. Sansuc, Principal homogeneous spaces under flasque tori : applications, J. of Algebra 106 (1987), 148-205.

[4] J.-L. Colliot-Thélène and J.-J. Sansuc, The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), Lecture notes, IX ELAM, Santiago de Chile, 1988.

[5] O. Debarre, Variétés de Fano, in Séminaire Bourbaki, Vol. 1996-97, Exp. 827, Astérisque 245, Société mathématique de France (1997), 197-221.

[6] P. Dèbes, Galois covers with prescribed fibers : the Beckmann-Black problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol. XXVIII (1999) 273-286.

[7] P. Dèbes and B. Deschamps, The regular inverse Galois problem over large fields, in Geometric Galois actions, L. Schneps and P. Lochak ed., London Mathematical Society Lecture Notes Series 243, Cambridge University Press (1997), 119-138.

[8] [EGA IV_{3}] A. Grothendieck, Éléments de géométrie algébrique, rédigés avec la collaboration de J. Dieudonné. IV. Étude locale des schémas et des morphismes de schémas (Troisième Partie), Inst. Hautes Études Sci., Publ. Math. No. 28, 1966.

[9] D. Harbater, Galois coverings of the arithmetic line, in Number Theory Seminar, New York 1984/85, LNM 1240, 165-195, Springer Verlag, 1987.

[10] D. Harbater, Fundamental groups of curves in characteristic p, in Proceedings of the 1994 International Congress of Mathematicians (Zürich), vol. 1, 656-666, Birkhäuser Verlag, Basel, 1995.

[11] J. Kollár, Rational Curves on Algebraic Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 32, Springer Verlag, 1996.

[12] J. Kollár, Rationally connected varieties over local fields, Annals of Mathematics 150 (1999) 357-367.

[13] J. Kollár, Y. Miyaoka and S. Mori, Rationally connected varieties, J. Alg. Geom. 1 (1992), 429-448.

[14] L. Moret-Bailly, Groupes de Picard et problèmes de Skolem II, Ann. scient. Éc. Norm. Sup., 4ème série, 22 (1989) 181-194.

[15] F. Pop, Embedding problems over large fields, Annals of Mathematics 144 (1996), 1-34.

[16] Recent developments in the inverse Galois problem, Proceedings of a 1993 Seattle research conference, M. Fried et al. ed., Contemp. Math. 186 (1995), 217-238.

[17] D. J. Saltman, Generic Galois extensions and problems in field theory, Adv. in Math. 43 (1982) 250-283.

[18] D. J. Saltman, Noether’s problem over an algebraically closed field, Invent. math. 77 (1984) 71-84.

[19] J-P. Serre, Topics in Galois Theory, Notes written by H. Darmon, Research Notes in Mathematics, Jones and Bartlett Publishers, 1992.

[20] R. G. Swan, Noether’s Problem in Galois Theory, in Emmy Noether in Bryn Mawr, Proc. Symp., Bryn Mawr 1982, Bh. Srinivasan and J. Sally ed., Springer Verlag (1983), 21-40.

[21] V. E. Voskresenskiǐ, Algebraic groups and their birational transformations, Translations of Mathematical Monographs 179, Amer. Math. Soc., Providence, Rhode Island, 1998.

(Received January 8, 1999; revised appendix received November 30, 1999)