Unramified Skolem Problems and

Unramified Arithmetic Bertini Theorems

in Positive Characteristic

Dedicated to Professor Kazuya Kato

on the occasion of his fiftieth birthday

Akio Tamagawa

Received: November 9, 2002 Revised: June 18, 2003

Abstract. In this paper, we prove unramified, positive-characteristic versions of theorems of Rumely and Moret-Bailly that generalized Skolem’s classical problems, and unramified, positive-characteristic versions of arithmetic Bertini theorems. We also give several applications of these results.

2000 Mathematics Subject Classification: Primary 11G35; Secondary 11G25, 14G15, 14G25, 14H25 Keywords and Phrases: Skolem’s problem, arithmetic Bertini theorem, positive characteristic.

§0. Introduction. Let f : X → S be a morphism between schemes X and S. We refer to an S-morphism σ : S′ → X from an S-scheme S′ to X as a quasi-section of f, if the structure morphism π : S′ → S is surjective. Moreover, for each property P of morphisms of schemes, we say that σ is a P quasi-section, if π is P. In this terminology, Rumely’s theorem, which generalized Skolem’s classical problems and was augmented by the work of Moret-Bailly, can be stated as follows:

Theorem ([Ru1], [Mo2]). Let S be a non-empty, affine, open subscheme of either the spectrum of the integer ring of an algebraic number field K or a proper, smooth, geometrically connected curve over a finite field with function field K. Let X be a scheme and f : X → S a morphism of schemes, such that X is irreducible, that X_{K} =^{def} X ×_{S} Spec(K) is geometrically irreducible over K, and that f is of finite type and surjective. Then, f admits a finite quasi-section.

On the other hand, the following is a well-known fact in algebraic geometry:

Theorem ([EGA4], Corollaire (17.16.3)(ii)). Let f : X → S be a morphism of schemes (with S arbitrary) which is smooth and surjective. Then, f admits an étale quasi-section.

In the present paper, we prove, among other things, the following theorem in positive characteristic, which is a sort of mixture of the above two theorems:

Theorem (0.1). (See (3.1).) Let S be a non-empty, affine, open subscheme of a proper, smooth, geometrically connected curve over a finite field with function field K. Let X be a scheme and f : X → S a morphism of schemes, such that X_{K} is geometrically irreducible over K, and that f is smooth and surjective. Then, f admits a finite étale quasi-section.

Here, we would like to note that the validity of this theorem is typical of positive characteristic. For example, it is easy to observe that P_{ℤ}^{1}−{0, 1, ∞} → Spec(ℤ) does not admit a finite étale quasi-section.

In the work of Rumely and Moret-Bailly, they also proved certain refined versions of the above theorem, which involve local conditions at a finite number of primes. To state these refined versions, let S and K be as in the theorem of Rumely and Moret-Bailly. Thus, K is either an algebraic number field or an algebraic function field of one variable over a finite field. We denote by Σ_{K} the set of primes of K, and we denote by Σ_{S} the set of closed points of S, which may be regarded as a subset of Σ_{K}. Moreover, let Σ be a (an automatically finite) subset of Σ_{K} − Σ_{S}, which is not the whole of Σ_{K} − Σ_{S}. (This last assumption is referred to as incompleteness hypothesis.) For each v ∈ Σ, let K_{v} denote the v-adic completion of K, and assume that a normal algebraic extension L_{v}/K_{v} (possibly of infinite degree) is given. Let X and f : X → S be as in the theorem of Rumely and Moret-Bailly, and assume that, for each v ∈ Σ, a non-empty, v-adically open, Gal(K_{v}^{sep}/K_{v})-stable subset Ω_{v} of X(L_{v}) is given.

Theorem ([Ru1], [Mo3]). Notations and assumptions being as above, assume, moreover, either L_{v} = \overline{K}_{v} ([Ru1]) or L_{v} is Galois over K_{v} ([Mo3]). Then, there exists a finite quasi-section S′ → X of f : X → S, such that, for each v ∈ Σ, S′_{L_{v}} =^{def} S′ ×_{S} Spec(L_{v}) is a direct sum of (a finite number of) copies of Spec(L_{v}), and the image of S′_{L_{v}} in X(L_{v}) = X_{L_{v}}(L_{v}) is contained in Ω_{v}.

Remark (0.2). In fact, Moret-Bailly’s version implies Rumely’s version. See [Mo3], Remarque 1.6 for this. Indeed, Moret-Bailly’s version implies more, namely, that it suffices to assume that L_{v} is a normal algebraic extension of K_{v} such that L_{v} ∩ K_{v}^{sep} is (v-adically) dense in L_{v}. (See the proof of [Mo3], Lemme 1.6.1, case (b).)

Remark (0.3). Here is a brief summary of the history (in the modern terminology) of Skolem’s problems and its generalizations. Skolem [S] proved the existence of finite quasi-sections for rational varieties. Cantor and Roquette [CR] proved it for unirational varieties. (A similar result was slightly later obtained in [EG].) Then, Rumely [Ru1] gave the first proof for arbitrary varieties (in the case of rings of algebraic integers). (See also [Ro].) Moret-Bailly (and Szpiro) [Mo2,3] gave an alternative proof of Rumely’s result in stronger forms. (Another alternative proof was later given in [GPR].) Moret-Bailly also proved the existence of finite quasi-sections for algebraic stacks ([Mo5]).

We also prove the following refined version in the unramified setting. Unfortunately, in the unramified setting, our version for the present is weaker than Moret-Bailly’s version (though it is stronger than Rumely’s version). To state this, let S and K be as in (0.1). Thus, K is an algebraic function field of one variable over a finite field. We denote by Σ_{K} the set of primes of K, and we denote by Σ_{S} the set of closed points of S. Moreover, let Σ be a subset of Σ_{K} −Σ_{S}, which is not the whole of Σ_{K} −Σ_{S}. For each v ∈ Σ, let K_{v} denote the v-adic completion of K, and assume that a normal algebraic extension L_{v}/K_{v} is given. Let f : X → S be as in (0.1), and assume that, for each v ∈ Σ, a non-empty, v-adically open, Gal(K_{v}^{sep}/K_{v})-stable subset Ω_{v} of X(L_{v}) is given.

Theorem (0.4). (See (3.1).) Notations and assumptions being as above, assume, moreover, that, for each v ∈ Σ, L_{v} ∩ K_{v}^{sep} is dense in L_{v}, and that the residue field of L_{v} is infinite. Then, there exists a finite étale quasi-section S′ → X of f : X → S, such that, for each v ∈ Σ, S′_{L_{v}} is a direct sum of copies of Spec(L_{v}), and the image of S′_{L_{v}} in X(L_{v}) = X_{L_{v}}(L_{v}) is contained in Ω_{v}.

Roughly speaking, the proof of (0.4) goes as follows. Via some reduction steps, we may assume that X is quasi-projective over S. Then, by means of a version of arithmetic Bertini theorem, we take hyperplane sections successively to obtain a suitable quasi-section finally. More precisely, we use the following unramified version of arithmetic Bertini theorem, which is another main result of the present paper:

Theorem (0.5). (See (3.2).) Let S, Σ, L_{v} be as in (0.4). Moreover, let Y_{1}, …, Y_{r} be irreducible, reduced, closed subschemes of P_{S}^{n}. For each v ∈ Σ, let Ω̌_{v} be a non-empty, v-adically open, Gal(K_{v}^{sep}/K_{v})-stable subset of P̌_{S}^{n}(L_{v}). Then, there exist a connected, finite, étale covering S′ → S such that, for each v ∈ Σ, S′_{L_{v}} is a direct sum of copies of Spec(L_{v}), and a hyperplane H ⊂ P_{S′}^{n}, such that the following hold: (a) for each i = 1, …, r, each geometric point \overline{s} of S′ and each irreducible component P of Y_{i,\overline{s}}, we have P ∩ H\overline{s} ⊊ P; (b) for each i = 1, …, r, the scheme-theoretic intersection (Y_{i}^{sm})_{S′} ∩ H (in P_{S′}^{n}) is smooth over S′ (Here, Y_{i}^{sm} denotes the set of points of Y_{i} at which Y_{i} → S is smooth. This is an open subset of Y_{i}, and we regard it as an open subscheme of Y_{i}.); (c) for each i = 1, …, r and each irreducible component P of Y_{i,\overline{K}} (where we identify the algebraic closure of the function field of S′ with that of S) with dim(P) ≥ 2, P ∩ H\overline{K} is irreducible; and (d) for each v ∈ Σ, the image of S′_{L_{v}} in P̌^{n}(L_{v}) by the base change to L_{v} of the classifying morphism [H] : S′ → P̌_{S}^{n} over S is contained in Ω̌_{v}.

There remain, however, the following non-trivial problems. Firstly, a Bertini-type theorem is, after all, to find a (quasi-)section in an open subset of the (dual) projective space, which requires a Rumely-type theorem. Secondly, in the (most essential) case where X is of relative dimension 1 over S, the boundary of X (i.e., the closure of X minus X in the projective space) may admit vertical irreducible components (of dimension 1). Since a hyperplane intersects non-trivially with every positive-dimensional irreducible component, the hyperplane section does not yield a finite quasi-section of X (but merely of the closure of X).

To overcome the first problem, we have to prove a Rumely-type theorem for projective n-spaces directly. It is not difficult to reduce this problem to the case n = 1. First, we shall explain the proof of this last case assuming Σ = ∅. So, we have to construct a finite, étale quasi-section in an open subscheme X of P_{S}^{1}. Moreover, for simplicity, we assume that X is a complement of the zero locus W of w(T) ∈ R[T] in A_{S}^{1} = Spec(R[T]), where R =^{def} Γ(S, O_{S}). (Since X is assumed to be surjectively mapped onto S, w(T) is primitive.) The original theorem of Rumely and Moret-Bailly, together with some arguments from Moret-Bailly’s proof, implies that there exists a monic polynomial g(T) ∈ R[T] of positive degree, such that the zero locus of g in A_{S}^{1} is contained in X. Now, if the zero locus of g is étale over S, we are done. In general, we shall consider the following polynomial: F (T) = g(T)^{pm} + w(T)^{p}T for sufficiently large m > 0. Then, F (T) is a monic polynomial in R[T], and its zero locus S′ gives a closed subscheme of A_{S}^{1} which is finite, flat over S. Since g(T) (resp. w(T)) is a unit (resp. zero) on W, F (T) is a unit on W, or, equivalently, S′ is contained in X. Moreover, since F′(T) = w(T)^{p}, the zero locus of F′ coincides with W, hence is disjoint from the zero locus S′ of F. This means that S′ is étale over S, as desired. (This argument is inspired by an argument of Gabber in [G].) Next, assume Σ ≠ ∅. Then, to find a finite, étale quasi-section with prescribed local conditions at Σ, we need to investigate local behaviors of roots of polynomials like the above F. Since it is easy to reduce the problem to the case where the above w is v-adically close to 1 (by means of a coordinate change), we see that it is essential to consider local behaviors of roots of polynomials in the form of

a_{1}T + ∑_{i=0}^{m} a_{ip}T^{ip}

with a_{1} ≠ 0. (In the present paper, we refer to a polynomial in this form as a superseparable polynomial.) As a result of this investigation, we see that we can take the above F so that, for each v ∈ Σ, every root of F is contained in the given Ω_{v}. Also, in this investigation, the (hopefully temporary) condition that the residue field of L_{v} is infinite for each v ∈ Σ arises.

To overcome the second problem, we take a finite, flat quasi-section with local conditions by means of Moret-Bailly’s version of Rumely’s theorem. Then, by using this (horizontal) divisor, we construct a (new) quasi-projective embedding of X. Now, in this projective space, we can construct a finite, étale quasi-section of X as a hyperplane section.

Here is one more ingredient of our proof that we have not yet mentioned:

Theorem (0.6). (See (2.1) and (2.2).) Let S and Σ be as in (0.4). Assume, moreover, that, for each v ∈ Σ, a finite Galois extension L_{v}/K_{v} is given. Then, there exists a connected, finite, étale, Galois covering S′ → S, such that, for each v ∈ Σ, S′ ×_{S} Spec(K_{v}) is isomorphic to a disjoint sum of copies of Spec(L_{v}) over K_{v}.

We use this result in some reduction steps. See §3 for more details.

The author’s original motivation to prove results like (0.1) arises from the study of coverings of curves in positive characteristic. For example, in the forthcoming paper, we shall prove the following result as an application of (0.1):

Theorem (0.7). For each pair of affine, smooth, connected curves X, Y over \overline{𝔽}_{p}, there exists an affine, smooth, connected curve Z over \overline{𝔽}_{p} that admits finite, étale morphisms Z → X and Z → Y over \overline{𝔽}_{p}.

In other words, there exists an \overline{𝔽}_{p}-scheme H, such that, for every affine, smooth, connected curve X over \overline{𝔽}_{p}, the ‘pro-finite-étale universal covering’ X̃ of X is isomorphic to H over \overline{𝔽}_{p}.

For other applications of the above main results, see §4.

Finally, we shall explain the content of each § briefly. In §1, we investigate the above-mentioned class of polynomials in positive characteristic, namely, superseparable polynomials. The aim here is to control how a superseparable polynomial over a complete discrete valuation field in positive characteristic decomposes. Here, (1.18) is a final result, on which the arguments in §2 and §3 are based. In §2, we prove the existence of unramified extensions with prescribed local extensions, such as (0.6) above. The main results are (2.1) and (2.2). In the former, we treat an arbitrary Dedekind domain in positive characteristic, while, in the latter, we only treat a curve over a field of positive characteristic but we can impose (weaker) conditions on all the primes of the function field. The proofs of both results rely on the results of §1. In §3, we prove the main results of the present paper, namely, an unramified version of the theorem of Rumely and Moret-Bailly in positive characteristic (3.1), and an unramified version of the arithmetic Bertini theorem in positive characteristic (3.2). In §4, we give several remarks and applications of the main results. Some of these applications are essentially new features that only arise after our unramified versions.

Acknowledgment. Professor Kazuya Kato was the advisor of the author’s master’s thesis [T] at the University of Tokyo (May 1990 – March 1992). The author would like to express his sincerest gratitude to Professor Kato for having encouraged and stimulated the author constantly since then. Also, the author would like to thank Bjorn Poonen and Pierre Dèbes (resp. Osamu Fujino and Hiromichi Takagi) very much for various helpful information concerning the work of Rumely and Moret-Bailly (resp. [CFHR]). Finally, the author would like to thank one of the referees for a simplification of Step 2 of §3. (See (3.8).)

§1. Superseparable polynomials. Throughout this §, we let K denote a field.

Definition. Let f(T) be a polynomial in K[T]. We say that f is superseparable, if the derivative f′(T) of f(T) falls in K[T]^{×} = K^{×}.

Lemma (1.1). For each f(T) ∈ K[T], the following (a)–(c) are equivalent.

(a) f is superseparable.

(b) The K-morphism A_{K}^{1} → A_{K}^{1} associated to f is étale everywhere.

(c) f is in the form of

f(T) = \{ \begin{matrix}a_{1}T + a_{0}, & if char(K) = 0, \\ a_{1}T + ∑_{i=0}^{m}a_{ip}T^{ip}, & if char(K) = p > 0, \end{matrix}

where a_{j} ∈ K and a_{1} ≠ 0.

Proof. Immediate. □

Remark (1.2). f is separable (i.e., (f, f′) = 1) if and only if the associated K-morphism from A_{upper,K}^{1} =^{def} A_{K}^{1} to A_{lower,K}^{1} =^{def} A_{K}^{1} is étale at 0 ∈ A_{lower,K}^{1}.

From now on, let p denote a prime number, and we assume that K is of characteristic p and is equipped with a complete discrete valuation v, normalized as v(K^{×}) = ℤ. We denote by R, 𝔪, k, and t the valuation ring of v, the maximal ideal of R, the residue field R/𝔪, and a prime element of R, respectively. We fix an algebraic closure \overline{K} of K, and we denote again by v the unique valuation \overline{K} → ℚ ∪ \{∞\} that extends v. Moreover, for each subfield L of \overline{K} containing K, we denote by R_{L}, 𝔪_{L}, and k_{L} the integral closure of R in L, the maximal ideal of R_{L}, and the residue field R_{L}/𝔪_{L}, respectively.

Now, consider a superseparable polynomial

(1.3)  f(T) = aT + h(T^{p}),

where a ∈ K^{×}, h ∈ K[T], and we put m =^{def} deg(h). (We put m = 0 if h = 0.) The aim of this § is to describe how f decomposes and what is the Galois group associated with f.

Definition. (i) We say that a polynomial g in \overline{K}[T] is integral, if all the coefficients of g belong to R\overline{K}.

(ii) Let g be a non-zero polynomial in \overline{K}[T]. We denote by roots(g) the set of roots of g in \overline{K}. This is a finite subset of \overline{K}. (iii) Let g be a separable polynomial in K[T]. Then, we denote by K_{g} the minimal splitting field of g in \overline{K}, i.e., the subfield of \overline{K} generated by roots(g) over K. This is a Galois extension of K, and we put G_{g} =^{def} Gal(K_{g}/K). (iii) Let g be a polynomial in \overline{K}[T], and α an element of roots(g). Then, we put

μ(g, α) =^{def} max{v(α′ − α) | α′ ∈ roots(g) − {α}}.

Here, we put max ∅ =^{def} −∞.

The following is a version of Krasner’s lemma.

Lemma (1.4). Let f be a monic, integral, superseparable polynomial in K[T] as in (1.3).

(i) For each α ∈ roots(f), we have μ(f, α) ≤ \frac{1}{p−1}v(a).

(ii) Let ϵ(T) = ∑_{j=0}^{mp}ϵ_{j}T^{j} be a polynomial in K[T] (with degree ≤ mp), such that

v(ϵ_{j}) > \frac{p}{p−1}v(a) holds for all j = 0, …, mp. We put f_{1} =^{def} f + ϵ. Then, f_{1} is separable and we have K_{f_{1}} = K_{f}.

Proof. (i) Observe the Newton polygon of f(T + α) (which is also a monic, integral, superseparable polynomial). (ii) For each α ∈ roots(f), put g_{α}(T) = f_{1}(T + α), which is an integral polynomial in \overline{K}[T]. Then, we have roots(g_{α}) = \{β − α | β ∈ roots(f_{1})\}. We have g_{α}(0) = f_{1}(α) = ϵ(α) and g′_{α}(0) = f′_{1}(α) = a+ϵ′(α), hence v(g_{α}(0)) > \frac{p}{p−1}v(a) and v(g′_{α}(0)) = v(a). Thus, by observing the Newton polygon of g_{α}, we see that there exists a unique β = β_{α} ∈ roots(f_{1}), such that v(β − α) > \frac{1}{p−1}v(a). The map roots(f) → roots(f_{1}), α ↦ β_{α} is clearly Gal(K^{sep}/K)-equivariant. Moreover, this map is injective, since, for each pair α, α′ ∈ roots(f) with α ≠ α′, we have v(α − α′) ≤ \frac{1}{p−1}v(a) by (i). As ♯(roots(f)) = mp ≥ ♯(roots(f_{1})), this map must be a bijection. Thus, we obtain a Gal(K^{sep}/K)-equivariant bijection roots(f)→^{∼} roots(f_{1}), so that f_{1} is separable and K_{f} = K_{f_{1}}, as desired. □

Definition. Let m and n be natural numbers.

(i) We put I_{n} =^{def} {1, …, n}. (ii) We denote by S_{n} the symmetric group on the finite set I_{n}. Moreover, identifying I_{n} with ℤ/nℤ naturally, we define

B_{n} =^{def} {σ ∈ S_{n} | ∃a ∈ (ℤ/nℤ)^{×}, ∃b ∈ ℤ/nℤ, ∀i ∈ ℤ/nℤ, σ(i) = ai + b}

and

C_{n} =^{def} {σ ∈ S_{n} | ∃b ∈ ℤ/nℤ, ∀i ∈ ℤ/nℤ, σ(i) = i + b}.

Thus, S_{n} ⊃ B_{n} ▷ C_{n}, and B_{n} (resp. C_{n}) can be naturally identified with the semi-direct product (ℤ/nℤ)^{×} ⋉ (ℤ/nℤ) (resp. the cyclic group ℤ/nℤ).

(ii) We denote by S_{m×n} the symmetric group on the finite set I_{m} × I_{n}. (Thus, S_{m×n} ≃ S_{mn}.) Let pr_{1} denote the first projection I_{m} × I_{n} → I_{m}. We define

S_{m⋉n} =^{def} \{σ ∈ S_{m×n} | ∃\overline{σ} ∈ S_{m}, ∀(i, j) ∈ I_{m} × I_{n}, pr_{1}(σ((i, j))) = \overline{σ}(i)\}.

Thus, S_{m⋉n} can be naturally identified with the semi-direct product S_{m} ⋉ (S_{n})^{I_{m}}. Here, for a group G and a positive integer r, G^{I_{r}} denotes the direct product \underbrace{G × ⋯ × G}_{r times}. We adopt this slightly unusual notation to save the

notation G^{r} for {g^{r} | g ∈ G} (for a commutative group G).

The following proposition is a mere exercise in Galois theory over local fields in positive characteristic, but it is the starting point of our proofs of main results in later §§.

Proposition (1.5). Let f be a superseparable polynomial as in (1.3) with m ≥ 1. Moreover, we assume that (a) h is separable, and (b) we have δ(a, h, α) > μ(h, α) for all α ∈ roots(h), where

δ(a, h, α) =^{def} min (v(a) − v(h′(α)) + \frac{1}{p}v(α), \frac{p}{p − 1}(v(a) − v(h′(α)))) .

Then, by choosing a suitable bijection between roots(f) and I_{m} × I_{p}:

(i) The Galois group G_{f} can be identified with a subgroup of S_{m⋉p} (⊂ S_{m×p}). (ii) G_{f} ∩ (S_{p})^{I_{m}} ⊂ (B_{p})^{I_{m}}. (iii) The group filtration

{1} ⊂ G_{f} ∩ (C_{p})^{I_{m}} ⊂ G_{f} ∩ (S_{p})^{I_{m}} ⊂ G_{f}

corresponds via Galois theory to the field filtration

K_{f} ⊃ M_{f} ⊃ K_{h} ⊃ K,

where M_{f} is the subfield of \overline{K} generated by \{(−a/h′(α))^{\frac{1}{p−1}} | α ∈ roots(h)\} over K_{h}.

Proof. (i) First, we shall prove the following:

Claim (1.6). (i) For each α ∈ roots(h), put

F_{α} =^{def} {β ∈ roots(f) | v(β^{p} − α) ≥ δ(a, h, α)}.

Then, F_{α} has cardinality p for each α ∈ roots(h). (ii) For each β ∈ roots(f), there exists a unique α = α_{β} ∈ roots(h), such that F_{α} ∋ β.

Proof. (i) Observe the Newton polygon of f(T +α^{1/p}) = h(T^{p}+α)+aT +aα^{1/p} by using δ(a, h, α) > μ(h, α). Then, we see that F_{α} has cardinality p, as desired (and that the subset {β ∈ roots(f) | v(β^{p} − α) = δ(a, h, α)} of F_{α} has cardinality ≥ p − 1). (ii) First, we shall prove the uniqueness. Suppose that there exist α_{1}, α_{2} ∈ roots(h), α_{1} ≠ α_{2}, such that v(β^{p} − α_{i}) ≥ δ(a, h, α_{i}) holds for i = 1, 2. Then, we have

v(α_{1} − α_{2}) = v((β^{p} − α_{2}) − (β^{p} − α_{1})) ≥ min(δ(a, h, α_{1}), δ(a, h, α_{2})),

while, by assumption, we have

min(δ(a, h, α_{1}), δ(a, h, α_{2})) > min(μ(h, α_{1}), μ(h, α_{2})) ≥ v(α_{1} − α_{2}).

This is absurd.

By this uniqueness and (i), we have

♯( ⋃_{α∈roots(h)}F_{α}) = ∑_{α∈roots(h)}♯(F_{α}) = mp = ♯(roots(f)),

hence ⋃_{α∈roots(h)}F_{α} = roots(f). This implies the existence of α = α_{β} for each

β ∈ roots(f). □

By (1.6)(ii), we obtain a well-defined map π : roots(f) → roots(h), β ↦ α_{β}. By (1.6)(i), π is surjective and each fiber of π has cardinality p. Since π is Gal(K^{sep}/K)-equivariant by definition, this implies (1.5)(i). (We may choose any bijections roots(h) ≃ I_{m} and F_{α} ≃ I_{p} (α ∈ roots(h)).)

Note that this construction already shows that the field extension of K corresponding to the subgroup G_{f} ∩ (S_{p})^{I_{m}} = Ker(G_{f} → S_{m}) coincides with K_{h}. (ii) We shall start with the following. From now on, for each x, x′ ∈ \overline{K}^{×}, we write x ∼ x′ if x′/x ∈ 1 + 𝔪\overline{K}, or, equivalently, v(x′ − x) > v(x).

Claim (1.7). (i) Let α, α′ ∈ roots(h) and β ∈ F_{α}. Assume α ≠ α′. Then, we have β^{p} − α′ ∼ α − α′. In particular, we have v(β^{p} − α′) = v(α − α′). (ii) Let α, α′ ∈ roots(h), β ∈ F_{α}, and β′ ∈ F_{α′}. Assume β ≠ β′. Then, we have

v((β′)^{p} − β^{p}) = \{ \begin{matrix}v(α′ − α) (≤ μ(h, α)), & if α ≠ α′, \\ \frac{p}{p−1}(v(a) − v(h′(α))) (≥ δ(a, h, α)), & if α = α′. \end{matrix}

Proof. (i) v((β^{p} −α′)−(α−α′)) = v(β^{p} −α) ≥ δ(a, h, α) > μ(h, α) ≥ v(α−α′). (ii) If α ≠ α′, we have v((β′)^{p} − β^{p}) = v(((β′)^{p} − α′) − (β^{p} − α′)) = v(β^{p} − α′), since v((β′)^{p}−α′) > v(β^{p}−α′) by the definition of F_{α′}. (Recall that F_{α}∩F_{α′} = ∅ holds by (1.6)(ii).) Thus, in this case, v((β′)^{p} − β^{p}) = v(α′ − α) holds by (i).

By using this and (i), observe the Newton polygon of f(T + β) = h(T^{p} + β^{p}) + aT + aβ and compare it with the Newton polygon of f(T + α^{1/p}). Then, we can read off the value v(β′ − β) for α = α′. □

For each α ∈ roots(h), the subgroup Gal(K_{f}/K(α)) of G_{f} acts on F_{α}. In order to prove (1.5)(ii), it suffices to prove that the image Gal(K(α)(F_{α})/K(α)) of this action is contained in B_{p} (⊂ S_{p}), after choosing a suitable bijection F_{α} ≃ I_{p}.

Claim (1.8). Let α ∈ roots(h) and β ∈ F_{α}.

(i) We have K(α)(F_{α}) = K(α)(β)((−a/h′(α))^{1/(p−1)}). (ii) Let β′ ∈ F_{α}, β′ ≠ β. Then, we have (β′ − β)^{p−1} ∼ −a/h′(α). More precisely, we have

\{(β′ − β)mod ∼| β′ ∈ F_{α}, β′ ≠ β\} = \{ζ(−a/h′(α))^{\frac{1}{p−1}} mod ∼| ζ ∈ 𝔽_{p}^{×}\}.

Proof. As in the proof of (1.7)(ii), observe the Newton polygon of f(T + β) = h(T^{p} + β^{p}) + aT + aβ. Then, observing the coefficients of T^{0}, T^{1}, …, T^{p}, we see that K(α)(F_{α}) = K(α)(β)((−a/h′(β^{p}))^{1/(p−1)}). Now, by (1.7)(i), we obtain h′(β^{p}) ∼ h′(α), which implies K(α)(β)((−a/h′(β^{p}))^{1/(p−1)}) = K(α)(β)((−a/h′(α))^{1/(p−1)}). These complete the proof of (i), and also show (ii). □

Lemma (1.9). Let G be a subgroup of S_{p}, and, for each i = 1, …, p, we denote by G_{i} the stabilizer of i in G. Moreover, let ϕ : G → 𝔽_{p}^{×} be a homomorphism, such that, for each i = 1, …, p, there exists an identification σ_{i} : I_{p} − {i}→^{∼}𝔽_{p}^{×}, such that σ_{i}g_{i}σ_{i}^{−1} coincides with the ϕ(g_{i})-multiplication map on 𝔽_{p}^{×} for each g_{i} ∈ G_{i}. Then, we have G ⊂ B_{p} via a suitable identification I_{p} ≃ 𝔽_{p}.

Proof. Put N =^{def} Ker(ϕ). Then, N is a normal subgroup of G. By using the identity ϕ(g_{i})· = σ_{i}g_{i}σ_{i}^{−1}, we see that N ∩G_{i} = {1} for all i = 1, …, p. Namely, the action of N on I_{p} is free. Since ♯(I_{p}) = p is a prime number, this implies that either N = {1} or N = C_{p} (via some identification I_{p} ≃ 𝔽_{p}). In the latter case, we obtain G ⊂ B_{p}, since the normalizer of C_{p} in S_{p} coincides with B_{p}. So, assume N = {1}. Then, G = G/N is abelian with ♯(G) | p − 1.

Let X be any G-orbit of I_{p}. Suppose that X is not a one-point set. Then, there exist i, j ∈ X, i ≠ j. Since G is abelian, this implies G_{i} = G_{j}. On the other hand, by the identity ϕ(g_{i})· = σ_{i}g_{i}σ_{i}^{−1}, we see that G_{i} ∩ G_{j} = {1}. Thus, we must have G_{i}(= G_{j}) = {1}.

By this consideration, we conclude that I_{p} is isomorphic as a G-set to a disjoint union of copies of G and copies of G/G. If a copy of G/G appears, then this means G = G_{i} for some i = 1, …, p, and, by using the unique extension of σ_{i} to I_{p}→^{∼}𝔽_{p}, we obtain G ⊂ 𝔽_{p}^{×} ⊂ B_{p}.

On the other hand, if no copy of G/G appears, we must have ♯(G) | p. As ♯(G) | p − 1 also holds, we conclude G = {1} ⊂ B_{p}. This completes the proof. □

By (1.8), we may apply (1.9) to G = Gal(K(α)(F_{α})/K(α)) and the Kummer character ϕ : G → 𝔽_{p}^{×} defined by (−a/h′(α))^{1/(p−1)}, and conclude G ⊂ B_{p}, as desired. (iii) This has been already done in the proofs of (i) and (ii). □

Corollary (1.10). Let m be an integer ≥ 1. Let R = R^{univ} be the completion of the discrete valuation ring 𝔽_{p}[s_{0}, s_{p}, s_{2p}, …, s_{(m−1)p}, s_{1}]_{(s_{1})} (where s_{i}’s are algebraically independent indeterminates), i.e., R = 𝔽_{p}(s_{0}, s_{p}, …, s_{(m−1)p})[[s_{1}]], and K = K^{univ} the field of fractions of R. Consider a superseparable polynomial f(T) as in (1.3), where a = s_{1} and

h(T) = ∑_{i=0}^{m}s_{ip}T^{i} (s_{mp} =^{def} 1). Then:

(i) By choosing a suitable bijection between roots(f) and I_{m} × I_{p}, the Galois group G_{f} can be identified with an extension group of S_{m} by a subgroup B of (B_{p})^{I_{m}}. Here, B is an extension of a subgroup E of (B_{p})^{I_{m}}/(C_{p})^{I_{m}} = (𝔽_{p}^{×})^{I_{m}} by (C_{p})^{I_{m}}, where E = (𝔽_{2}^{×})^{I_{m}} = {1} if p = 2,

E = \{ \begin{matrix}Ker((𝔽_{p}^{×})^{I_{2}} ↠ (𝔽_{p}^{×}/\{±1\})^{I_{2}}/Δ(𝔽_{p}^{×}/\{±1\})), & m = 2, \\ Ker((𝔽_{p}^{×})^{I_{m}} ↠ 𝔽_{p}^{×}/(𝔽_{p}^{×})^{2}), & m ≡ 0 (mod 2), m ≠ 2, \\ (𝔽_{p}^{×})^{I_{m}}, & m ≢ 0 (mod 2), \end{matrix}

if p ≡ 1 (mod 4), and

E = \{ \begin{matrix}Ker((𝔽_{p}^{×})^{I_{2}} ↠ (𝔽_{p}^{×}/\{±1\})^{I_{2}}/Δ(𝔽_{p}^{×}/\{±1\})), & m = 2, \\ Ker((𝔽_{p}^{×})^{I_{m}} ↠ 𝔽_{p}^{×}/(𝔽_{p}^{×})^{2}), & m ≡ 0 (mod 4), \\ (𝔽_{p}^{×})^{I_{m}}, & m ≢ 0 (mod 4), m ≠ 2, \end{matrix}

if p ≡ 3 (mod 4). Here, for a commutative group G and a positive integer r, we define subgroups Δ(G) and (G^{I_{r}})^{0} of G^{I_{r}} by Δ(G) = {(g, …, g) ∈ G^{I_{r}} | g ∈ G} and (G^{I_{r}})^{0} = Ker(G^{I_{r}} → G, (g_{1}, …, g_{r}) ↦ g_{1} ⋯ g_{r}), respectively, and, in the case where either p ≡ 1 (mod 4), m ≡ 0 (mod 2), m ≠ 2 or p ≡ 3 (mod 4), m ≡ 0 (mod 4) holds, the surjective homomorphism (𝔽_{p}^{×})^{I_{m}} ↠ 𝔽_{p}^{×}/(𝔽_{p}^{×})^{2} is the composite of (𝔽_{p}^{×})^{I_{m}} ↠ (𝔽_{p}^{×}/(𝔽_{p}^{×})^{2})^{I_{m}} and (𝔽_{p}^{×}/(𝔽_{p}^{×})^{2})^{I_{m}} ↠ (𝔽_{p}^{×}/(𝔽_{p}^{×})^{2})^{I_{m}}/((𝔽_{p}^{×}/(𝔽_{p}^{×})^{2})^{I_{m}})^{0} = 𝔽_{p}^{×}/(𝔽_{p}^{×})^{2}. Moreover, the inertia subgroup of G_{f} corresponds to Δ(𝔽_{p}^{×}) ⋉ (𝔽_{p})^{I_{m}}. (ii) k_{K_{f}} is generated by {(α mod 𝔪_{K_{f}})^{1/p} | α ∈ roots(h)} ∪ \{(h′(α)/h′(α′))^{\frac{1}{p−1}} mod 𝔪_{K_{f}} | α, α′ ∈ roots(h)\} over k. Moreover, the algebraic closure 𝔽 of 𝔽_{p} in k_{K_{f}} coincides with 𝔽_{2} if p = 2,

𝔽 = \{ \begin{matrix}𝔽_{p^{2}}, & m = 2, \\ 𝔽_{p}, & m ≠ 2, \end{matrix}

if p ≡ 1 (mod 4), and

𝔽 = \{ \begin{matrix}𝔽_{p^{2}}, & m ≡ 2 (mod 4), \\ 𝔽_{p}, & m ≢ 2 (mod 4), \end{matrix}

if p ≡ 3 (mod 4).

Proof. (i) In order to apply (1.5), we have to check that conditions (a) and (b) of (1.5) hold. It is easy to see that (a) holds. Next, since K_{h} = 𝔽_{p}(α_{1}, …, α_{m})((s_{1})), where roots(h) = {α_{1}, …, α_{m}}, we have μ(h, α) = 0 for each α ∈ roots(h), while δ(a, h, α) = v(s_{1}) = 1. Thus (b) holds, and we may apply (1.5).

It is easy to see that K_{h}/K is an unramified S_{m}-extension. Next we have M_{f} = K_{h}((−s_{1}/h′(α_{1}))^{1/(p−1)}, …, (−s_{1}/h′(α_{m}))^{1/(p−1)}). Since −1/h′(α_{i}) is a unit of R_{K_{h}} and s_{1} is a prime element of R_{K_{h}}, the inertia subgroup of Gal(M_{f}/K_{h}) corresponds to Δ(𝔽_{p}^{×}), and the maximal unramified subextension M_{0,f}/K_{h} in M_{f}/K_{h} is K_{h}((h′(α_{i})/h′(α_{j}))^{1/(p−1)} | i, j = 1, …, m) = K_{h}((h′(α_{i})/h′(α_{1}))^{1/(p−1)} | i = 2, …, m).

Now, observing the subgroup of K_{h}^{×}/(K_{h}^{×})^{p−1} generated by the classes of −a/h′(α) (α ∈ roots) by using the divisor group of (the spectrum of) the polynomial ring 𝔽_{p}[α_{1}, …, α_{m}] over 𝔽_{p}, we obtain the desired description of E. (We leave the details to the readers.)

Finally, by (1.6)(i), k_{K_{f}} contains {(α mod 𝔪_{K_{f}})^{1/p} | α ∈ roots(h)}. Since k_{K_{h}} is a purely transcendental extension of 𝔽_{p} generated by α mod 𝔪_{K_{f}} and k_{M_{f}} is separable over k_{K_{h}}, the inseparable degree of the extension k_{K_{f}} /k_{M_{f}} is at least p^{m}. Thus, the ramification index of the extension K_{f}/M_{f} is at least p^{m}. Therefore, K_{f}/M_{f} must be totally ramified with degree p^{m} and the Galois group Gal(K_{f}/M_{f}) must coincide with the whole of (C_{p})^{I_{m}}.

These complete the proof of (i). (ii) The above proof shows that k_{K_{h}} = k(α mod 𝔪_{K_{h}} | α ∈ roots(h)), and that k_{K_{f}} contains the field k′_{K_{f}} generated by {(α mod 𝔪_{K_{f}})^{1/p} | α ∈ roots(h)} ∪

\{(h′(α)/h′(α′))^{\frac{1}{p−1}} mod 𝔪_{K_{f}} | α, α′ ∈ roots(h)\} over k_{K_{h}} (or, equivalently, over k, as α = (α^{1/p})^{p}). Moreover, we can check [k_{K_{f}} : k_{K_{h}}] = [k′_{K_{f}} : k_{K_{h}}], which implies k_{K_{f}} = k′_{K_{f}}, as desired.

Finally, k_{K_{h}} is a purely transcendental extension of 𝔽_{p} generated by {α mod 𝔪_{K_{h}} | α ∈ roots(h)}. Moreover, observing the subgroup of (K_{h}\overline{𝔽}_{p})^{×}/((K_{h}\overline{𝔽}_{p})^{×})^{p−1} generated by the classes of −a/h′(α) (α ∈ roots(h)) by using the divisor group of (the spectrum of) the polynomial ring \overline{𝔽}_{p}[α_{1}, …, α_{m}] over \overline{𝔽}_{p}, and comparing the result with the above description of the subgroup of K_{h}^{×}/(K_{h}^{×})^{p−1} generated by the classes of −a/h′(α) (α ∈ roots(h)), we see that the algebraic closure of 𝔽_{p} in k_{K_{h}} is as described in the assertion. Since k_{K_{f}} /k_{M_{f}} is purely inseparable, this completes the proof. □

So far, we have only investigated superseparable polynomials over complete discrete valuation fields. Here, we shall introduce the following global situation and study superseparable polynomials in a moduli-theoretic fashion. We put

A_{upper}^{mp} =^{def} Spec(𝔽_{p}[t_{1}, …, t_{mp}]) ≃ A_{𝔽_{p}}^{mp}

and

A_{lower}^{mp} =^{def} Spec(𝔽_{p}[s_{0}, …, s_{mp−1}]) ≃ A_{𝔽_{p}}^{mp}.

Moreover, consider the morphism E : A_{upper}^{mp} → A_{lower}^{mp}, defined by

∏_{i=1}^{mp}(T − t_{i}) = ∑_{i=0}^{mp}s_{i}T^{i},

where s_{mp} =^{def} 1. Namely, for i = 0, …, mp − 1, s_{i} is (−1)^{mp−i} times the (mp − i)-th elementary symmetric polynomial in t_{1}, …, t_{mp}. It is well-known that E is finite flat of degree (mp)!, and that, if we delete the discriminant locus D_{lower} from A_{lower}^{mp} and the union D_{upper} of weak diagonals from A_{upper}^{mp}, E gives a finite, étale, Galois covering with Galois group S_{mp}.

Let A_{lower}^{m+1} be the closed subscheme of A_{lower}^{mp} defined by s_{i} = 0 for all i with p ∤ i and i ≠ 1. We define a divisor A_{lower}^{m} of A_{lower}^{m+1} by s_{1} = 0. Observe that A_{lower}^{m} coincides with the non-étale locus of E|_{A_{lower}^{m+1}}, and that we have A_{lower}^{m} = A_{lower}^{m+1} ∩ D_{lower} set-theoretically. We also have A_{lower}^{m+1} ≃ A_{𝔽_{p}}^{m+1} and A_{lower}^{m} ≃ A_{𝔽_{p}}^{m} naturally.

Now, we have the following diagram:

A_{upper}^{mp} − D_{upper} →^{E} A_{lower}^{mp} − D_{lower}
↑ c.i.  □  ↑ c.i.
U_{m}  → A_{lower}^{m+1} − A_{lower}^{m}

where □ means a fiber product diagram, →^{c.i.} means a closed immersion, and U_{m} =^{def} A_{upper}^{mp} ×_{A_{lower}^{mp}} (A_{lower}^{m+1} − A_{lower}^{m}).

We shall apply this moduli-theoretic situation to the study of superseparable polynomials over a (an arbitrary) complete discrete valuation field K of characteristic p > 0. From now, for each finite subset S of \overline{K}, we put

ϕ_{S}(T) =^{def} ∏_{α∈S}(T − α).

Proposition (1.11). Let m be an integer ≥ 1. Assume that there exists a finite subset S of K with cardinality m, such that ϕ_{S} satisfies

(1.12)  ϕ′_{S}(γ)/ϕ′_{S}(γ′) ∈ (K^{×})^{p−1} for all γ, γ′ ∈ S.

Then, there exists a monic, superseparable polynomial f(T) ∈ K[T] with deg(f) = mp, such that f is completely splittable in K.

Proof. We consider the above moduli-theoretic situation E : A_{upper}^{mp} → A_{lower}^{mp}

and A_{lower}^{m} →^{c.i.} A_{lower}^{m+1} →^{c.i.} A_{lower}^{mp}. We have to show that U =^{def} U_{m} admits a Krational point. Recall that E induces a finite, étale (not necessarily connected) S_{mp}-Galois covering U → A_{lower}^{m+1} −A_{lower}^{m}. However, first we need to investigate the non-étale loci of E.

We put A_{upper}^{m} =^{def} Spec(𝔽_{p}[u_{1}, …, u_{m}]) ≃ A_{𝔽_{p}}^{m}, and define a morphism D : A_{upper}^{m} → A_{upper}^{mp} by

(u_{1}, …, u_{m}) ↦ (\underbrace{u_{1}, …, u_{1}}_{p times}, …, \underbrace{u_{m}, …, u_{m}}_{p times}),

which is clearly a closed immersion. It is easy to see that E∘D : A_{upper}^{m} → A_{lower}^{mp}

factors through A_{lower}^{m} →^{c.i.} A_{lower}^{mp}. More explicitly, E ∘ D induces a morphism A_{upper}^{m} → A_{lower}^{m}, (u_{1}, …, u_{m}) ↦ ((v_{1})^{p}, …, (v_{m})^{p}), where v_{i} is (−1)^{m−i} times the (m − i)-th elementary symmetric polynomial in u_{1}, …, u_{m}.

Now, we obtain the following diagram:

A_{upper}^{mp}  →^{E} A_{lower}^{mp}
↑ c.i. □  ↑ c.i.
X̃  →  X  → A_{lower}^{m+1}
↑ c.i. □  ↑ c.i. □  ↑ c.i.
Z′  →  Z  → A_{lower}^{m}
↑ c.i.  ↑ c.i.
W̃ →  W  → A_{upper}^{m}.

Here, X =^{def} A_{upper}^{mp} ×_{A_{lower}^{mp}} A_{lower}^{m+1}, X̃ denotes the normalization of X in U,

Z =^{def} X ×_{A_{lower}^{m+1}} A_{lower}^{m}, Z′ =^{def} X̃ ×_{X} Z, W denotes an irreducible component of

X̃ ×_{X} A_{upper}^{m} (regarded as a reduced closed subscheme of X̃) that is surjectively mapped onto A_{upper}^{m}, and W̃ is the normalization of the integral scheme W.

Now, we are in the situation of (1.10). More explicitly, in the notation of (1.10), K^{univ} is just the field of fractions of the completed local ring of A_{lower}^{m+1} at the generic point of A_{lower}^{m}, k_{K^{univ}} = 𝔽_{p}(A_{lower}^{m}), and k_{K_{f}^{univ}} = 𝔽_{p}(W). Moreover, we see that 𝔽_{p}(A_{upper}^{m}) = K_{h}^{univ}((α_{1})^{1/p}, …, (α_{m})^{1/p}). Thus, (1.10)(ii) implies that 𝔽_{p}(W) is generated by {(h′(α)/h′(α′))^{1/(p−1)} | α, α′ ∈ roots(h)} over 𝔽_{p}(A_{upper}^{m}). Note that u_{i} = α_{i}^{1/p} holds for each i = 1, …, m. So, if we put

𝕊 =^{def} {u_{1}, …, u_{m}}, we have ϕ′_{𝕊}(u_{i})^{p} = h′(α_{i}), hence

(\frac{(h′(α_{i})/h′(α_{j}))^{1/(p−1)}}{(ϕ′_{𝕊}(u_{i})/ϕ′_{𝕊}(u_{j}))} )^{p−1} = ϕ′_{𝕊}(u_{i})/ϕ′_{𝕊}(u_{j}).

Thus, we see that 𝔽_{p}(W) is generated by {(ϕ′_{𝕊}(u_{i})/ϕ′_{𝕊}(u_{j}))^{1/(p−1)} | i, j = 1, …, m} over 𝔽_{p}(A_{upper}^{m}).

Let V be the complement of the union of weak diagonals defined by u_{i}−u_{j} = 0 for i, j = 1, …, m, i ≠ j in A_{upper}^{m}. Since W̃ coincides with the integral

closure of A_{upper}^{m} in 𝔽_{p}(W), we now see that W̃_{V} =^{def} W̃ ×_{A_{upper}^{m}} V is finite étale covering generated by {(ϕ′_{𝕊}(u_{i})/ϕ′_{𝕊}(u_{j}))^{1/(p−1)} | i, j = 1, …, m} over V.

Now, take a finite set S as in our assumption, and put S = roots(ϕ_{S}) = {γ_{1}, …, γ_{m}}. Then, x = (γ_{1}, …, γ_{m}) gives an element of V (K) ⊂ A_{upper}^{m}(K). Moreover, condition (1.12), together with the above description of W̃_{V}, implies that the fiber of W̃_{V} → V at x consists of K-rational points. In particular, we have W̃_{V} (K) ≠ ∅. Note that W̃_{V} is smooth over 𝔽_{p}, as being étale over A_{upper}^{m}. Or, equivalently, W̃_{V} is contained in the smooth locus W̃^{sm} of W̃. Now, as K is large, we conclude that W̃ (K) is Zariski dense in W̃. (See [Pop] for the definition and properties of large fields.) Accordingly, W (K) is dense in W, a fortiori.

On the other hand, since X̃ is normal (and 𝔽_{p} is perfect), the complement of X̃^{sm} is of codimension ≥ 2 in X̃. It follows from this that W ∩ X̃^{sm} is non-empty (and open in W). Moreover, since W is integral (and 𝔽_{p} is perfect), we have W^{sm} is also non-empty and open, hence so is W′ =^{def} W^{sm} ∩ X̃^{sm}. As we have already seen, W (K) is dense in W. Accordingly, we have W′(K) ≠ ∅, hence, a fortiori, X̃^{sm}(K) ≠ ∅. As K is large, this implies that there exists a connected (or, equivalently, irreducible) component Y of X̃, such that Y (K) is dense in Y. Now, observe that Y → A_{lower}^{m+1} is (finite and) surjective. From this, Y_{U} =^{def} Y ×_{X} U = Y ∩ U (where the last intersection is taken in X̃) is nonempty (and open in Y). Thus, Y_{U}(K) is non-empty, hence, a fortiori, U(K) is non-empty. This completes the proof. □

Lemma (1.13). Let m be an integer ≥ 1. Assume that (K, m) satisfies:

(1.14)

At least one of the following holds:

K ⊃ 𝔽_{p^{2}}; p = 2; m ≡ ϵ (mod p + 1) for some ϵ ∈ {0, ±1}.

Then, there exists a finite subset S of K with cardinality m, such that ϕ_{S} satisfies (1.12).

Proof. Let s denote any element of 𝔪 ∩ (K^{×})^{p−1} (e.g., s = t^{p−1}).

Firstly, assume that either K ⊃ 𝔽_{p^{2}} or p = 2 holds. In this case, we take

any integers i_{1}, …, i_{m} with i_{1} < ⋯ < i_{m} and put S =^{def} {s^{i_{k}} | k = 1, …, m}.

Then, ♯(S) = m clearly holds. Now, for γ = s^{i_{k}} ∈ S, we have

ϕ′_{S}(γ) = ∏_{j=1}^{k−1}(s^{i_{k}} − s^{i_{j}}) ∏_{j=k+1}^{m}(s^{i_{k}} − s^{i_{j}})

= (−1)^{k−1}s^{i_{1}+···+i_{k−1}+(m−k)i_{k}} ∏_{j≠k}(1 − s^{|i_{k}−i_{j}|}).

Note that we have −1, s ∈ (K^{×})^{p−1} and 1 + 𝔪 ⊂ (K^{×})^{p−1}. (For −1, use the assumption that either K ⊃ 𝔽_{p^{2}} or p = 2 holds.) Thus, (1.12) holds.

Secondly, assume m ≡ ϵ (mod p + 1) with ϵ ∈ {0, ±1}. We may put m = (p + 1)n + ϵ. Moreover, we take any integers i_{1}, j_{1}, i_{2}, j_{2}, …, i_{n}, j_{n}, i_{n+1} with i_{1} < j_{1} < i_{2} < j_{2} < ⋯ < i_{n} < j_{n} < i_{n+1}. Now, we put S_{ϵ} = {s^{i_{k}} | k ∈ I_{ϵ}} ∪ {s^{j_{k}} + cs^{j_{k}+1} | k = 1, …, n, c ∈ 𝔽_{p}}, where

I_{ϵ} = \{ \begin{matrix}\{2, …, n\}, & ϵ = −1, \\ \{1, …, n\}, & ϵ = 0, \\ \{1, …, n + 1\}, & ϵ = 1. \end{matrix}

Then, by using s ∈ (K^{×})^{p−1}, 1 + 𝔪 ⊂ (K^{×})^{p−1}, and the fact ∏_{j∈𝔽_{p}^{×}}j = −1,

we can elementarily check that ϕ′_{S_{ϵ}}(γ) ∈ (K^{×})^{p−1} (resp. ϕ′_{S_{ϵ}}(γ) ∈ −(K^{×})^{p−1}) holds for each γ ∈ S_{ϵ}, if ϵ = 0, 1 (resp. ϵ = −1). Thus, (1.12) holds. □

Definition. Let f(T) = a_{1}T +∑_{i=0}^{m}a_{ip}T^{ip} be a superseparable polynomial (over

some field of characteristic p).

(i) We say that f is of special type, if a_{mp} = a_{1} = 1, a_{0} = 0 holds. (ii) We put def(f) =^{def} sup{r > 0 | a_{j} = 0 for all j with mp > j > mp − r} and call it the defect of f. (Thus, we have 0 < def(f) ≤ mp − 1, unless m = 0.)

Corollary (1.15). Let m be a positive integer.

(i) Assume that (K, m) satisfies (1.14). Then, there exists a monic, integral, superseparable polynomial f(T) ∈ K[T] with f(0) = 0 and deg(f) = mp, such that f is completely splittable in K. (ii) Assume that (K, m) satisfies one of the following: K ⊃ 𝔽_{p^{2}} and (p−1, m− 1) = (p + 1, m + 1) = 1; p = 2; m ≡ ϵ (mod p + 1) for some ϵ ∈ {0, ±1} and (p − 1, m − 1) = 1. Then, there exists a superseparable polynomial f(T) ∈ K[T] of special type with deg(f) = mp, such that f is completely splittable in K.

Proof. (i) By (1.11) and (1.13), there exists a monic superseparable polynomial f(T) ∈ K[T] with deg(f) = mp, such that f is completely splittable in K. Replacing f(T) by f_{c}(T) =^{def} c^{mp}f(c^{−1}T) with c ∈ K^{×}, v(c) ≫ 0, we may assume that f is integral. (Observe that roots(f_{c}) = c roots(f).) Finally, replacing f(T) by f(T + α), where α ∈ roots(f), we may assume f(0) = 0. (ii) We have K ⊃ 𝔽_{q}((t)) with q = p^{2} (resp. q = p), if K ⊃ 𝔽_{p^{2}} (resp. either p = 2 or m ≡ ϵ (mod p + 1) with ϵ ∈ {0, ±1}). Thus, it suffices to prove the assertion in the case K = 𝔽_{q}((t)). So, from now on, we assume that K = 𝔽_{q}((t)).

By (1.11) and (1.13), there exists a monic, superseparable polynomial f_{1}(T) ∈ K[T] with deg(f_{1}) = mp, such that f_{1} is completely splittable in K. Replacing f_{1}(T) by f_{1}(T + α), where α ∈ roots(f_{1}), we may assume

that f_{1}(0) = 0. Moreover, we may put f_{1}(T) = a_{1}(t)T + ∑_{i=0}^{m}a_{ip}(t)T^{ip}, where a_{ip}(t) ∈ K = 𝔽_{q}((t)), a_{1}(t) ∈ K^{×} = 𝔽_{q}((t))^{×}, and a_{mp}(t) = 1, a_{0}(t) = 0. Next,

we put f_{2}(T) =^{def} a_{1}(t^{mp−1})T + ∑_{i=1}^{m}a_{ip}(t^{mp−1})T^{ip}. Then, f_{2}(T) is completely

splittable in 𝔽_{q}((t)) ⊃ 𝔽_{q}((t^{mp−1})).

Put a_{1}(t) = ct^{r} + ⋯, where c ∈ 𝔽_{q}^{×}, r ∈ ℤ, and ⋯ ’ means the higher order terms. Then, we have a_{1}(t^{mp−1}) = ct^{r(mp−1)} + · · ·. Here, observe that (p−1, m−1) = (p+1, m+1) = 1 (resp. (p−1, m−1) = 1) is equivalent to saying (q − 1, mp − 1) = 1, for q = p^{2} (resp. q = p). So, we have 𝔽_{q}^{×} = (𝔽_{q}^{×})^{mp−1}. By using this fact (and the fact that 1 + 𝔪 ⊂ (K^{×})^{mp−1} as p ∤ mp − 1), we see that a_{1}(t^{mp−1}) ∈ (K^{×})^{mp−1}. So, write a_{1}(t^{mp−1}) = b(t)^{mp−1}. Now, it is easy to check that f(T) =^{def} b(t)^{−mp}f_{2}(b(t)T) satisfies the desired conditions. This completes the proof. □

Corollary (1.16). (i) There exists a positive integer m_{1} (which depends only on p), such that, for each positive integer m with m_{1} | m, there exists a monic, integral, superseparable polynomial f in K[T] with f(0) = 0 and deg(f) = mp, such that f is completely splittable in K. (ii) There exists a positive integer m_{2} (which depends only on p), such that, for each positive integer m with m_{2} | m, there exists a superseparable polynomial f in K[T] of special type and with deg(f) = mp, such that f is completely splittable in K.

Proof. (i) (resp. (ii)) is a direct corollary of (1.15)(i) (resp. (ii)). We can take, for example, m_{1} = p + 1 (resp. m_{2} = (p + 1)(p − 1)). □

Lemma (1.17). Let F be a field of characteristic p, and L a Galois extension of F. Let A be an 𝔽_{p}[Gal(L/F)]-submodule of L with dim_{𝔽_{p}}(A) = r < ∞, and

put ϕ_{A}(T) =^{def} ∏_{α∈A}(T − α). Then:

(i) ϕ_{A}(T) is a monic superseparable polynomial in F [T]. (ii) F_{ϕ_{A}} = F (A). (iii) deg(ϕ_{A}) = p^{r} and def(ϕ_{A}) ≥ p^{r} − p^{r−1}.

(iv) ϕ′_{A}(T) = ∏_{α∈A−{0}}α.

(v) If, moreover, F = K and A ⊂ R_{L}, then ϕ_{A} is integral.

Proof. (i) By definition, ϕ_{A} is monic and separable. It is well-known that ϕ_{A} is an additive polynomial, hence a superseparable polynomial. Since A is Gal(L/F)-stable, ϕ_{A}(T) ∈ F [T]. (ii) Clear. (iii) The first assertion is clear. The second assertion follows from the fact that ϕ_{A} is an additive polynomial. (iv) Since ϕ_{A} is monic and superseparable, we obtain

ϕ′_{A}(T) = ϕ′_{A}(0) = ∏_{α∈A−{0}}(−α) = ∏_{α∈A−{0}}α,

as desired.

(v) Clear. □

The following corollary is a final result of this §, of which (i) (resp. (ii)) will play a key role in §2 (resp. §3). Note that one of the main differences between (i) and (ii) consists in the fact that, in (ii), the defect of the superseparable polynomial is estimated from below.

Corollary (1.18). (i) Let L be a finite Galois extension of K. Then, there exists a positive integer m_{L/K}, such that, for each positive integer m with m_{L/K} | m, there exists a superseparable polynomial f(T) ∈ K[T] of special type with deg(f) = mp and K_{f} = L. (ii) We have:

∀n: positive integer,

∃m_{n}: positive integer (depending only on p and n),

∀m: positive integer with m_{K,n} | m,

∃c = c_{K,n,m}: positive real number,

∀L: (possibly infinite) Galois extension of K,

∀A: finite 𝔽_{p}[Gal(L/K)]-submodule of R_{L} with A ∩ 𝔪_{L} = {0},

∀r: integer > dim_{𝔽_{p}}(A),

∀ν: integer,

∃δ: positive integer with δ ≤ cmp^{r+1}/♯(A) and δ ≡ ν (mod n),

∀a ∈ K^{×} with v(a) = δ,

∃f(T): monic, integral, superseparable polynomial in K[T],

s.t. deg(f) = mp^{r+1}, def(f) ≥ (p − 1)p^{r}, f′(T) = a and K_{f} = K(A) ⊂ L.

Proof. (i) Since L/K is finite, we see that there exists a finite 𝔽_{p}[Gal(L/K)]- submodule A_{0} ≠ {0} of L, such that L = K(A_{0}). We put q =^{def} ♯(A_{0}), which is a power of p. Then, by (1.17), ϕ_{1} =^{def} ϕ_{A_{0}} is a monic, superseparable polynomial in K[T] with deg(ϕ_{1}) = q, ϕ_{1}(0) = 0, and ϕ′_{1}(T) = a_{0} =^{def} ∏_{α∈A_{0}−{0}} α, and L = K_{ϕ_{1}}. On the other hand, take m_{2} as in (1.16)(ii). Now, we put

m_{L/K} =^{def} q(q − 1)m_{2}.

Let m be any positive integer divisible by m_{L/K}, and put n =^{def} m/m_{L/K}. Then, by (1.16)(ii), there exists a superseparable polynomial f_{1}(T) ∈ K[T] of special type and with degree n(q − 1)m_{2}p, such that f_{1} is completely splittable in K.

For each b ∈ K^{×}, we put f_{b}(T) =^{def} b^{n(q−1)m_{2}p}f_{1}(b^{−1}T) (resp. ϕ_{b}(T) =^{def} b^{q}ϕ_{1}(b^{−1}T)), so that f_{b} (resp. ϕ_{b}) is a monic, superseparable polynomial with deg(f_{b}) = n(q − 1)m_{2}p (resp. deg(ϕ_{b}) = q), f_{b}(0) = 0 (resp. ϕ_{b}(0) = 0), and f′_{b}(T) = b^{n(q−1)m_{2}p−1} (resp. ϕ′_{b}(T) = b^{q−1}a_{0}).

Now, by (1.4)(ii), every polynomial in L[T] with degree q which is sufficiently close to ϕ_{b}(T) (b ∈ K^{×}) is completely splittable in L. By using this, we see that F_{b,b′} =^{def} f_{b′} ∘ ϕ_{b} ∈ K[T] satisfies K_{F_{b,b′}} = L for all b′ ∈ K^{×} with v(b′) ≥ C(b), where C(b) denotes a constant depending on b. Observe that F_{b,b′} is a monic, superseparable polynomial with deg(F_{b,b′}) = n(q −1)m_{2}p×q = mp, F_{b,b′}(0) = 0, and F′_{b,b′}(T) = (b′)^{n(q−1)m_{2}p−1}b^{q−1}a_{0}.

Now, take b = a_{0}^{−nm_{2}p} and b′ = a_{0}d^{mp−1} for any d ∈ K^{×} with v(d) sufficiently large, then f(T) =^{def} D^{−mp}F_{b,b′}(DT) with D =^{def} d^{n(q−1)m_{2}p−1} satisfies all the desired properties. (ii) Let m_{1} be as in (1.16)(i), and choose any common multiple m_{n} > 1 of m_{1}, p − 1, and n.

Let m be any positive integer with m_{n} | m. Then, by (1.16)(i), there exists a monic, integral, superseparable polynomial f_{1}(T) ∈ K[T] with deg(f_{1}) = mp, f_{1}(0) = 0, and K_{f_{1}} = K. Now, put f′_{1}(T) = a_{1} ∈ R and c =^{def} max(\frac{v(a_{1})}{m}, \frac{np}{p−1}).

Let L be any Galois extension of K, A any finite 𝔽_{p}[Gal(L/K)]-submodule of R_{L} with A ∩ 𝔪_{L} = {0}, and r any integer > r_{0} =^{def} dim_{𝔽_{p}}(A). Let ν be any integer. We define μ to be the unique integer with 0 < μ ≤ n, such that μ ≡

v(a_{1})−(r −r_{0})−ν (mod n). We put δ =^{def} v(a_{1})+μ(mp−1)+ ∑_{j=1}^{r−r_{0}}(mp^{j+1} −1).

Then, we have

δ ≤ v(a_{1}) + ∑_{j=0}^{r−r_{0}}nmp^{j+1}

= m (\frac{v(a_{1})}{m} + \frac{np}{p − 1}(p^{r−r_{0}+1} − 1))

≤ m(c + c(p^{r−r_{0}+1} − 1))

= cmp^{r+1}/♯(A)

and

δ ≡ v(a_{1}) − μ − (r − r_{0}) ≡ ν (mod n),

as desired.

Let a be any element of K^{×} with v(a) = δ. For j = 0, …, r − r_{0}, we shall inductively define a monic, integral, superseparable polynomial f_{2,j}(T) with deg(f_{2,j}) = mp^{j+1}, f_{2,j}(0) = 0, f_{2,j}(T) ≡ T^{mp^{j+1}} (mod 𝔪), and K_{f_{2,j}} = K, as

follows. First, for j = 0, we put g_{0}(T) =^{def} f_{1}(T) and f_{2,0}(T) =^{def} t^{μmp}g_{0}(t^{−μ}T). Next, for j with 0 < j < r−r_{0}, we put g_{j} =^{def} f_{2,j−1}∘ϕ_{𝔽_{p}}, where ϕ_{𝔽_{p}}(T) = T^{p}−T,

and f_{2,j}(T) =^{def} t^{mp^{j+1}}g_{j}(t^{−1}T). Finally, for j = r − r_{0}, let u and u′ be elements of R^{×}, which we shall fix later, and we put g_{r−r_{0}} =^{def} f_{2,r−r_{0}−1} ∘ ϕ_{u𝔽_{p}}, where

ϕ_{u𝔽_{p}}(T) = T^{p} − u^{p−1}T, and f_{2,r−r_{0}}(T) =^{def} (u′t)^{mp^{r−r_{0}+1}}g_{r−r_{0}}((u′t)^{−1}T).

We can check inductively that f_{2,j}(T) is a monic, integral, superseparable polynomial with deg(f_{2,j}) = mp^{j+1}, f_{2,j}(0) = 0, f_{2,j}(T) ≡ T^{mp^{j+1}} (mod 𝔪), and K_{f_{2,j}} = K. Moreover, since f′_{2,0} = t^{μ(mp−1)}a_{1}, f′_{2,j} = (−t^{mp^{j+1}−1})f′_{2,j−1} (0 < j < r − r_{0}), and f′_{2,r−r_{0}} = (−u^{p−1})(u′t)^{mp^{r−r_{0}+1}−1}, we obtain

f′_{2,r−r_{0}} = u^{p−1}(u′)^{mp^{r−r_{0}+1}−1}(−1)^{r−r_{0}}a_{1}t^{μ(mp−1)+∑_{j=1}^{r−r_{0}} (mp^{j+1}−1)}
= u^{p−1}(u′)^{mp^{r−r_{0}+1}−1}(−1)^{r−r_{0}}(a_{1}/t^{v(a_{1})})t^{δ}.

So, put u′ = (−1)^{r−r_{0}}(a_{1}/t^{v(a_{1})})(t^{δ}/a)w, where w =^{def} ∏_{α∈A−{0}}α ∈ R^{×}, and u =

(u′)^{− \frac{m}{p−1} p^{r−r_{0}+1}}, then we have f′_{2,r−r_{0}}(T) = aw^{−1}. Now, we put f_{2} =^{def} f_{2,r−r_{0}}.

Finally, put f =^{def} f_{2} ∘ ϕ_{A}. Then, f is a monic, integral, superseparable polynomial in K[T] with deg(f) = deg(f_{2}) deg(ϕ_{A}) = mp^{r−r_{0}+1}♯(A) = mp^{r+1}, f′ = f′_{2}ϕ′_{A} = (aw^{−1})w = a, and K_{f} = K(A) ⊂ L. Finally, by the above construction, we see that f is in the form of (a superseparable polynomial with degree mp) ∘ (an additive polynomial with degree p^{r}). As m ≥ m_{K,n} > 1 and r > r_{0} ≥ 0, this implies def(f) ≥ p^{r+1} − p^{r}. This completes the proof. □

Remark (1.19). So far, we have assumed that K is a complete discrete valuation field (of characteristic p). However, this assumption is superfluous. More specifically, (1.4), (1.5), (1.11), (1.13), (1.15), (1.16), and (1.18) remain valid if we replace this assumption by the weaker assumption that K is henselian (of characteristic p), and (1.10) remains valid if we replace the phrase ‘completion’ by ‘henselization’. Indeed, the proof of the henselian case is just similar to the complete case.

Moreover, among these, (1.11), (1.13), (1.15) (except that we need to delete the phrase ‘integral’ in (i)), (1.16) (except that we need to delete the phrase ‘integral’ in (i)), and (1.18)(i) remain valid, if we only assume that K is a large field (of characteristic p) in the sense of [Pop]. (In particular, we do not have to assume that K is equipped with a discrete valuation.) Indeed, we see that these statements can be formulated in terms of the existence of K-rational points of K-varieties. The validity of the complete case implies that these varieties admit K((t))-rational points. Now, the large case follows directly from one of the equivalent definitions of large fields (see [Pop], Proposition 1.1, (5)).

§2. Unramified extensions with prescribed local extensions. In this §, we use the following new notation. Let C be a noetherian, normal, integral, separated 𝔽_{p}-scheme of dimension 1. We denote by K the rational function field of C, and fix an algebraic closure \overline{K} of K. We denote by K^{sep} and G = G_{K} the separable closure of K in \overline{K} and the absolute Galois group Gal(K^{sep}/K) of K, respectively. Let Σ_{C} be the set of closed points of C. For each v ∈ Σ_{C}, we denote by R_{v} the completion of the local ring O_{C,v}. This is a complete discrete valuation ring. We denote by K_{v}, 𝔪_{v} and k_{v} the field of fractions of R_{v}, the maximal ideal of R_{v} and the residue field R_{v}/𝔪_{v} of R_{v}, respectively. We fix an algebraic closure \overline{K}_{v} of K_{v}, and denote by K_{v}^{sep} and G_{v} = G_{K_{v}} the separable closure of of K_{v} in \overline{K}_{v} and the absolute Galois group Gal(K_{v}^{sep}/K_{v}), respectively.

Definition. We refer to a tuple C = (C, Σ, {L_{v}}_{v∈Σ}) as a base scheme data, if C is as above, Σ is a (possibly empty) finite subset of Σ_{C}; and, for each v ∈ Σ, L_{v} is a (possibly infinite) normal subextension of \overline{K}_{v} over K_{v}, such that L_{v} ∩ K_{v}^{sep} is v-adically dense in L_{v}. (For example, this last condition is satisfied if either L_{v}/K_{v} is Galois or L_{v} = \overline{K}_{v}.)

If, moreover, C is a normal, geometrically integral curve over a field k of characteristic p, we refer to C as a base curve data over k.

For a base scheme data C = (C, Σ, {L_{v}}_{v∈Σ}), we put B = B_{C} =^{def} C − Σ. If, moreover, B is affine, then we put R = R_{C} =^{def} Γ(B, O_{B}), so that R is a Dedekind domain and that B = Spec(R).

We say that a base scheme data C = (C, Σ, {L_{v}}_{v∈Σ}) is finite, if L_{v} is a finite extension of K_{v} for each v ∈ Σ. (In this case, L_{v} is automatically Galois over K_{v}.)

Definition. Let C = (C, Σ, {L_{v}}_{v∈Σ}) be a base scheme data. Let K′ be an extension of K contained in \overline{K}. Then, we say that K′ is C-distinguished (resp. C-admissible), if the integral closure C′ of C in K′ is étale over B; and, for each v ∈ Σ and each embedding ι : \overline{K} ↪ \overline{K}_{v} over K, we have ι(K′)K_{v} = L_{v} (resp. ι(K′)K_{v} ⊂ L_{v}).

Theorem (2.1). Let C = (C, Σ, {L_{v}}_{v∈Σ}) be a finite base scheme data, and assume that C is affine. Then, there exists a C-distinguished finite Galois extension K′/K.

Proof. For each v ∈ Σ, take a positive integer m_{L_{v}/K_{v}} as in (1.18)(i), and let m be any common multiple of m_{L_{v}/K_{v}} (v ∈ Σ). Then, for each v ∈ Σ, there exists a superseparable polynomial f_{v}(T) ∈ K_{v}[T] of special type and with degree mp, such that L_{v} = (K_{v})_{f_{v}}.

Now, observe that R is dense in ∏_{v∈Σ} K_{v}. (This follows essentially from the Chinese Remainder Theorem for the Dedekind domain Γ(C, O_{C}).) From this, we can take a superseparable polynomial f(T) ∈ R[T] of special type and with degree mp, which is arbitrarily close to f_{v} for each v ∈ Σ. Then, we have (K_{v})_{f} = (K_{v})_{f_{v}} = L_{v}. Or, equivalently, K_{f} ⊗_{K} K_{v} is isomorphic to a direct product of copies of L_{v} over K_{v}. On the other hand, for each v ∈ Σ_{C} − Σ, f mod 𝔪_{v} is a separable polynomial over k_{v}, since f is of special type. From this, we see that K_{f} is unramified at v. Thus, K′ =^{def} K_{f} satisfies all the desired properties. □

Definition. Let F be a field. We denote by F^{sep} and G_{F} a separable closure of F and the absolute Galois group Gal(F^{sep}/F) of F, respectively. For each prime number l, we define F (l) to be the union of finite Galois extensions F′ of F in F^{sep} with Gal(F′/F) ≃ (ℤ/lℤ)^{n} for some n. Thus, F (l) corresponds via Galois theory to the closed subgroup G_{F} (l) =^{def} \overline{[G_{F} , G_{F}](G_{F})^{l}} of G_{F} (which coincides with the kernel of G_{F} ↠ G_{F}^{ab}/(G_{F}^{ab})^{l}).

Definition. Let C = (C, Σ, {L_{v}}_{v∈Σ}) be a base scheme data and Σ_{∞} a subset of Σ. Let K′ be an extension of K contained in \overline{K}. Then, we say that K′ is nearly C-distinguished (resp. nearly C-admissible) with respect to Σ_{∞}, if the integral closure C′ of C in K is étale over B; for each v ∈ Σ − Σ_{∞} and each embedding ι : \overline{K} ↪ \overline{K}_{v} over K, we have ι(K′)K_{v} = L_{v} (resp. ι(K′)K_{v} ⊂ L_{v}); and, for each v ∈ Σ_{∞} and each embedding ι : \overline{K} ↪ \overline{K}_{v} over K, we have L_{v} ⊂ ι(K′)K_{v} ⊂ L_{v}(p) (resp. ι(K′)K_{v} ⊂ L_{v}(p)).

Theorem (2.2). Let k be a field of characteristic p, and C = (C, Σ, {L_{v}}_{v∈Σ}) a finite base curve data over k. Let Σ_{∞} be a subset of Σ, and assume that C − Σ_{∞} is affine. Then, there exists a finite Galois extension K′/K that is nearly C-distinguished with respect to Σ_{∞}.

Proof. Let C^{∗} be the normal, geometrically integral compactification of C, and put Σ^{∗} =^{def} Σ ∪ (Σ_{C^{∗}} − Σ_{C}) and Σ_{∞}^{∗} =^{def} Σ_{∞} ∪ (Σ_{C^{∗}} − Σ_{C}). Moreover, for each v ∈ Σ_{C^{∗}} − Σ_{C}, we choose any finite Galois extension L_{v} of K_{v} (say, L_{v} = K_{v}). Then, replacing C = (C, Σ, {L_{v}}_{v∈Σ}) by (C^{∗}, Σ^{∗}, {L_{v}}_{v∈Σ^{∗}}) and Σ_{∞} by Σ_{∞}^{∗}, we may assume that C is proper over k. In this case, we have Σ_{∞} ≠ ∅, since C − Σ_{∞} is affine.

For each v ∈ Σ, take a positive integer m_{L_{v}/K_{v}} as in (1.18)(i), and let m be any common multiple of m_{L_{v}/K_{v}} (v ∈ Σ − Σ_{∞}), pm_{L_{v}/K_{v}} (v ∈ Σ_{∞}) and 2. Then, for each v ∈ Σ − Σ_{∞} (resp. v ∈ Σ_{∞}), there exists a superseparable polynomial f_{v}(T) ∈ K_{v}[T] of special type and with degree mp (resp. m), such that L_{v} = (K_{v})_{f_{v}}.

Now, let v ∈ Σ−Σ_{∞}. Then, for each polynomial f_{1,v}(T) ∈ K_{v}[T] with degree mp which is sufficiently close to f_{v}(T), we have (K_{v})_{f_{1,v}} = (K_{v})_{f_{v}} = L_{v}. More precisely, we can take (sufficiently small) n_{v} ∈ ℤ and (sufficiently large) m_{v} ∈ ℤ with n_{v} < m_{v}, so that every coefficient of f_{v} belongs to 𝔪_{v}^{n_{v}}, and that, if every coefficient of f_{1,v} − f_{v} belongs to 𝔪_{v}^{m_{v}}, then we have (K_{v})_{f_{1,v}} = (K_{v})_{f_{v}}.

On the other hand, let v ∈ Σ_{∞}. Then, similarly as above, we can take (sufficiently small) n_{v} ∈ ℤ and (sufficiently large) m_{v} ∈ ℤ with n_{v} < m_{v}, so that every coefficient of f_{v} belongs to 𝔪_{v}^{n_{v}}, and that, for each polynomial f_{1,v}(T) ∈ K_{v}[T] with degree m, if every coefficient of f_{1,v} − f_{v} belongs to 𝔪_{v}^{m_{v}}, then we have (K_{v})_{f_{1,v}} = (K_{v})_{f_{v}}. Moreover, replacing n_{v} and m_{v} if necessary, we may assume that, for each monic polynomial f_{1,v}(T) ∈ K_{v}[T] with degree m whose constant term is 0, if every coefficient of f_{1,v}−f_{v} belongs to 𝔪_{v}^{m_{v}}, then we have (K_{v})_{f_{1,v}} = (K_{v})_{f_{v}} and there exists a bijection ι : roots(f_{v})→^{∼} roots(f_{1,v}), such that, for each α ∈ roots(f_{v}), v(ια) = v(α), μ(f_{1,v}, ι(α)) = μ(f_{v}, α), and f′_{1,v}(ι(α)) ∼ f′_{v}(α) = 1. (For the notations μ(−, −) and ∼, see §1.) Now, we let d_{v} denote the minimal non-negative integer satisfying

μ(f_{v}, α) < min (d_{v} − v(f′_{v}(α)) + \frac{1}{p}v(α), \frac{p}{p − 1}(d_{v} − v(f′_{v}(α))))

= min (d_{v} + \frac{1}{p}v(α), \frac{p}{p − 1}d_{v})

for all α ∈ roots(f_{v}).

Lemma (2.3). Let P_{1}, …, P_{r} be distinct closed points of C, and, for each i = 1, …, r, let a_{i} and b_{i} be integers with a_{i} ≥ b_{i}. If b_{1}[k_{P_{1}} : k] + ⋯ + b_{r}[k_{P_{r}} : k] > 2p_{a}(C) − 2, then the natural map Γ(C, O_{C}(a_{1}P_{1} + ⋯ + a_{r}P_{r})) → 𝔪_{P_{1}}^{−a_{1}}/𝔪_{P_{1}}^{−b_{1}} ⊕⋯⊕𝔪_{P_{r}}^{−a_{r}}/𝔪_{P_{r}}^{−b_{r}} is surjective. Here, p_{a}(C) denotes the arithmetic genus of C.

Proof. This follows from [CFHR], Theorem 1.1. □

We fix a sufficiently large integer N satisfying

(2.4)  (p(p − 1) ∑_{v∈Σ_{∞}}[k_{v} : k]) N − ∑_{v∈Σ}[k_{v} : k]m_{v} > 2p_{a}(C) − 2

and

(2.5)  (mp − 1)(p − 1)N ≥ max{d_{v} | v ∈ Σ_{∞}}(≥ 0),

and, for each v ∈ Σ_{∞}, choose any e_{v} ∈ K_{v} with v(e_{v}) = N.

Now, put

g_{v}(T) =^{def} \{ \begin{matrix}f_{v}(T), & v ∈ Σ − Σ_{∞}, \\ e_{v}^{−mp(p−1)}f_{v}(−e_{v}^{p(p−1)}T^{p}) + T, & v ∈ Σ_{∞}. \end{matrix}

Then, g_{v}(T) is a superseparable polynomial in K_{v}[T] of special type and with degree mp. (For v ∈ Σ_{∞}, use the assumption that 2 | m.) So, by (2.3) and (2.4), we see that there exists a superseparable polynomial g(T) ∈ R[T] of special type and with degree mp, such that every coefficient of g(T) − g_{v}(T) belongs to 𝔪_{v}^{m_{v}} (resp. 𝔪_{v}^{−p(p−1)N+m_{v}}) for v ∈ Σ − Σ_{∞} (resp. v ∈ Σ_{∞}).

We put K′ =^{def} K_{g}. Just as in the proof of (2.1), K′ satisfies the desired property for v ∈ Σ − Σ_{∞} and v ∈ Σ_{C} − Σ. So, we shall observe what happens at v ∈ Σ_{∞}. We put g_{e_{v}}(T) =^{def} e_{v}^{mp(p−1)}g(−e_{v}^{−(p−1)}T). Or,

writing g(T) = T + k(T^{p}), we have g_{e_{v}}(T) =^{def} −e_{v}^{(mp−1)(p−1)}T + k_{v}(T^{p}),

where k_{v}(T) =^{def} e_{v}^{mp(p−1)}k(−e_{v}^{−(p−1)p}T^{p}). By the choice of g, every coefficient of g(T) − g_{v}(T) belongs to 𝔪_{v}^{−p(p−1)N+m_{v}}, hence every coefficient of g_{e_{v}}(T)−(f_{v}(T^{p})−e_{v}^{(mp−1)(p−1)}T) belongs to e_{v}^{p(p−1)}𝔪_{v}^{−p(p−1)N+m_{v}} = 𝔪_{v}^{m_{v}}. Or, equivalently, every coefficient of k_{v}−f_{v} belongs to 𝔪_{v}^{m_{v}}. Thus, we may apply the preceding argument to f_{1,v} = k_{v}. Since, moreover, v(−e_{v}^{(mp−1)(p−1)}) = (mp − 1)(p − 1)N ≥ d_{v} by (2.5), we may apply (1.5) to g_{e_{v}}(T) = −e_{v}^{(mp−1)(p−1)}T + k_{v}(T^{p}). Then, firstly, we have (K_{v})_{g_{ev}} ⊃ (K_{v})_{k_{v}} = (K_{v})_{f_{v}} = L_{v}. Secondly, for each α_{1} ∈ roots(k_{v}), (−(−e_{v}^{(mp−1)(p−1)})/k′_{v}(α_{1})) ∼ (e_{v}^{mp−1})^{p−1} belongs to ((K_{v})_{k_{v}}^{×})^{p−1}. Thus, we have M_{g_{ev}} = (K_{v})_{k_{v}}. Thirdly, since Gal((K_{v})_{g_{ev}} /M_{g_{ev}}) is a subgroup of (C_{p})^{I_{m}}, we have (K_{v})_{g_{ev}} ⊂ M_{g_{ev}} (p). Combining these, we obtain L_{v} ⊂ (K_{v})_{g_{ev}} ⊂ L_{v}(p). Finally, since roots(g_{e_{v}}) = −e_{v}^{p−1} roots(g), we have (K_{v})_{g_{ev}} = (K_{v})_{g}. Thus, K′ = K_{g} satisfies the desired property at v ∈ Σ_{∞}. This completes the proof. □

§3. Main results. In this §, we use the following notation. Let k be an (a possibly infinite) algebraic extension of 𝔽_{p} and C a smooth, geometrically connected (or, equivalently, normal, geometrically integral) curve over k. In particular, C is a noetherian, normal, integral, separated 𝔽_{p}-scheme of dimension 1, and we use the notations introduced at the beginning of §2 for this C. Among other things, see §2 for the definition of base curve data.

Definition. (i) We refer to a tuple S = (C, f : X → B, {Ω_{v}}_{v∈Σ}) as a (smooth) Skolem data, if C = (C, Σ, {L_{v}}_{v∈Σ}) is a base curve data; B = B_{C}; f : X → B is a smooth, surjective morphism whose generic fiber X_{K} is geometrically irreducible; and, for each v ∈ Σ, Ω_{v} is a non-empty, v-adically open, G_{v}-stable subset of X(L_{v}). (Observe that X is automatically irreducible.) (ii) We refer to a tuple B = (C, Y_{1}, …, Y_{r} ⊂ P(E), {Ω̌_{v}}_{v∈Σ}) as a Bertini data, if C = (C, Σ, {L_{v}}_{v∈Σ}) is a base curve data; E is a locally free sheaf of finite rank ≠ 0 on B; r ≥ 0; Y_{i} is an irreducible, reduced, closed subscheme of P(E); and, for each v ∈ Σ, Ω̌_{v} is a non-empty, v-adically open, G_{v}-stable subset of P(Ě)(L_{v}), where Ě =^{def} Hom_{O_{B}}(E, O_{B}).

For a Bertini data B = (C, Y_{1}, …, Y_{r} ⊂ P(E), {Ω̌_{v}}_{v∈Σ}), we define Y_{i}^{sm} (i = 1, …, r) to be the set of points of Y_{i} at which Y_{i} → B is smooth. This is an (a possibly empty) open subset of Y_{i}, and we regard it as an open subscheme of Y_{i}.

Definition. (i) Let S = (C, f : X → B, {Ω_{v}}_{v∈Σ}) be a Skolem data with C = (C, Σ, {L_{v}}_{v∈Σ}). Then, an S-admissible quasi-section is a B-morphism s : B′ → X, where B′ is the integral closure of B in a finite, C-admissible extension K′ of K, such that, for each v ∈ Σ, the image of B′_{L_{v}} =^{def} B′ ×_{B} L_{v}

in X_{L_{v}} =^{def} X ×_{B} L_{v} is contained in Ω_{v} (⊂ X(L_{v}) = X_{L_{v}}(L_{v})). (ii) Let B = (C, Y_{1}, …, Y_{r} ⊂ P(E), {Ω̌_{v}}_{v∈Σ}) be a Bertini data with C = (C, Σ, {L_{v}}_{v∈Σ}). Then, a B-admissible quasi-hyperplane is a hyperplane H in P(E)_{B′}, where B′ is the integral closure of B in a finite, C-admissible extension K′ of K, such that (a) for each i = 1, …, r, each geometric point \overline{b} of B′ and each irreducible component P of Y_{i,\overline{b}}, we have P ∩ H\overline{b} ⊊ P; (b) for each i = 1, …, r, the scheme-theoretic intersection (Y_{i}^{sm})_{B′} ∩ H (in P(E)_{B′}) is smooth over B′; (c) for each i = 1, …, r and each irreducible component P of Y_{i,\overline{K}} with dim(P) ≥ 2, P ∩ H\overline{K} is irreducible; and (d) for each v ∈ Σ, the image of B′_{L_{v}} in P(Ě)_{L_{v}} by the base change to L_{v} of the classifying morphism [H] : B′ → P(Ě) over B is contained in Ω̌_{v} (⊂ P(Ě)(L_{v}) = P(Ě)_{L_{v}}(L_{v})).

Definition. Let C = (C, Σ, {L_{v}}_{v∈Σ}) be a base curve data over k.

(i) We denote by R_{L_{v}}, 𝔪_{L_{v}}, and k_{L_{v}} the integral closure of R_{v} in L_{v}, the maximal ideal of R_{L_{v}}, and the residue field R_{L_{v}}/𝔪_{L_{v}}, respectively. (ii) We say that C satisfies condition (RI), if [k_{L_{v}} : 𝔽_{p}] = ∞ for all v ∈ Σ. (Here, ‘RI’ means ‘residually infinite’.)

Now, the following are the main results of the present paper.

Theorem (3.1). Let S = (C, f : X → B, {Ω_{v}}_{v∈Σ}) be a Skolem data with C = (C, Σ, {L_{v}}_{v∈Σ}), and assume that C is affine and that (RI) holds. Then, there exists an S-admissible quasi-section.

Theorem (3.2). Let B = (C, Y_{1}, …, Y_{r} ⊂ P(E), {Ω̌_{v}}_{v∈Σ}) be a Bertini data with C = (C, Σ, {L_{v}}_{v∈Σ}), and assume that C is affine and that (RI) holds. Then, there exists a B-admissible quasi-hyperplane.

The aim of the rest of this § is to prove these theorems, together and step by step. From now on, we put C = (C, Σ, {L_{v}}_{v∈Σ}), S = (C, f : X → B, {Ω_{v}}_{v∈Σ}), and B = (C, Y_{1}, …, Y_{r} ⊂ P(E), {Ω̌_{v}}_{v∈Σ}), and assume always that C is affine and that (RI) holds.

Definition. We say that a Skolem data S = (C, f : X → B, {Ω_{v}}_{v∈Σ}) is essentially rational, if Ω_{v} ∩ X(K_{v}) ≠ ∅ for each v ∈ Σ.

Step 1. Assume that S is essentially rational, and that X is an open subscheme of P_{B}^{1}. Then, there exists an S-admissible quasi-section.

Proof. We put W =^{def} P_{B}^{1} − X. By shrinking X if necessary, we may assume that W is purely of codimension 1 in P_{B}^{1} and that W contains the infinity section ∞_{B} of P_{B}^{1}. Next, we put R̃ =^{def} Γ(C, O_{C}), which is a Dedekind domain contained in R = Γ(B, O_{B}), such that C = Spec(R̃).

Since Pic(C) is a torsion group (cf. [Mo2], 1.9), there exists n > 0, such that (𝔪_{v} ∩ R̃)^{n} is a principal ideal of R̃ for each v ∈ Σ. In particular, there exists ϖ ∈ R̃, such that (∏_{v∈Σ}(𝔪_{v} ∩ R̃))^{n} = R̃ϖ. On the other hand, since A^{1}(R) is dense in ∏_{v∈Σ} P^{1}(K_{v}), there exists x ∈ A^{1}(R)(= R), such that x ∈ Ω_{v} ∩ X(K_{v}) for each v ∈ Σ. (Here, we have used the assumption that S is essentially rational.) Since Ω_{v} is v-adically open in X(L_{v}), there exists l_{v} ≥ 0, such that x + (𝔪_{v}R_{L_{v}})^{l_{v}} ⊂ Ω_{v}. Finally, take a sufficiently large integer M, such that nM ≥ l_{v} for each v ∈ Σ and that nM > v(ω − x) for each v ∈ Σ and each ω ∈ W (\overline{K}_{v}) − \{∞\}.

Now, let S denote the coordinate of A_{B}^{1} that we are using. Since ϖ ∈ R^{×} and x ∈ R, the coordinate change S → T =^{def} (S −x)/ϖ^{M} gives an automorphism of P_{B}^{1} that fixes the infinity section ∞_{B}. (More sophisticatedly, this corresponds to a certain blowing-up(-and-down) process in the fibers of P_{C}^{1} → C at Σ.)

From now, we shall use this new coordinate T. Then, by the choice of (ϖ, x, M), we have R_{L_{v}} = A^{1}(R_{L_{v}}) ⊂ Ω_{v} for each v ∈ Σ and v(ω) < 0 for each v ∈ Σ and ω ∈ W (\overline{K}_{v}) − \{∞\}.

We define W̃ to be the closure of W in P_{C}^{1}, which contains the infinity section ∞_{C} of P_{C}^{1}. By the above choice of coordinate, we have W̃ ∩ P_{k_{v}}^{1} ⊂ ∞_{k_{v}} for each v ∈ Σ. From now, we regard W̃ as a reduced closed subscheme (or, as a divisor) of P_{C}^{1}. By [Mo2], Théorème 1.3, Pic(W̃) is a torsion group. So, let s_{0} be the order of the class of the line bundle O_{P_{C}^{1}} (1)|W̃ on W̃. On the other hand, let e be the degree of W̃ over C. Now, choose a positive integer s which is divisible by s_{0} and greater than e − 2. As in [Mo2], proof of Théorème 1.7, Étape VIII, consider the exact sequence

0 → O_{P_{C}^{1}} (s)(−W̃) → O_{P_{C}^{1}} (s) → O_{P_{C}^{1}} (s)|W̃ → 0,

which induces the following long exact sequence:

⋯ → H^{0}(P_{C}^{1}, O_{P_{C}^{1}} (s)) → H^{0}(W̃ , O_{P_{C}^{1}} (s)|W̃)
→ H^{1}(P_{C}^{1}, O_{P_{C}^{1}} (s)(−W̃)) → · · · .

Since s_{0} | s, we have O_{P_{C}^{1}} (s)|W̃ ≃ OW̃, so that there exists an element g_{0} ∈ H^{0}(W̃ , O_{P_{C}^{1}} (s)|W̃) which generates O_{P_{C}^{1}} (s)|W̃. On the other hand, since s > e − 2, we see that H^{1}(P_{C}^{1}, O_{P_{C}^{1}} (s)(−W̃)) (which is the dual of H^{0}(P_{C}^{1}, O_{P_{C}^{1}} (−2 − s)(W̃))) vanishes. Thus, there exists an element g ∈ H^{0}(P_{C}^{1}, O_{P_{C}^{1}} (s)) that maps to g_{0}. Then, we have Supp(g) ∩ W̃ = ∅. In particular, we have Supp(g) ∩ ∞_{C} = ∅.

We may identify H^{0}(P_{C}^{1}, O_{P_{C}^{1}} (s)) with the set of polynomials in R̃[T] with degree ≤ s. Then, since Supp(g) ∩ ∞_{C} = ∅, we see that g is strictly of degree s and that the coefficient u of T^{s} in g = g(T) is an element of R̃^{×}. So, replacing g by u^{−1}g (and g_{0} by u^{−1}g_{0}), we may assume that g is monic.

Next, since Pic(A_{C}^{1}) = Pic(C) is a torsion group, there exists an element w(T) ∈ R̃[T], such that the zero locus of w(T) in A_{C}^{1} coincides (settheoretically) with W̃ ∩ A_{C}^{1}. Recall that, for each v ∈ Σ and each root ω of w in \overline{K}_{v}, we have v(ω) < 0. From this fact (and the fact that W̃ ∩ A_{k_{v}}^{1} ⊊ A_{k_{v}}^{1}), we see that w(0) is a unit in R_{v} and that w(T) ≡ w(0) (mod 𝔪_{v}). Moreover, since k_{v}^{×} is a torsion group, we may assume that w(0) ≡ 1 (mod 𝔪_{v}) for each v ∈ Σ, replacing w by a suitable power. Now, we define d to be the degree of w.

First, assume Σ ≠ ∅, and we shall apply (1.18)(ii) carefully. Let n be as in the beginning of the proof. Then, there exists a positive integer m_{n}. We choose a positive integer m to be a common multiple of m_{n} and s. For this m, we obtain a positive real number c_{K_{v},n,m}. We put c_{m} =^{def} max{c_{K_{v},n,m} | v ∈ Σ}.

We put D =^{def} \frac{d}{p−1} and E =^{def} \frac{p}{p−1}c_{m}mp. We take a positive integer t, such that p^{t} > D. Next, since we are assuming the condition (RI) that k_{L_{v}} is an infinite algebraic extension of 𝔽_{p}, there exists a finite subfield of k_{L_{v}} with arbitrarily large cardinality. So, we may take a finite subfield 𝔽_{v} of k_{L_{v}}, such that p^{r_{v}} =^{def} ♯(𝔽_{v}) > Ep^{t}. Since R_{v} is complete, 𝔽_{v} admits a canonical lifting in R_{L_{v}}, to which we refer again as 𝔽_{v}. Now, take a positive integer r > max{r_{v}(v ∈ Σ), t}.

Applying (1.18)(ii) to L = L_{v}, A = 𝔽_{v}, r as above, and ν = 0, we see that there exists a positive integer δ_{v}, such that δ_{v} ≤ c_{m}mp^{r+1}/♯(𝔽_{v}) = c_{m}mp^{r−r_{v}+1} and that δ_{v} ≡ 0 (mod n). Since δ_{v} is divisible by n, 𝔞 =^{def} ∏_{v∈Σ}(𝔪_{v} ∩ R̃)^{δ_{v}} is a principal ideal of R̃. So, let a ∈ R̃ be a generator of 𝔞. Then, v(a) = δ_{v} for each v ∈ Σ. Now, the conclusion of (1.18)(ii) is that there exists a monic, integral, superseparable polynomial f_{v}(T) ∈ K_{v}[T], such that deg(f_{v}) = mp^{r+1}, def(f_{v}) ≥ (p−1)p^{r}, f′_{v}(T) = a and (K_{v})_{f_{v}} = K_{v}𝔽_{v} ⊂ L_{v}. Moreover, by using (1.4)(ii) and the Chinese Remainder Theorem (for the Dedekind domain R̃), we may assume that f_{v}(T) ∈ R̃[T] and f_{v}(T) does not depend on v. So, put f(T) =^{def} f_{v}(T) for some (or, equivalently, all) v ∈ Σ, then, f is monic, superseparable polynomial in R̃[T], such that deg(f) = mp^{r+1}, def(f) ≥ (p − 1)p^{r}, f′(T) = a and (K_{v})_{f} = K_{v}𝔽_{v} ⊂ L_{v} for each v ∈ Σ.

Next, assume Σ = ∅. In this case, we define m to be any multiple of s, put D =^{def} \frac{d}{p−1}, take a positive integer t with p^{t} > D and a positive integer r with r > t, and let a be any element of R̃^{×}. Now, we choose a monic superseparable polynomial f(T) ∈ R̃[T] = R[T], such that deg(f) = mp^{r+1}, def(f) ≥ (p−1)p^{r}, and f′(T) = a. (For example, put f(T) = T^{mp^{r+1}} + aT.)

Finally, we put

F (T) =^{def} g(T)^{\frac{m}{s} p^{r+1}} + w(T)^{p^{r−t}}(f(T) − g(T)^{\frac{m}{s} p^{r+1}}) ∈ R̃[T].

Claim (3.3). F is monic of degree mp^{r+1}.

Proof. f is monic with deg(f) = mp^{r+1} and def(f) ≥ (p − 1)p^{r}. On the other hand, since g is monic of degree s, g^{\frac{m}{s}} is monic of degree m, hence g^{\frac{m}{s} p^{r+1}} is monic of degree mp^{r+1} and with ‘defect’ ≥ p^{r+1} ≥ (p − 1)p^{r}. From these, we see that f − g^{\frac{m}{s} p^{r+1}} has degree ≤ mp^{r+1} − (p − 1)p^{r}. Thus,

deg(w^{p^{r−t}}(f − g^{\frac{m}{s} p^{r+1}})) ≤ p^{r−t}d + mp^{r+1} − (p − 1)p^{r}
< p^{r}dD^{−1} + mp^{r+1} − (p − 1)p^{r} = mp^{r+1}.

Since g^{\frac{m}{s} p^{r+1}} is monic of degree mp^{r+1} as we have already seen, we conclude that F is monic of degree mp^{r+1}. □

Claim (3.4). For each v ∈ Σ, any root α of F in \overline{K}_{v} is contained in R_{L_{v}}.

Proof. Since w(T) ≡ 1 (mod 𝔪_{v}), we have w(T)^{p^{r−t}} ≡ 1 (mod 𝔪_{v}^{p^{r−t}}). Thus,

F (T) ≡ g(T)^{\frac{m}{s} p^{r+1}} + (f(T) − g(T)^{\frac{m}{s} p^{r+1}}) = f(T) (mod 𝔪_{v}^{p^{r−t}}).

Now, since p^{r−t} > Ep^{r−r_{v}} = \frac{p}{p−1}c_{m}mp^{r−r_{v}+1} ≥ \frac{p}{p−1}δ_{v}, we have (K_{v})_{F} = (K_{v})_{f} ⊂ L_{v} by (1.4)(ii). This implies α ∈ L_{v}. Since F (T) is a monic polynomial in R̃[T] ⊂ R_{v}[T], we have α ∈ R_{L_{v}}, as desired. □

Let Z̃ be the zero locus of F (T) in A_{C}^{1}. By (3.3), Z̃ is closed in P_{C}^{1}. We

put Z =^{def} Z̃ ∩ P_{B}^{1}.

Claim (3.5). (i) Z ⊂ X. (ii) Z is finite étale over B.

Proof. (i) On (W ∩ A_{B}^{1})^{red}, we have F (T) ≡ g(T)^{\frac{m}{s} p^{r+1}}. Now, since the zero locus of g in A_{B}^{1} is disjoint from W ∩ A_{B}^{1}, so is that of F, as desired. (ii) By (3.3), Z = Spec(R[T]/(F (T))) is finite (and flat) over R. Since F′ = w^{p^{r−t}}f′ = w^{p^{r−t}}a and a ∈ R^{×}, the zero locus of F′ in A_{B}^{1} is (set-theoretically) contained in W. This, together with (i), implies that the zero loci of F and F′ are disjoint from each other, as desired. □

Take an irreducible (or, equivalently, connected) component B′ of Z. By (3.5), we have a natural immersion B′ ↪ X over B, which we regard as a finite étale quasi-section of X → B. Since R_{L_{v}} ⊂ Ω_{v}, (3.4) implies that this quasi-section is S-admissible. This completes the proof. □

Step 1 is the main step, and, roughly speaking, the rest of proof is only concerning how to reduce general cases to Step 1.

Step 2. Assume that S is essentially rational, and that X is an open subscheme of P_{B}^{n} for some n ≥ 0. Then, there exists an S-admissible quasi-section.

Proof. If n = 0, we must have X = B, and the assertion clearly holds. So, assume n ≥ 1.

Let A be a commutative ring. We define P^{n}(A)^{0} to be (A^{n+1} − 0)(A)/A^{×}, where 0 denotes the section (0, …, 0), regarded as a closed subscheme of A^{n+1}. We define P^{n}(A)^{00} to be ∪_{i=0}^{n}U_{i}(A), where U_{i}(≃ A^{n}) is the standard open subset of P^{n}. Then, we have P^{n}(A)^{00} ⊂ P^{n}(A)^{0} ⊂ P^{n}(A). If Pic(A) = {0} (resp. A is a local ring), then we have P^{n}(A)^{0} = P^{n}(A) (resp. P^{n}(A)^{00} = P^{n}(A)^{0} = P^{n}(A)). If A is a Dedekind domain, we see that P^{n}(A)^{0} forms a GL_{n+1}(A)-orbit.

Now, observe that P^{n}(R)^{00} ∩ X(K) is dense in ∏_{v∈Σ} P^{n}(K_{v}). (X(R) may be empty, though.) So, there exists x ∈ P^{n}(R)^{0} ∩ X(K), such that x ∈ Ω_{v} ∩ X(K_{v}) for each v ∈ Σ. (Note that S is essentially rational.) By changing the coordinates via the GL_{n+1}(R)-action, we may assume x = [1 : 0 : ⋯: 0] (∈ U_{0}).

Let e_{1}, …, e_{n−1} be positive integers, and consider the B-morphism i_{e_{1},...,e_{n−1}} : A_{B}^{1} → U_{0} = A_{B}^{n}, t ↦ (t^{e_{1}}, …, t^{e_{n−1}}, t). It is easy to see that i_{e_{1},...,e_{n−1}} is a closed immersion.

Claim (3.6). For some choice of e_{1}, …, e_{n−1}, (i_{e_{1},...,e_{n−1}})^{−1}(X) surjects onto B.

Proof. Denote by T_{1}, …, T_{n} the coordinates of U_{0} = A_{B}^{n}. Then, there exist a finite number of polynomials f_{1}, …, f_{r} ∈ R[T_{1}, …, T_{n}], such that the closed subset A_{B}^{n} −X of A_{B}^{n} coincides with the common zero locus of f_{1}, …, f_{r}. Since A_{B}^{n} ∩ X surjects onto B (as X surjects onto B), we see that, for each b ∈ B, there exists i = i_{b} ∈ {1, …, r}, such that the image of f_{i} in k_{b}[T_{1}, …, T_{r}] is non-zero.

Lemma (3.7). Let S be a finite subset of ℤ^{n}. Then, there exist positive integers e_{1}, …, e_{n−1}, such that the map S → ℤ, (k_{1}, …, k_{n}) ↦ e_{1}k_{1} +⋯+e_{n−1}k_{n−1} + k_{n} is injective.

Proof. Put T =^{def} {s − s′ | s, s′ ∈ S, s ≠ s′}. This is a finite subset of ℤ^{n} that does not contain 0 = (0, …, 0). For each t = (l_{1}, …, l_{n}) ∈ T, consider the linear subspace W_{t} of A_{ℚ}^{n} defined by l_{1}x_{1} + … l_{n}x_{n} = 0. On the other hand, consider the hyperplane H = {(x_{1}, …, x_{n}) | x_{n} = 1} of A_{ℚ}^{n}. As H ⊄

W_{t}, H′ =^{def} H − ∪_{t∈T} W_{t} is a non-empty open subset of H. Since the set {(e_{1}, …, e_{n−1}, 1) | e_{1}, …, e_{n−1} ∈ ℤ_{>0}} is Zariski dense in H (as ℤ_{>0} is an infinite set), it must intersect non-trivially with H′. Take (e_{1}, …, e_{n−1}, 1) in this intersection, then e_{1}, …, e_{n−1} satisfies the desired property. □

We define S to be the set of elements (k_{1}, …, k_{n}) ∈ (ℤ_{≥0})^{n} such that the coefficient of T_{1}^{k_{1}} ⋯ T_{n}^{k_{n}} in f_{i} is non-zero for some i = 1, …, r. Applying (3.7) to this S, we obtain e_{1}, …, e_{n−1} ∈ ℤ_{>0}. Then, we see that, for each b ∈ B, there exists i = i_{b} ∈ {1, …, r}, such that the image of f_{i}(T^{e_{1}}, …, T^{e_{n−1}}, T) in k_{b}[T] is non-zero. This means that (i_{e_{1},...,e_{n−1}})^{−1}(X) surjects onto B, as desired. □

Take e_{1}, …, e_{n−1} as in (3.6), and put S′ =^{def} (C, (i_{e_{1},...,e_{n−1}})^{−1}(X) →

B, {(i_{e_{1},...,e_{n−1}}(L_{v}))^{−1}(Ω_{v})}), where i_{e_{1},...,e_{n−1}}(L_{v}) denotes the map A^{1}(L_{v}) → U_{0}(L_{v}) = A^{n}(L_{v}) induced by i_{e_{1},...,e_{n−1}}. Then, S′ is an essentially rational Skolem data. (Observe that 0 ∈ A^{1}(K) lies in (i_{e_{1},...,e_{n−1}}(L_{v}))^{−1}(Ω_{v}).)

Now, by Step 1, there exists an S′-admissible quasi-section. By composing this quasi-section with i_{e_{1},...,e_{n−1}}, we obtain an S-admissible quasi-section. This completes the proof. □

Remark (3.8). The above argument that involves rational curves with higher degree was communicated to the author by a referee. The author’s original argument, which is slightly more complicated, uses lines over finite extensions.

Step 3. Assume that X is an open subscheme of P_{B}^{n} for some n ≥ 0. Then, there exists an S-admissible quasi-section.

Proof. For each v ∈ Σ, P^{n}(L_{v} ∩ K_{v}^{sep}) is v-adically dense in P^{n}(L_{v}). Accordingly, we have Ω_{v} ∩ P^{n}(L_{v} ∩ K_{v}^{sep}) ≠ ∅. So, there exists a finite Galois subextension M_{v}/K_{v} of L_{v}/K_{v}, such that Ω_{v} ∩X(M_{v}) is non-empty. Now, put C_{1} =^{def} (C, Σ, {M_{v}}_{v∈Σ}), which is a finite base curve data. By (2.1), there exists a C_{1}-distinguished finite Galois extension K′ of K. We define C′ (resp. B′) to be the integral closure of C (resp. B) in K′, and put Σ′ =^{def} C′ − B′, which is the inverse image of Σ in C′. Let v′ be an element of Σ′ and v the image of v′ in Σ. Then, we have (K′)_{v′} = M_{v} ⊂ L_{v}. So, put C′ =^{def} (C′, Σ′, {L_{v}}_{v′∈Σ′}) and S′ =^{def} (C′, X_{B′} → B′, {Ω_{v}}_{v′∈Σ′}). Then, C′ is a base curve data (over the algebraic closure of k in K′) such that C′ is affine and that (RI) holds, and S′ is an essentially rational Skolem data such that X_{B′} is an open subscheme of P_{B′}^{n}. So, by Step 2, there exists an S′-admissible quasi-section B′′ → X_{B′}. Now, the composite of this morphism and the natural projection X_{B′} → X gives an S-admissible quasi-section. This completes the proof. □

Step 4. Assume that E ≃ O_{B}^{n+1}, where n + 1 is the rank of E. Then, there exists a B-admissible quasi-hyperplane.

Proof. For simplicity, we put P = P(E) and P̌ = P(Ě). Let I denote the incidence subscheme of P ×_{B} P̌, and let p and p̌ be the natural projections P ×_{B} P̌ → P and P × P̌ → P̌, respectively. Both p|_{I} and p̌|_{I} are P^{N−1}- bundles, hence, a fortiori, smooth.

Let i = 1, …, r. Since Y_{i} is an integral scheme and B is a smooth curve, the morphism Y_{i} → B is either flat over B or flat over b_{i} for some closed point b_{i} ∈ B. We shall refer to the former (resp. latter) case as case 1 (resp. 2).

In case 1, let Ỹ_{i} be the normalization of Y_{i} and B_{i} the integral closure of B in Ỹ_{i}. Then, since the generic fiber of Ỹ_{i} → B_{i} is geometrically irreducible, there exists a non-empty open subset B′_{i} of B_{i}, such that each fiber of Ỹ_{i} ×_{B_{i}} B′_{i} → B′_{i} is geometrically irreducible ([EGA4], Théorème (9.7.7)). We denote by Σ_{i} the image of B_{i} − B′_{i} in B, which is a finite set. We define U_{i,1} to be the image of (Ỹ_{i} ×_{B_{i}} P̌_{B_{i}} minus the inverse image of I_{B_{i}}) in P̌_{B_{i}}, which is an open subset of P̌_{B_{i}}, and define U_{i,2} to be the complement in P̌ of the image of P̌_{B_{i}} − U_{i,1}, which is an open subset of P̌. Moreover, for each b ∈ Σ_{i}, fix a geometric point \overline{b} on b. Then, for each irreducible component P of Y_{i,\overline{b}}, we put U_{P,1} the image of (P ×\overline{b} P̌\overline{b} minus the inverse image of I\overline{b}) in P̌\overline{b}, and define T_{P,2} to be the image of P̌\overline{b} − U_{P,1} in P̌_{b}, which is a closed subset of P̌_{b}. Now, put

U_{i} =^{def} U_{i,2} − ∪_{b∈Σ_{i}} ∪_{P} T_{P,2}, which is an open subset of P̌. In case 2, for each irreducible component P of Y_{i,\overline{b}_{i}}, we define a closed subset T_{P,2} of P̌_{b_{i}} just as

above, and put U_{i} =^{def} P̌ − ∪_{P} T_{P,2}.

Now, we put U =^{def} ∩_{i=1}^{r}U_{i}. Let \overline{b} be a geometric point on B, then, we see that a point of U\overline{b} corresponds to a hyperplane H\overline{b} of P\overline{b} that satisfies condition (a) in the definition of B-admissible quasi-hyperplane. In particular, U_{b} is a non-empty open subset of P̌_{b} for each b ∈ B.

Next, for each i = 1, …, r, let ((p|_{I})^{−1}(Y_{i}^{sm}))^{non-sm} be the set of points of (p|_{I})^{−1}(Y_{i}^{sm}) at which (p|_{I})^{−1}(Y_{i}^{sm}) → P̌ is not smooth. This is a closed subset of (p|_{I})^{−1}(Y_{i}^{sm}). Let Z_{i} be the image of ((p|_{I})^{−1}(Y_{i}^{sm}))^{non-sm} in P̌. Chevalley’s theorem implies that Z_{i} is a constructible subset of P̌, and the usual Bertini theorem implies that, for each b ∈ B, P̌_{b}−Z_{i} contains a non-empty open subset of P̌_{b}. From these, we observe that V_{i} =^{def} P̌ − \overline{Z_{i}} satisfies that, for each b ∈ B, (V_{i})_{b} is a non-empty open subset of P̌_{b}. We put V =^{def} ∩_{i=1}^{r}V_{i}. Then, for each b ∈ B, V_{b} is a non-empty open subset of P̌_{b}.

Next, let P be an irreducible component of Y_{i,\overline{K}} with dim(P) ≥ 2. Then, by a version of Bertini theorem ([J], Théorème 6.11, 3), there exists a non-empty open subset W_{P,1} of P̌\overline{K}, such that, for each hyperplane H\overline{K} corresponding to a point of W_{P,1}, P ∩ H\overline{K} is irreducible. We define W_{1} to be the intersection of W_{P,1} for irreducible components P of Y_{i,\overline{K}} with dim(P) ≥ 2, which is a non-empty open subset of P̌\overline{K}, and T_{2} the image of P̌\overline{K} − W_{1} in P̌_{K}, which is a proper closed subset of P̌_{K}. Moreover, we denote by \overline{T}_{2} the closure of T_{2} in P̌. We see that (\overline{T}_{2})_{b} is a proper closed subset of P̌_{b} for each b ∈ B. Now, we put W = P̌ − \overline{T}_{2}. Then, for each b ∈ B, W_{b} is a non-empty open subset of P_{b}.

Now, we put X̌ =^{def} U ∩V ∩W. This is an open subset of P̌ that is surjectively mapped onto B. Put S′ =^{def} (C, X̌ → B, {Ω̌_{v}∩X̌(L_{v})}_{v∈Σ}). Then, S′ is a Skolem data.

So, by Step 3 and the assumption that E ≃ O_{B}^{n+1}, there exists an S′- admissible quasi-section B′ → X̌. By the choice of S′, this section corresponds to a hyperplane H of P_{B′}, which satisfies all the conditions (a)–(d) in the definition of B-admissible quasi-hyperplane. This completes the proof. □

Step 5. Assume that X is quasi-projective of relative dimension 1 over B. Then, there exists an S-admissible quasi-section.

Proof. By assumption, we may assume that X is a subscheme of P_{B}^{n} for some n ≥ 1. We define \overline{X}_{1} to be the closure of X in P_{B}^{n}, regarded as a reduced scheme. \overline{X}_{1} is a projective flat integral curve over B, and X is an open subscheme of \overline{X}_{1}. It is well-known that, after normalizations and blowing-ups outside X, \overline{X}_{1} can be desingularized. Namely, there exists a birational projective morphism π : \overline{X}_{2} → \overline{X}_{1}, where \overline{X}_{2} is a regular, integral scheme, such that π^{−1}(X)→^{∼}X. Since X\overline{K} is irreducible, so is (\overline{X}_{2})\overline{K}. Hence, by [EGA4], Théorème (9.7.7), there exists a non-empty open subset B_{1} of B, such that each fiber of X_{B_{1}} → B_{1} is geometrically irreducible, and, in particular, irreducible. We put Σ_{1} =^{def} B − B_{1}, which is a finite set.

Now, we introduce a new base curve data C_{1} =^{def} (C, Σ ∪ Σ_{1}, {L_{v}}_{v∈Σ} ∪ {K_{v}^{ur}}_{v∈Σ_{1}}), where K_{v}^{ur} denotes the maximal unramified extension of the complete discrete valuation field K_{v}. Note that C_{1} satisfies (RI) and that C is affine. Moreover, we put S_{1} = {C_{1}, X_{B_{1}} → B_{1}, {Ω_{v}}_{v∈Σ} ∪ {X(R_{K_{v}^{ur}})}_{v∈Σ_{1}}}. Since X → B is smooth surjective, we see that X(R_{K_{v}^{ur}}) is non-empty. Thus, S_{1} becomes a Skolem data. Now, suppose that there exists an S_{1}-admissible quasi-section B′_{1} → X_{B_{1}}. Then, firstly, the integral closure B′ of B in B′_{1} is finite étale over B. Secondly, as \overline{X}_{2} → B is proper, B′_{1} → X_{B_{1}} extends to B′ → \overline{X}_{2}. Now, since B′_{1} → X_{B_{1}} is S_{1}-admissible, we see that the image of B′ → \overline{X}_{2} must be contained in X. Thus, we obtain an S-admissible quasi-section B′ → X.

So, replacing S by S_{1}, we may assume that each fiber of \overline{X}_{2} → B is geometrically irreducible. Now, we put \overline{X} =^{def} \overline{X}_{2}.

Lemma (3.9). Let F be a field and X a projective, geometrically integral F - scheme of dimension 1. We denote by X′ the normalization of X\overline{F}, so that we have a natural morphism π : X′ → X. Then, there exists a natural number N, such that each invertible sheaf L on X with deg(L) ≥ N is very ample, where deg(L) =^{def} deg(π^{∗}(L)).

Proof. This follows from [CFHR], Theorem 1.1. (We may take N = 2p_{a}(X) + 1.) □

Now, take a natural number N for \overline{X}_{K} as in (3.9). We shall choose horizontal divisors Y_{1}, Y_{2}, … of \overline{X} inductively, as follows. Firstly, by [Mo3], Théorème 1.3, there exists a horizontal divisor Y_{1} of \overline{X}, such that Y_{1} is contained in X and that, for each v ∈ Σ, Y_{1} ×_{B} Spec(L_{v}) is a disjoint union of copies of Spec(L_{v}) and is contained in Ω_{v}. Next, assume that we have defined Y_{1}, …, Y_{r}. Then, again by [Mo3], Théorème 1.3, there exists a horizontal divisor Y_{r+1} of \overline{X}, such that Y_{r+1} is contained in X − ∪_{i=1}^{r}Y_{i} and that, for each v ∈ Σ, Y ×_{B} Spec(L_{v}) is a disjoint union of copies of Spec(L_{v}) and is contained in Ω_{v} − ∪_{i=1}^{r}Y_{i}(L_{v}). By construction, Y_{1}, Y_{2}, … are disjoint from one another. Now, take n so large that deg(Y_{1,K} +⋯+Y_{n,K}) ≥ N, and we put Y =^{def} Y_{1}+⋯+Y_{n}. (Note that each Y_{i} defines an invertible sheaf on \overline{X}, since it lies in the smooth locus.) Then, by (3.9), Y_{K} is very ample. On the other hand, since each fiber of \overline{X} → B is geometrically irreducible, Y itself is ample (cf. [Mo2], Proposition 4.3), hence there exists a natural number m such that mY is very ample. So, consider an embedding \overline{X} ↪ P_{B}^{n} with respect to the very ample divisor mY.

We put D =^{def} \overline{X} − X. Let E_{1}, …, E_{h} be the irreducible components of D, which must be either an isolated point or a horizontal divisor, as X → B is surjective and each fiber of \overline{X} → B is irreducible. Next, for each v ∈ Σ, we define Ω̌′_{v} to be the subset of P̌^{n}(L_{v}) consisting of points corresponding to L_{v}- rational hyperplanes H such that \overline{X}_{L_{v}} ∩ H is a disjoint union of L_{v}-rational points in Ω_{v} (whose cardinality must coincide with deg(mY_{K})). It is easy to show that Ω̌′_{v} is a v-adically open subset of P̌(L_{v}). Moreover, by using the fact that (not only mY_{K} but also) Y_{K} is very ample and that Y_{L_{v}} is a disjoint union of L_{v}-rational points in Ω_{v}, we see that Ω̌′_{v} is non-empty.

Now, we put B′ =^{def} (C, \overline{X}, E_{1}, …, E_{h} ⊂ P_{B}^{n}, \{Ω̌′_{v}\}_{v∈Σ}), which becomes a Bertini data. As P_{B}^{n} = P(O_{B}^{n+1}), we may apply Step 4 to this Bertini data B′, to conclude that there exists a B′-admissible quasi-hyperplane H ⊂ P_{B′}^{n}. By condition (a), we see that \overline{X}_{B′} ∩ H is finite (as proper and quasi-finite) over B, and that E_{i,B′} ∩ H = ∅ for each i = 1, …, h, hence D_{B′} ∩ H = ∅, or, equivalently, \overline{X}_{B′} ∩ H = X_{B′} ∩ H. By condition (b), we see that X_{B′} ∩ H is smooth over B′. From these, X_{B′} ∩ H is finite étale over B′, hence over B. Moreover, by condition (d), each component of (\overline{X}_{B′} ∩ H)_{L_{v}} is a disjoint union of L_{v}-rational point in Ω_{v}. Thus, any connected component of X_{B′} ∩ H gives an S-admissible quasi-section. This completes the proof. □

Step 6. Assume that X is quasi-projective over B. Then, there exists an Sadmissible quasi-section.

Proof. We shall prove this by using induction on the relative dimension d of X over B. If d = 0, this is clear. If d = 1, this is just the content of Step 5. So, assume d > 1. Since X is quasi-projective, we may choose an embedding X ↪ P_{B}^{n}. We denote by \overline{X} the closure of X in P_{B}^{n}, and put W =^{def} \overline{X}−X. Next, for each v ∈ Σ, we define Ω̌′_{v} to be the subset of P̌^{n}(L_{v}) consisting of points corresponding to L_{v}-rational hyperplanes that meet transversally with a point of Ω_{v}. Then, it is easy to see that Ω̌′_{v} is a non-empty, v-adically open, G_{v}-stable subset of P̌^{n}(L_{v}). Thus, B′ =^{def} (C, \overline{X}, W ⊂ P_{B}^{n}, \{Ω̌′_{v}\}_{v∈Σ}) becomes a Bertini data, and, by Step 4, there exists a B′-admissible quasi-hyperplane H ⊂ P_{B′}^{n}, where B′ is the integral closure of B in some finite C-admissible extension K′ of K. By conditions (a) and (b) in the definition of B′-admissibility, we see that X′_{B′} =^{def} X_{B′} ∩ H is smooth, surjective over B′. By condition (c), X′_{B′} ×_{B′} \overline{K′} is irreducible. Moreover, by condition (d), Ω′_{v} =^{def} Ω_{v} ∩ H(L_{v}) is non-empty.

Now, we define C′ (resp. B′) to be the integral closure of C (resp. B) in K′, and put Σ′ =^{def} C′ − B′, which is the inverse image of Σ in C′. Let v′ be an element of Σ′ and v the image of v′ in Σ. Then, we have (K′)_{v′} ⊂ L_{v}. So,

put C′ =^{def} (C′, Σ′, {L_{v}}_{v′∈Σ′}) and S′ =^{def} (C′, X′_{B′} → B′, {Ω′_{v}}_{v′∈Σ′}). Then, C′ is

a base curve data (over the algebraic closure of k in K′) such that C′ is affine and that (RI) holds, and S′ is a Skolem data such that the relative dimension of X_{B′} over B′ is d − 1. Thus, by the assumption of induction, there exists an S′-admissible quasi-section B′′ → X′_{B′}. Composing this quasi-section with the natural map X′_{B′} → X, we obtain an S-admissible quasi-section, as desired. This completes the proof. □

Step 7. There exists an S-admissible quasi-section. Namely, (3.1) holds.

Proof. Let X′ be a non-empty affine open subset of X, and let B′ denote the image of X′ in B, which is a non-empty open subset of B. Put Σ′ =^{def} B − B′.

Then, S′ =^{def} (C′, X′ → B′, {Ω_{v} ∩ X′(L_{v})}_{v∈Σ} ∪ {X′(K_{v}^{ur}) ∩ X(R_{K_{v}^{ur}})}_{v∈Σ′}),

where C′ =^{def} (C, Σ ∪ Σ′, {L_{v}}_{v∈Σ} ∪ {K_{v}^{ur}}_{v∈Σ′}), becomes a Skolem data. Now, just as in the proof of Step 5, an S′-admissible quasi-section (whose existence is assured by Step 6) induces an S-admissible quasi-section. This completes the proof. □

Step 8. There exists a B-admissible quasi-hyperplane. Namely, (3.2) holds.

Proof. This is just similar to the proof of Step 4, except that we use Step 7 instead of Step 3. □

§4. Some remarks and applications.

4.1. On condition (RI). It is desirable to remove the disgusting condition (RI) in the main results (3.1) and (3.2). The main (and the only) technical difficulty in doing so appears in Step 1 of §3. More specifically, recall that we have applied (1.18)(ii) in Step 1. However, to apply (1.18)(ii), we need a finite submodule A of R_{L_{v}} with A ∩ 𝔪_{L_{v}} = {0} and with ♯(A) sufficiently large, which requires the infiniteness of the residue field of L_{v}. In fact, it is possible to modify (1.18)(ii) to include the case where A ∩ 𝔪_{L_{v}} = {0} does not hold, but then we cannot expect the valuation δ of a is sufficiently small compared to deg(f), and the proof of (3.4) fails when we try to apply (1.4)(ii).

4.2. On the incompleteness hypothesis. In the main results (3.1) and (3.2), we have assumed the incompleteness hypothesis that the base curve C is affine. It is impossible to remove this condition entirely, but it is desirable to be able to control the objects at the points at infinity, even in some weaker sense. In this direction, we have a capacitytheoretic approach due to Rumely ([Ru1], [Ru2]) and another approach via small codimension arguments due to Moret-Bailly ([Mo5]). The author hopes for the following third approach (though it is only applicable to positive characteristic). More precisely, let C = (C, Σ, {L_{v}}_{v∈Σ}) be a base curve data over an algebraic extension k of 𝔽_{p} (with C not necessarily affine), and Σ_{∞} a nonempty subset of Σ. Then, even if C is proper over k, we might expect that the following version of (3.1) and (3.2) hold.

For (3.1), let S = (C, f : X → B, {Ω_{v}}_{v∈Σ−Σ_{∞}} ∪{X(L_{v})}_{v∈Σ_{∞}}) be a Skolem data. (Thus, for v ∈ Σ_{∞}, we just assume X(L_{v}) ≠ ∅.) Then, we might expect that there exists a quasi-section s : B′ → X of f : X → B which is nearly S-admissible with respect to Σ_{∞} in the following sense: K′ is a finite extension of K which is nearly C-admissible with respect to Σ_{∞}; B′ is the integral closure of B in K′; and for each v ∈ Σ − Σ_{∞}, the image of

B′_{L_{v}} =^{def} B′ ×_{B} L_{v} in X_{L_{v}} =^{def} X ×_{B} L_{v} is contained in Ω_{v} (⊂ X(L_{v}) = X_{L_{v}}(L_{v})).

(For v ∈ Σ_{∞}, the image of B′_{L_{v}(p)} in X_{L_{v}(p)} is automatically contained in X(L_{v}(p)) = X_{L_{v}(p)}(L_{v}(p)), and we do not impose any more condition.)

For (3.2), let B = (C, Y_{1}, …, Y_{r} ⊂ P(E), {Ω̌_{v}}_{v∈Σ−Σ_{∞}} ∪ {P(Ě)(L_{v})}_{v∈Σ_{∞}}) be a Bertini data. Then, we might expect that there exists a quasi-hyperplane H ⊂ P(E)_{B′} which is nearly B-admissible with respect to Σ_{∞} in the following sense: K′ is a finite extension of K which is nearly C-admissible with respect to Σ_{∞}; B′ is the integral closure of B in K′; (a), (b), (c) as in the definition of Badmissibility; and (d) as in the definition of B-admissibility only for v ∈ Σ−Σ_{∞}. (For v ∈ Σ_{∞}, the image of B′_{L_{v}(p)} in P(Ě)_{L_{v}(p)} is automatically contained in P(Ě)(L_{v}(p)) = P(Ě)_{L_{v}(p)}(L_{v}(p)), and we do not impose any more condition.)

We might consider (2.2) as a weak evidence for this expectation.

4.3. Hopeful generalizations (mild). Firstly, it should be possible to generalize the main results (3.1) and (3.2) to the case of algebraic spaces or even algebraic stacks, along the lines of [Mo5].

Secondly, it should be possible to prove qualitative versions of (3.1) and (3.2), in terms of heights (cf. [U], and [A1,2]) and/or degrees (cf. [Mi], [E1,2]). (See also [Poo].)

Thirdly, it is desirable to be able to prove that, in (3.1), we can choose an Sadmissible quasi-section B′ → X which is a closed immersion (cf., e.g., [Mo2], Définition 1.4.), and, similarly, that, in (3.2), we can choose a B-admissible quasi-hyperplane H ⊂ P(E)_{B′} such that the classifying morphism [H] : B′ → P(Ě) is a closed immersion. This third possible generalization was suggested to the author by the referee. Indeed, this generalization is possible in Steps 1 and 2 of §3. (For Step 2, this is possible by means of the simplification of the proof due to him or her. See (3.8).)

4.4. Hopeful generalizations (ambitious). As we have mentioned in the Introduction, word-for-word translations of the main results (3.1) and (3.2) to the number field case, namely, to the case where C in the base scheme data is (an open subscheme of) the spectrum of the integer ring of an algebraic number field are false. However, it is very interesting (at least to the author) to ask if we might hope for any (modified) unramified versions of (3.1) and (3.2) also in the number field case.

Also, it might be interesting to investigate what happens in the case where C is a higher-dimensional (affine) scheme, even in positive characteristic. One of the main obstacles of this direction consists in the fact that the Picard group of C is no longer a torsion group, and word-for-word translations of (3.1) and (3.2) to the higher-dimensional case are false. However, there might exist some reasonable restrictions on the (Skolem or Bertini) data, with which (3.1) and (3.2) are valid.

4.5. An application to local-global principle and largeness in field theory.

Definition. Let C = (C, Σ, {L_{v}}_{v∈Σ}) be a base scheme data. Then, we define K^{C} to be the maximal C-admissible extension of K contained in the algebraic closure \overline{K} of K, C^{C} (resp. B^{C}) the integral closure of C (resp. B) in K^{C}, and Σ^{C} to be the inverse image of Σ in C^{C}. (Thus, Σ^{C} = C^{C} − B^{C}.)

As an application of (3.1), we obtain the following local-global principle in field theory (cf. [Mo4]).

Theorem (4.1). Let C = (C, Σ, {L_{v}}_{v∈Σ}) be a base curve data over an algebraic extension of 𝔽_{p}, and assume that C is affine and that (RI) holds. Then, K^{C} satisfies the local-global principle in the sense that, for each smooth, geometrically connected scheme X over K^{C}, X(K^{C}) ≠ ∅ holds if and only if X((K^{C})_{w}) ≠ ∅ holds for every prime w of K^{C}. Here, (K^{C})_{w} denotes the algebraic closure of K_{v} in the completion of K^{C} at w, where v is the prime of K that is below w.

Proof. The ‘only if’ part is trivial. To prove the ‘if’ part, assume that X((K^{C})_{w}) ≠ ∅ holds for every prime w of K^{C}. First, replacing X by any non-empty quasi-compact open subset, we may assume that X is of finite type over K^{C}. (Observe that the image of X((K^{C})_{w}) in X is Zariski dense.) Then, replacing K and C by a suitable finite C-admissible extension and a suitable base curve data, respectively, we may assume that X comes from a (smooth, geometrically connected) K-scheme X_{K}. Now, the following (4.2) implies X(K^{C}) = X_{K}(K^{C}) ≠ ∅, as desired. (Put Ω_{v} = X_{K}(L_{v}).) □

Theorem (4.2). Notations and assumptions being as in (4.1), let X_{K} be a smooth, geometrically connected K-scheme. Assume that X_{K}(K_{b}^{ur}) ≠ ∅ holds for each closed point b ∈ B, and that a non-empty, v-adically open, G_{v}-stable subset Ω_{v} of X_{K}(L_{v}) is given for each v ∈ Σ. Then, there exists a finite Cadmissible extension K′ of K and s_{K} ∈ X_{K}(K′), such that, for each v ∈ Σ, the image by s_{K} ×_{K} L_{v} of Spec(K′) ×_{K} L_{v} in X_{L_{v}} =^{def} X_{K} ×_{K} L_{v} is contained in Ω_{v} (⊂ X_{K}(L_{v}) = X_{L_{v}}(L_{v})).

Proof. First, assume that there exist a regular, integral scheme \overline{X} proper, flat over B and an open immersion X_{K} ↪ \overline{X}_{K} over K. (We shall refer to such an \overline{X} as a regular, relative compactification over B.) Put W_{K} =^{def} \overline{X}_{K} − X_{K} and denote by W the closure of W_{K} in \overline{X}. Then, we see easily that W ∩ \overline{X}_{K} = W_{K} and that the fiber W_{b} of W at each closed point b of B has dimension strictly smaller than the dimension of the whole fiber \overline{X}_{b} (which is automatically equidimensional). On the other hand, let \overline{X}^{sm} denote the set of points of \overline{X} at which \overline{X} → B is smooth. This is an open subset of \overline{X}. Since \overline{X} is regular and \overline{X}(K_{b}^{ur})(⊃ X_{K}(K_{b}^{ur})) ≠ ∅ for each closed point b ∈ B, we see that \overline{X}^{sm} → B is surjective. From these, we conclude that X =^{def} \overline{X}^{sm} − W surjects onto B and that X ×_{B} K = X_{K} holds. Now, applying (3.1) to the Skolem data S =^{def} (C, X → B, {Ω_{v}}_{v∈Σ}), we obtain an S-admissible quasisection s : B′ → X. Then, s_{K} =^{def} s ×_{B} K satisfies the desired properties.

In general, the above desingularization result may not be available, but we can proceed by using induction on d =^{def} dim(X_{K}), as follows. The case d = 0 is trivial. In the case d = 1, the existence of a regular, relative compactification as above is well-known. So, we may assume d ≥ 2. Replacing X_{K} by a suitable (say, non-empty and affine) open subset, we may also assume that X_{K} is quasi-projective over K. (Observe that the image of Ω_{v} in X_{K} is Zariski dense.) We choose an embedding X_{K} ↪ P_{K}^{n}. Then, as in the proof of Step 4 of §3, there exists a non-empty open subset Ǔ_{K} of the dual projective space P̌_{K}^{n}, such that, for each hyperplane H\overline{K} that corresponds to a point of Ǔ_{K}(\overline{K}), X\overline{K} ∩ H\overline{K} is smooth, (geometrically) connected of dimension d − 1. Moreover, as in the proof of Step 6 of §3, for each v ∈ Σ, we define Ω̌′_{v} to be the subset of P̌^{n}(L_{v}) consisting of points corresponding to L_{v}-rational hyperplanes that meet transversally with a point of Ω_{v}. Then, it is easy to see that Ω̌′_{v} is a nonempty, v-adically open, G_{v}-stable subset of P̌^{n}(L_{v}). Now, since Ǔ_{K} admits a regular, relative compactification P̌_{B}^{n}, we may apply the above argument to (Ǔ_{K}, {Ω′_{v} ∩ Ǔ_{K}(L_{v})}_{v∈Σ}) to obtain a suitable quasi-section s′_{K} of Ǔ_{K}. (Note also that Ǔ_{K}(K_{b}^{ur}) ≠ ∅ holds for each closed point b of B.) Now, as in the proof of Step 6 of §3, we may reduce the problem to the case d − 1 by cutting (the base change of) X_{K} with the quasi-hyperplane corresponding to s′_{K}. Thus, the proof by induction is completed. □

Corollary (4.3). K^{C} is large (in the sense of [Pop]). □

Proof. Immediate from (4.1). (See [Pop], Proposition 3.1.) □

This corollary gives an interesting new example of large fields. Indeed, as far as the author knows, in all the known examples of large fields which are algebraic extensions of either number fields K or function fields K over finite fields, we can control only finitely many primes of K. On the other hand, our K^{C} is defined by imposing restrictions at almost all primes of K.

In this sense, this corollary may be regarded as the first example of large fields which are not so large! (See also (4.11) below.)

4.6. An application to principal ideal theorem. As an application of (3.1), we obtain the following (cf. [Mo1], 3.1):

Theorem (4.4). Let C = (C, Σ, {L_{v}}_{v∈Σ}) be a base curve data over an algebraic extension k of 𝔽_{p}, and assume that C is affine and that (RI) holds. Then, we have Pic(C^{C}) = {0}. In particular, we have Pic(B^{C}) = {0}.

Proof. Let L^{C} be any invertible sheaf on C^{C}. Then, there exists a finite subextension K_{1} of K^{C} over K, such that L^{C} = L_{1}⊗_{O_{C_{1}}} O_{C^{C}} holds for some invertible sheaf L_{1} on C_{1}, where C_{1} is the integral closure of C in K_{1}. We define B_{1} and Σ_{1} to be the integral closure of B in K_{1} and the inverse image of Σ in C_{1}, respectively, and, for each v_{1} ∈ Σ_{1}, we put L_{v_{1}} =^{def} L_{v}, where v is the image of v_{1} in Σ. Then, observe that C_{1} =^{def} (C_{1}, Σ_{1}, {L_{v_{1}}}_{v_{1}∈Σ_{1}}) becomes a base curve data (over the algebraic closure of k in K_{1}), such that (C_{1})^{C_{1}} = C^{C}. Now, consider the Skolem data S_{1} = (C_{1}, (V(Ľ_{1})−0_{C_{1}})×_{C_{1}}B_{1} → B_{1}, {(V(Ľ_{1})−0_{C_{1}})(R_{L_{v}})}_{v_{1}∈Σ_{1}}) (such that C_{1} is affine and that (RI) holds), where V(Ľ_{1}) denotes the (geometric) line bundle on C_{1} defined by the dual Ľ_{1} of L_{1}, and 0_{C_{1}} denotes the zero section of V(Ľ_{1}). Now, by (3.1), there exists an S_{1}-admissible quasi-section B′ → (V(Ľ_{1}) − 0_{C_{1}}) ×_{C_{1}} B_{1}. By the choice of S_{1}, this quasi-section extends to a (unique) quasi-section C′ → V(Ľ_{1}) − 0_{C_{1}} of V(Ľ_{1}) − 0_{C_{1}} → C_{1}, where C′ is the integral closure of C_{1} in B′. This implies that L_{1} admits an everywhere non-vanishing section over C′, or, equivalently, L_{1} becomes trivial after the base change to C′. Therefore, L^{C} is trivial, a fortiori. This completes the proof of the first assertion.

The second assertion follows from the first, as the natural map Pic(C^{C}) → Pic(B^{C}) is surjective. □

In the case where Σ = ∅, (4.4) directly follows from the principal ideal theorem in class field theory. In general, however, there exists an invertible sheaf that cannot be trivialized if we only consider abelian C-admissible extensions. (See (4.11) below.) In this sense, we may regard (4.4) as a new (non-abelian) type of principal ideal theorem which cannot be covered by class field theory.

4.7. An application to torsors. More generally than the second assertion of (4.4), we obtain the following:

Theorem (4.5). Let C = (C, Σ, {L_{v}}_{v∈Σ}) be a base curve data over an algebraic extension of 𝔽_{p}, and assume that C is affine and that (RI) holds. Let G^{C} be a smooth, separated group scheme of finite type over B^{C}, such that the generic fiber G_{K^{C}}^{C} is connected. Then, we have Ker(Ȟ_{fpqc}^{1}(B^{C}, G^{C}) → ∏_{w∈Σ^{C}} Ȟ_{fpqc}^{1}(Spec((K^{C})_{w}), G^{C}))) = {1}.

Proof. By [Ra], Théorème XI, 3.1, each class x of Ȟ_{fpqc}^{1}(B^{C}, G^{C}) corresponds to a representable B^{C}-torsor X^{C}. If, moreover, x belongs to the kernel in question, X^{C} admits a (K^{C})_{w}-rational point for each w ∈ Σ^{C}. Since X^{C} is of finite presentation over B^{C}, it comes from a scheme over a finite (C-admissible) extension of B. Now, as in the proof of (4.4), we can prove that X^{C} admits a B^{C}-section, by using (3.1). This completes the proof. □

Note that the second assertion of (4.4) is a special case of (4.5), where G^{C} = 𝔾_{m,B^{C}}. (The first assertion of (4.4) can also be generalized in a suitable sense. We leave it to the readers.)

As an interesting corollary of (4.5), we obtain:

Corollary (4.6). Let C be an affine, smooth curve over an algebraic extension of 𝔽_{p}, and K the function field of C. Let G be a smooth, separated, commutative group scheme of finite type over C, such that the generic fiber G_{K} is connected.

Then, we have H_{ét}^{1}(C, G) = H^{1}(π_{1}(C), G(C̃)), where C̃ =^{def} B^{C}(= C^{C}) for the base curve data C =^{def} (C, ∅, ∅).

Proof. This follows from (4.5), together with the Hochschild-Serre spectral sequence. □

4.8. A group-theoretical remark. Recall that a quasi-p group (for a prime number p) is a finite group that does not admit a non-trivial quotient group of order prime to p.

Proposition (4.7). Let B be a smooth, geometrically connected curve over a finite field k of characteristic p. We denote by B^{∗} the smooth compactification of B and put Σ =^{def} B^{∗} − B (which we regard as a reduced closed subscheme of B^{∗}). Then, there exists a natural number N (depending only on the genus g of B^{∗} and the cardinality n of Σ(\overline{k})), such that, for each finite extension l of k with ♯(l) ≥ N, there is no non-trivial finite étale covering of B_{l} =^{def} B ×_{k} l at most tamely ramified over Σ_{l} =^{def} Σ ×_{k} l in which every point of B(l) = B_{l}(l) splits completely.

Proof. This is a rather well-known application of the Weil bound on the cardinality of rational points over finite fields. More specifically, suppose that l is a finite extension of k with cardinality q and that B′ is a connected finite étale covering with degree d of B_{l} at most tamely ramified over Σ_{l} in which every point of B(l) splits completely. We denote by l′ the integral closure of l in B′, and define (B′)^{∗}, g′ and n′ for B′ just similarly to B^{∗}, g and n, respectively, for B. Now, firstly, the tamely ramified condition, together with the Hurwitz’ formula, implies

2g′ − 2 ≤ d_{geom}(2g − 2) + (d_{geom} − 1)n ≤ d(2g − 2) + (d − 1)n,

where d_{geom} =^{def} d/[l′ : l]. Secondly, the complete splitting condition, together with the Weil (lower) bound for B_{l}, implies that

♯(B′(l)) = d♯(B_{l}(l)) ≥ d(♯((B′)^{∗}(l)) − n) ≥ d(1 + q − 2g\sqrt{q} − n).

Thirdly, the Weil (upper) bound for B′ implies

♯(B′(l)) ≤ ♯((B′)^{∗}(l)) ≤ 1 + q + 2g′\sqrt{q}.

(Note that this holds (trivially) even if [l′ : l] > 1.) Combining these three inequalities together, we obtain

d(q − (4g + n − 2)\sqrt{q} − (n − 1)) ≤ q − (n − 2)\sqrt{q} + 1.

From this, we see that d < 2 (or, equivalently, d = 1) must hold for sufficiently large q, as desired. □

Proposition (4.8). Let k and B be as in (4.7). Then, there exist finite sets Σ_{1} and Σ_{2} of closed points of B, disjoint from each other, such that the following holds: For each v ∈ Σ_{1} (resp. v ∈ Σ_{2}), let L_{v} be any (possibly infinite) pro-p Galois extension of K_{v} (resp. Galois extension such that Gal(L_{v}∩K_{v}^{ur}/K_{v}) is a pro-prime-to-p group) and put C = (B, Σ_{1} ∪ Σ_{2}, {L_{v}}_{v∈Σ_{1}∪Σ_{2}}). Then, for every C-admissible, finite, Galois extension K′ of K, Gal(K′/K) is a quasi-p group and the constant field of K′ (i.e., the algebraic closure of k in K′) coincides with k.

Proof. Take a natural number N as in (4.7), and choose two finite extensions l_{1} and l′_{1} of k with ♯(l_{1}) ≥ N and ♯(l′_{1}) ≥ N, such that l_{1} ∩ l′_{1} = k. We define Σ_{1} to be the union of the images of B(l_{1}) and B(l′_{1}) in B. Next, the Weil bound implies that there exists a natural number N′, such that, for each finite extension l of k with ♯(l) ≥ N′, (B − Σ_{1})(l) ≠ ∅ holds. (Note that B − Σ_{1} is geometrically connected over k.) So, we can choose a finite extension l_{2} of k, such that (B − Σ_{1})(l_{2}) ≠ ∅ and that [l_{2} : k] is not divisible by p. We define Σ_{2} to be any non-empty subset of the image of (B − Σ_{1})(l_{2}) in B − Σ_{1}. Now, for each v ∈ Σ_{1} (resp. v ∈ Σ_{2}), let L_{v} be any pro-p Galois extension of K_{v} (resp.

Galois extension such that Gal(L_{v} ∩ K_{v}^{ur}/K_{v}) is a pro-prime-to-p group), and put C = (B, Σ_{1} ∪ Σ_{2}, {L_{v}}_{v∈Σ_{1}∪Σ_{2}}).

Let K′ be any C-admissible finite Galois extension of K. We define Gal(K′/K)^{p′} (resp. Gal(K′/K)^{p}) to be the maximal quotient group of Gal(K′/K) with order prime to p (resp. with order a power of p), and denote by M_{1} (resp. M_{2}) the subextension of K′ over K that corresponds via Galois theory the kernel of the natural surjective map Gal(K′/K) ↠ Gal(K′/K)^{p′} (resp. Gal(K′/K) ↠ Gal(K′/K)^{p}). Thus we have Gal(M_{1}/K) = Gal(K′/K)^{p′} (resp. Gal(M_{2}/K) = Gal(K′/K)^{p}) .

We shall first prove that Gal(K′/K) is a quasi-p group, or, equivalently, that M_{1} = K holds. As K′ is C-admissible, so is M_{1}. Since, moreover, Gal(M_{1}/K) has order prime to p and Gal(L_{v}/K_{v}) is pro-p for each v ∈ Σ_{1}, M_{1}/K must split completely at each v ∈ Σ_{1}. Now, by (4.7), we obtain M_{1} ⊂ Kl_{1} ∩ Kl′_{1} = K, as desired.

Next, we shall prove that the constant field of K′ is k. Since the Galois group over k of a finite extension of k is cyclic (hence nilpotent, a fortiori), we see that the constant field of K′ is the compositum of those of M_{1} and M_{2}. Since we have already proved M_{1} = K, it suffices to prove that the constant field of M_{2} is k. As K′ is C-admissible, so is M_{2}. Since, moreover, Gal(M_{2}/K) has p-power order and Gal(L_{v} ∩ K_{v}^{ur}/K_{v}) is pro-prime-to-p for each v ∈ Σ_{2}, M_{2}/K does not admit a non-trivial residue field extension over Σ_{2}. In particular, the constant field of M_{2} is contained in the residue field of each v ∈ Σ_{2}, hence in l_{2} by the choice of Σ_{2}. Now, since Gal(M_{2}/K) has p-power order and [l_{2} : k] is prime to p, the constant field of M_{2} must coincide with k, as desired.

This completes the proof. □

Proposition (4.9). Let the notations and the assumptions be as in (2.2), and assume, moreover, that k is an algebraic extension of 𝔽_{p}. Then, in the conclusion of (2.2), we may assume that Gal(K′/K) is a quasi-p group and that the constant field of K′ coincides with k.

Proof. We can choose a finite subextension k_{0} of 𝔽_{p} in k, such that the curve C and the (reduced) closed subscheme Σ of C descend to C_{0} and Σ_{0}, respectively. Replacing k_{0} by a suitable finite extension (in k), we may assume ♯(Σ) = ♯(Σ_{0}). Moreover, again replacing k_{0} by a suitable finite extension, we may assume that, for each v ∈ Σ, the finite Galois extension L_{v}/K_{v} descends to a finite Galois extension L_{0,v_{0}}/K_{0,v_{0}}, where K_{0} is the function field of C_{0} and v_{0} is the image of v in Σ_{0}. We define Σ_{0,∞} to be the image of Σ_{∞} in Σ_{0}. We also put

B_{0} =^{def} C_{0} − Σ_{0}.

Now, take finite sets Σ_{1} and Σ_{2} of closed points of B_{0} as in (4.8), and put

C_{0} =^{def} (C_{0}, Σ_{0} ∪ Σ_{1} ∪ Σ_{2}, {L_{0,v_{0}}}_{v_{0}∈Σ_{0}} ∪ {K_{0,v_{0}}}_{v_{0}∈Σ_{1}∪Σ_{2}}). Applying (2.2) to

C_{0} and Σ_{∞,0}, we obtain a finite Galois extension K′_{0} of K_{0} that is nearly C_{0}- distinguished with respect to Σ_{∞,0}. By (4.8), Gal(K′_{0}/K_{0}) is a quasi-p group and the constant field of K′_{0} coincides with k_{0}.

Finally, put K′ =^{def} K′_{0}k. Then, we easily see that K′ is a finite Galois extension of K = K_{0}k that is nearly C-distinguished with respect to Σ_{∞}, that Gal(K′/K) is a quasi-p group (as Gal(K′/K)→^{∼} Gal(K′_{0}/K_{0})) and that the constant field of K′ coincides with k. This completes the proof. □

Proposition (4.10). In (3.1) and (3.2), we may choose B′ such that K′ is a finite Galois extension of K with quasi-p Galois group and that the constant field of K′ coincides with k.

Proof. The proof of this fact goes rather similarly as that of (4.9). The main difference between them consists in the fact that, in the proof of (4.9), we can use the trivial extension K_{0,v_{0}}/K_{0,v_{0}} for each v ∈ Σ_{1} ∪ Σ_{2}, while, in the proof of (4.10), this is impossible, since we have to require that condition (RI) also holds for the (enlarged) base curve data. Here, however, we may take K_{0,v_{0}}𝔽_{p}(p^{∞})/K_{0,v_{0}} (resp. K_{0,v_{0}}𝔽_{p}(p′)/K_{0,v_{0}}) for v_{0} ∈ Σ_{1} (resp. v_{0} ∈ Σ_{2}) instead of K_{0,v_{0}}/K_{0,v_{0}}, where 𝔽_{p}(p^{∞}) (resp. 𝔽_{p}(p′)) denotes the maximal prop (resp. pro-prime-to-p) extension of 𝔽_{p}. Details are left to the readers. □

Remark (4.11). The group-theoretical results of this subsection are also applicable to other results in this section.

For example, (4.8) implies that, for some base curve data C, the field K^{C} that appears in (4.1) and (4.3) satisfies the following property: Gal(K^{C}/K) is pro-quasi-p in the sense that its maximal pro-prime-to-p quotient is trivial.

A similar remark is also applicable to (4.4). Namely, for some base curve data C, Gal(K^{C}/K)(= Aut(B^{C}/B)) is pro-quasi-p. In particular, the abelianization Gal(K^{C}/K)^{ab} of Gal(K^{C}/K) is a pro-p group, or, equivalently, every finite abelian C-admissible extension of K has p-power degree. Accordingly, if, moreover, we start with C such that Pic(C) admits a non-trivial torsion element [L] whose order is prime to p, then L cannot be trivialized over any finite abelian C-admissible extension, while it can be trivialized over some finite (necessarily non-abelian) C-admissible extension by (4.4).

References

[A1] P. Autissier, Points entiers sur les surfaces arithmétiques, J. Reine Angew. Math. 531 (2001), 201–235.

[A2] , Points entiers et théorèmes de Bertini arithmétiques, Ann. Inst. Fourier (Grenoble) 51 (2001), 1507–1523; Corrigendum, ibid. 52 (2002), 303–304.

[CR] D. C. Cantor and P. Roquette, On Diophantine equations over the ring of all algebraic integers, J. Number Theory 18 (1984), 1–26.

[CFHR] F. Catanese, M. Franciosi, K. Hulek and M. Reid, Embeddings of curves and surfaces, Nagoya Math. J. 154 (1999), 185–220.

[EGA4] A. Grothendieck et al, Éléments de géométrie algébrique IV, Étude locale des schémas et des morphismes de schémas, Publications Mathématiques, 20, 24, 28, 32, IHES, 1964–1967.

[E1] R. Erné, The degree of an integral point in P^{s} minus a hypersurface, C. R. Acad. Sci. Paris, Sér. I, Math. 324 (1997), 1121–1126.

[E2] , On the degree of integral points of a projective space minus a horizontal hypersurface, J. Reine Angew. Math. 532 (2001), 151–177.

[EG] D. R. Estes and R. M. Guralnick, Module equivalences: local to global when primitive polynomials represent units, J. Algebra 77 (1982), 138–157.

[G] O. Gabber, On space filling curves and Albanese varieties, Geom. Funct. Anal. 11 (2001), 1192–1200.

[GPR] B. Green, F. Pop and P. Roquette, On Rumely’s local-global principle, Jahresber. Deutsch. Math.-Verein. 97 (1995), 43–74.

[J] J.-P. Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics, 42, Birkhäuser Boston, Inc., Boston, 1983.

[Mi] P. Mikkelsen, Effective bounds for integral points on arithmetic surfaces, J. Reine Angew. Math. 496 (1998), 55–72.

[Mo1] L. Moret-Bailly, Points entiers des variétés arithmétiques, in Séminaire de Théorie des Nombres, Paris 1985–86, Progr. Math., 71, Birkhäuser Boston, Boston, 1987, pp. 147–154.

[Mo2] , Groupes de Picard et problèmes de Skolem, I, Ann. Sci. École Norm. Sup. (4) 22 (1989), 161–179.

[Mo3] , Groupes de Picard et problèmes de Skolem, II, Ann. Sci. École Norm. Sup. (4) 22 (1989), 181–194.

[Mo4] , Applications of local-global principles to arithmetic and geometry, in Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), Contemp. Math., 270, Amer. Math. Soc., Providence, 2000, pp. 169–186.

[Mo5] , Problèmes de Skolem sur les champs algébriques, Compositio Math. 125 (2001), 1–30.

[Poo] B. Poonen, Bertini theorems over finite fields, Ann. of Math. (to appear).

[Pop] F. Pop, Embedding problems over large fields, Ann. of Math. (2) 144 (1996), 1–34.

[Ra] M. Raynaud, Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Mathematics, 119, Springer-Verlag, Berlin-New York, 1970.

[Ro] P. Roquette, Solving diophantine equations over the ring of all algebraic integers, in Atas da 8. Escola de Álgebra (Rio de Janeiro, 1984), 2, Coleção Atas, no. 16, Instituto de Mathemática Pura e Aplicada, 1985, pp. 1–26.

[Ru1] R. S. Rumely, Arithmetic over the ring of all algebraic integers, J. Reine Angew. Math. 368 (1986), 127–133.

[Ru2] , Capacity theory on algebraic curves, Lecture Notes in Mathematics, 1378, Springer-Verlag, Berlin, 1989.

[S] T. Skolem, Lösung gewisser Gleichungen in ganzen algebraischen Zahlen, insbesondere in Einheiten, Skr. Norske Vid.-Akad. Oslo I, Mat. Naturv. Kl. (1934), no. 10.

[T] A. Tamagawa, The Eisenstein quotient of the Jacobian variety of a Drinfeld modular curve, Publ. Res. Inst. Math. Sci. 31 (1995), 203– 246.

[U] E. Ullmo, Points entiers, points de torsion et amplitude arithmétique, Amer. J. Math. 117 (1995), 1039-1055.

Akio Tamagawa Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502, Japan 832

Documenta Mathematica · Extra Volume Kato (2003)