RECONSTRUCTION OF FUNCTION FIELDS

by

Fedor Bogomolov and Yuri Tschinkel

Abstract. — We study the structure of abelian subgroups of Galois groups of function fields of surfaces.

Contents

Introduction . . .............................................. 2
2. Overview . . .............................................. 3
3. Basic algebra and geometry of fields . . ...................... 6
4. Projective structures . . .................................... 15
5. Flag maps . . ...................... ...................... 19
6. Galois groups . . .......................................... 28
7. Valuations . . ...................... ...................... 30
8. A dictionary . . ..................... ..................... 32
9. Flag maps and valuations . . ................................ 34
10. Galois groups of curves . . ................................ 36
11. Valuations on surfaces . . ................ ................ 42
12. ℓ-adic analysis: generalities . . .............. .............. 43
13. ℓ-adic analysis: curves . . ................ ................ 51
14. ℓ-adic analysis: surfaces . . ................................ 52
15. Proof . . ................................................ 55
References . . .............................................. 57

Key words and phrases. — Galois groups, function fields.

Introduction

sect:introduction

We fix two distinct primes p and ℓ. Let k = 𝔽̄_{p} be an algebraic closure of the finite field 𝔽_{p}. Let X be an algebraic variety defined over k and K = k(X) its function field. Let G_{K}^{a} be the abelianization of the pro-ℓ-quotient G_{K} of the absolute Galois group of K. Under our assumptions on k, G_{K}^{a} is a torsionfree ℤ_{ℓ}-module. Let G_{K}^{c} be its canonical central extension - the second lower central series quotient of G_{K}. It determines the following structure on G_{K}^{a}: a set Σ_{K} of distinguished (primitive) subgroups which are isomorphic to finite rank ℤ_{ℓ}-modules. A topologically noncyclic subgroup σ ∈ Σ_{K} iff

– σ lifts to an abelian subgroup of G_{K}^{c};

– σ is maximal: there are no abelian subgroups σ′ ⊂ G_{K}^{a} which lift to an abelian subgroup of G_{K}^{c} and contain σ as a proper subgroup.

We will call Σ_{K} a fan. The main theorem of this paper is

Theorem 1. — Let K and L be function fields over algebraic closures of

thm:main

finite fields of characteristic ≠ ℓ. Assume that K = k(X) is a function field of a surface X/k and that there exists an isomorphism

Ψ = Ψ_{K,L} : G_{K}^{a} ≃ G_{L}^{a}

of abelian pro-ℓ-groups inducing a bijection of sets

Σ_{K} = Σ_{L}.

Then L is isomorphic to a finite purely inseparable extension of K.

We implement the program outlined in [1] and [2] describing the correspondence between higher-dimensional function fields and their abelianized Galois groups. For results concerning the reconstruction of function fields from their (full) Galois groups (the birational Grothendieck program) we refer to the works of Pop, Mochizuki and Efrat (see [9], [8],[5]).

Acknowledgments. Both authors were partially supported by the NSF. The second author was employed by the Clay Mathematics Institute. We are grateful to Laurent Lafforgue and Barry Mazur for their interest and the referee for many useful remarks. We thank the participants of the Algebraic Geometry Seminar at the University of Nice for their comments and suggestions.

2. Overview

sect:ober

In this section we outline our strategy of reconstruction, or rather recognition, of the function field K of an algebraic variety X over an algebraic closure k of a finite field from a certain quotient of its Galois group.

Let G_{K}^{a} be the pro-ℓ-quotient of the abelianization

G_{K}/[G_{K}, G_{K}],

of the absolute Galois group G_{K} = Gal(K̄/K) of K, ℓ ≠ char(k). By Kummer theory, G_{K}^{a} determines the pro-ℓ-completion K̂^{∗} of the multiplicative group K^{∗}.

A Galois-theoretic characterization of the field K involves the recognition of the subgroup K^{∗}/k^{∗} ⊂ K̂^{∗}, and of the canonical projective structure, the projectivization of the additive group K, considered as a vector space over k. This projective structure is invariant under multiplicative translations by elements of K^{∗}/k^{∗}.

For this we need Galois-theoretic information coming from G_{K}^{c}, the maximal pro-ℓ-quotient of

G_{K}/[[G_{K}, G_{K}], G_{K}].

This topological group parametrizes those ℓ-extensions of K whose Galois group is a central extension of an abelian ℓ-group.

Our main Galois-theoretic object is a pair (G_{K}^{a}, Σ_{K}), where the fan Σ_{K} is the set of all maximal (by inclusion) topologically noncyclic subgroups σ ⊂ G_{K}^{a} whose set-theoretic preimage in G_{K}^{c} is an abelian group. It turns out that such liftable subgroups σ are finite-dimensional ℤ_{ℓ}-subspaces in G_{K}^{a}. Moreover, the maximal ℤ_{ℓ}-rank of such σ is dim(X).

Theorem 1 states that if for two function fields K = k(X), L = l(Y), where X/k is an algebraic surface, Y/l an algebraic variety, k and l are algebraic closures of finite fields of characteristic ≠ ℓ there is an isomorphism

Ψ : (G_{K}^{a}, Σ_{K}) → (G_{L}^{a}, Σ_{L})

then k ≃ l, Y is a surface and L is isomorphic to a purely inseparable extension of K.

Define a subfan Σ_{K}^{div} ⊂ Σ_{K} as the set of those maximal liftable subgroups which have nontrivial intersection with at least one other subgroup in Σ_{K}. There is a geometric reason to distinguish Σ_{K}^{div}. Let K be the function field of a surface X over k, D an irreducible divisor on X and ν = ν_{D} the corresponding nonarchimedian divisorial valuation. Its abelian decomposition group D_{ν}^{a} ⊂ G_{K}^{a} is a (noncanonical) direct product of the inertia subgroup I_{ν}^{a} ≃ ℤ_{ℓ} and the group G_{k(D)}^{a} of the field k(D). Now a subgroup σ ⊂ D_{ν}^{a} of ℤ_{ℓ}-rank 2 is liftable if and only if it contains I_{ν}^{a}. Thus Σ_{K}^{div} contains all liftable subgroups of ℤ_{ℓ}-rank 2 which are contained in groups D_{ν}^{a}.

The first important result says that Σ_{K}^{div} exactly coincides with the set of all liftable subgroups of ℤ_{ℓ}-rank 2 contained in the groups D_{ν}^{a}, for different ν = ν_{D}. This gives an purely group-theoretic description of the groups D_{ν}^{a}: the nontrivial intersection of two liftable groups σ, σ′ is always I_{ν}^{a}, for some divisorial valuation ν = ν_{D}, and D_{ν}^{a} “centralizes” I_{ν}^{a}, it consists of all those elements in G_{K}^{a} which commute with I_{ν}^{a}, after lifting to G_{K}^{c}.

The proof is based on Kummer theory and the interpretation of G_{K}^{a} as a space of special (logarithmic) ℤ_{ℓ}-valued maps on the infinite-dimensional projective space ℙ_{k}(K) = K^{∗}/k^{∗} over k. The description of liftable subgroups is then reduced to questions in finite-dimensional projective geometry. Complete proofs of these results for K = 𝔽̄_{q}(X) are contained in Section 5. The case of arbitrary algebraically closed ground fields k is treated in [3],[2].

At this stage we characterized all pairs D_{ν_{D}}^{a}, I_{ν_{D}}^{a} inside G_{K}^{a}, or, vaguely speaking, we recovered “all curves” on all models X of K (and Y of L). Next we recover the “points” on D, as inertia groups I_{w}^{a} ⊂ G_{k(D)}^{a}, using various subgroups I_{ν_{D′}}^{a} as follows: the image of I_{w}^{a} under any homomorphism of G_{K}^{a} to a finite group, which is trivial on I_{ν_{D}}^{a}, coincides with the image of some I_{ν_{D′}}^{a} (see Section 10). Conversely, for any γ ∈ D_{ν_{D}}^{a} \ I_{w}^{a}, for some divisorial valuation on k(D) (a point), there exists such a homomorphism with the property that the image of γ is not contained in the image of any inertia subgroup I_{ν_{D′}}^{a}.

Now we can recover the genus of D and distinguish the set of divisorial valuations of K which on some model of K are represented by curves of genus > 0. Note that these valuations have 1-dimensional centers on every model of K.

We switch our attention to the dual space K̂^{∗} of G_{K}^{a}. We seek to recover the subset L^{∗}/l^{∗} ⊂ K̂^{∗} = L̂^{∗} using Galois-theoretic data. This is done in several steps, each time obtaining a smaller subgroup:

– FS(K) ⊂ K̂^{∗} - elements in K̂^{∗} with finite nonrational support,

– K^{∗}/k^{∗} ⊗ ℤ_{ℓ} ⊂ FS(K) and

– K^{∗}/k^{∗} ⊗ ℤ_{(ℓ)}.

Elements of K̂^{∗} can be thought of as infinite products of elements f_{i}^{ℓ^{i}} ∈ K^{∗}, modulo natural identifications, and they can be represented by, in general, infinite sums of irreducible divisors on a projective model X of the field with ℤ_{ℓ}-adic coefficients which converge to 0 in the ℓ-adic topology. The subgroup FS(K) ⊂ K̂^{∗} consists of elements whose support contains only finitely many nonrational divisors (characterized above). Of course, FS(K) contains K^{∗}/k^{∗} and L^{∗}/l^{∗} but it is still rather big - elements in FS(K) may have infinite rational “tails” if X contains infinitely many rational curves.

Next we use an ℓ-adic analog of a symbol (f, g) mod ℓ^{n} ∈ K_{2}(K)/ℓ^{n}. Notice that (f, g) mod ℓ^{n} = 0 for any n ∈ ℕ if f, g belong to the same one-dimensional subfield in K. In particular, for any f ∈ K \ k there is an element g which is not a power of f and such that (f, g) = 0 (we can take g = f + 1). This imposes a strong condition on f since for a generic element in K̂^{∗} the “commutator” of f consists of ℓ-adic powers of f only. This allows us to characterize

K^{∗}/k^{∗} ⊗ ℤ_{ℓ} = L^{∗}/l^{∗} ⊗ ℤ_{ℓ} ⊂ FS(K)

as the subgroup generated by elements satisfying this property and having a “sufficiently big” support.

The next step involves a normalization. Inside K^{∗}/k^{∗} ⊗ ℤ_{ℓ} we cannot Galois-theoretically distinguish L^{∗}/l^{∗}⊗ℤ_{(ℓ)} from a·K^{∗}/k^{∗}⊗ℤ_{(ℓ)}, for a ∈ ℤ_{ℓ}^{∗}. However, this conformal invariance is the only freedom there is. If we fix the values of f ∈ L^{∗}/l^{∗} ⊗ ℤ_{ℓ} on one (arbitary) irreducible divisor on a model X of K then L^{∗}/l^{∗} ⊗ ℤ_{(ℓ)} is naturally identified inside K^{∗}/k^{∗} ⊗ ℤ_{ℓ}. Thus, after multiplication by a ∈ ℤ_{ℓ}^{∗}, we can assume that L^{∗}/l^{∗} ⊂ K^{∗}/k^{∗} ⊗ ℤ_{(ℓ)}.

Now we have K^{∗}/k^{∗} and L^{∗}/l^{∗} inside K^{∗}/k^{∗}⊗ℤ_{(ℓ)} = L^{∗}/l^{∗}⊗ℤ_{(ℓ)}. We also know that subgroups generated by elements f, g with pairwise trivial symbol (f, g) = 0 correspond to one-dimensional subfields in K, L, respectively. Most one-dimensional subfields in K are isomorphic to k(x), for some x, and Galois data allow us to recognize these subfields. Hence if k(x) ⊂ K then k(x)^{∗}/k^{∗} ⊗ ℤ_{(ℓ)} = l^{∗}(t)/l^{∗} ⊗ ℤ_{(ℓ)} ⊂ K^{∗}/k^{∗} ⊗ ℤ_{(ℓ)}, for some t ∈ L.

Next we show that the corresponding groups k(x)^{∗}/k^{∗} and l^{∗}(x)/l^{∗} intersect in k(x)^{∗}/(k^{∗})^{r} = l^{∗}(x)/(l^{∗})^{s} for some rational r, s. This property implies that L^{∗}/l^{∗} is isomorphic (as a multiplicative group) to K_{1}^{∗}/k^{∗} where K/K_{1} is a purely inseparable extension.

Now we add the projective structure over k, l, respectively. We notice that the sets of lines {ℙ(k ⊕ kx)} and {ℙ(l ⊕ lt)} in K^{∗}/k^{∗} and L^{∗}/l^{∗}, over all x, t generating closed subfields k(x) ⊂ K and l(t) ⊂ L, are the same. It turns out that the sets of these lines and their (multiplicative) translations is compatible with a unique projective structure on the (multiplicative) groups K^{∗}/k^{∗} and L^{∗}/l^{∗} - namely the one coming from the field structure. This defines a unique additive structure and finishes the proof of our main result.

3. Basic algebra and geometry of fields

sect:basicalg

Notations 3.1. — Throughout, k is an algebraic closure of the finite field

nota:k

𝔽 = 𝔽_{p} and K = k(X) the function field of an algebraic variety X/k over k (its model).

In this paper we use extensively the fact that two-dimensional function fields K have “nice” models: smooth projective surfaces X over k with K = k(X), whose geometric properties play an important role in the recognition procedure. In this section we collect some technical results about function fields of curves and surfaces and their models.

We assume familiarity with – basic notions of field theory (transcendence degree, purely inseparable

extensions);

– basic notions of algebraic geometry: k-rational points X(k), Picard group Pic(X), Néron-Severi group NS(X).

Lemma 3.2. — Let C/k be a smooth curve and Q ⊂ C(k) a finite set. Then

lemm:C

there exists an n = n_{Q} ∈ ℕ such that for every degree zero divisor D with support in Q the divisor nD is principal.

Proof. — Finitely generated subgroups of torsion groups are finite. The group of degree zero divisors Pic^{0}(C) (over any finite field) is torsion and every subgroup of divisors with support in a finite set Q ⊂ C(k) is finitely generated.

Lemma 3.3. — Let X/k be a surface, C_{1}, …, C_{s} a finite set of (pairwise

lemm:ress

distinct) curves on X and f_{i} ∈ k(C_{i})^{∗}, for i = 1, …, s. Then there exists an f ∈ k(X)^{∗} whose restriction to C_{i} is defined and equal to f_{i}, for all i.

Proof. — Well known.

Lemma 3.4. — For some ample smooth curve i : C ↪ X the restriction

lemm:new

i^{∗} : Pic^{0}(X) → Pic^{0}(C) is an injection of abstract groups (of k-points). In particular, every element in Pic^{0}(X) is torsion.

Proof. — Let H be a polarization on X. There exists an n ∈ ℕ such that for all pairs L, L′ ∈ Pic^{0}(X) one has H^{1}(X, −(L − L′ + nH)) = 0. Indeed, the property

H^{1}(X, −(L − L′ + mH)) = 0

is open in Pic^{0}(X) × Pic^{0}(X) \ Δ (where Δ the diagonal subgroup), since Pic^{0}(X) is an algebraic group scheme. Denote by

U_{mH} ⊂ (Pic^{0}(X) × Pic^{0}(X) \ Δ)

the corresponding subset. If we consider a increasing sequence

U_{n_{i}H}, n_{i} ∈ ℤ, U_{mH} ⊂ U_{nH}, for m < n,

the union of all U_{n_{i}H} is equal to Pic^{0}(X) × Pic^{0}(X) \ Δ. Then there is an n ∈ ℕ such that U_{nH} = Pic^{0}(X) × Pic^{0}(X) \ Δ (due to algebraicity of Pic^{0}(X) × Pic^{0}(X) \ Δ and all U_{n_{i}H}).

Exact sequence in cohomology implies that:

H^{0}(X, −L + L′) = H^{0}(C_{n}, −L + L′),

where C_{n} is a smooth curve in the class [nH]. Since H^{0}(X, −L + L′) = 0, for L ≠ L′, the same holds for their restrictions. In particular,

i^{∗} : Pic^{0}(X) → Pic^{0}(nH)

is a set-theoretic embedding (on the set of k-points).

Remark 3.5. — A more delicate analysis shows that for n ≫ 0 the map

rem:stra

i^{∗} : Pic^{0}(X) → Pic^{0}(nH) is an embedding of algebraic groups. Note that over a closure of a finite field the map

i : Pic(X) → Pic(C)

is never an embedding if rk NS(X) > 1 (in contrast with characteristic zero).

Lemma 3.6. — Let X/k be a smooth projective surface, C ⊂ X an irre-

lemm:gogo

ducible curve and Q a finite set of points on C. Then there exists a diagram

\xymatrix{C̃ ⊂ & X̃ \ar[d]^{π̃} \ar[r]^{π} & Y \\ C ⊂ & X}

where X̃ = Bl(X) → X is a blowup of X with center supported in C \ Q and π is an isomorphism on X̃ \ C̃ (the strict transform of C under π̃) which maps C̃ to a point on Y.

Proof. — There is a polarization H on X such that H −C restricted to X \C is very ample (induces an embedding of X \ C into a projective space). Let ℙ_{C}^{r} be the projective space spanned by C under the embedding X ⊂ ℙ^{n} by H. By our choice of H, r < n. A generic hyperplane ℙ^{r−1} ⊂ ℙ_{C}^{r} intersects C transversally in finitely many smooth points q_{1}, …, q_{s} which are contained in C \ Q (here we use Bertini’s theorem for embedded curves, which in this case is evident over any algebraically closed field). The projection from this ℙ^{r−1} (inside ℙ^{n}) induces a proper map from the blowup X̃ of X with center in ∪q_{i} onto a projective surface Y ⊂ ℙ^{n−r}. Note that the image of C under the projection is a point q ∈ Y.

By construction, ℙ_{C}^{r} intersects X exactly in C. Hence, the proper preimage of q in X̃ is C̃. Any other ℙ^{r} ⊂ ℙ^{n} intersects X \ C in at most one point and transversally (by assumption on H^{0}(X, H −C)). It follows that the projection induces an isomorphism between X̃ \ C̃ and Y \ q.

Lemma 3.7. — Let X/k be a smooth projective surface, C ⊂ X a curve

lemm:go

and Q ⊂ C(k) a finite set. Let L be a line bundle on X whose restriction to C is trivial (L_{C} ≃ O_{C}). Then there exists a diagram

\xymatrix{C̃ ⊂ & X̃ \ar[d]^{π̃} \ar[r]^{π} & Y \\ C ⊂ & X}

where X̃ = Bl(X) → X is a blowup with center supported in finitely many points on C \ Q and π is a proper map as in Lemma 3.6 (contracting C̃) such that the pullback L̃ = π̃^{∗}L is induced from Y.

Proof. — By Lemma 3.6, we may assume that C is (already) contractible. Since L is trivial on C we have L ≃ O(R_{1} − R_{2}), where R_{1}, R_{2} are divisors on X intersecting C transversally, and

R_{1} ∩ C = R_{2} ∩ C ⊂ C \ Q.

Indeed, we can find a polarization H, so that L + H is also a polarization, giving surjective maps

\begin{aligned}H^{0}(X, L + H) & → & H^{0}(C, (L + H)_{C}) \\ H^{0}(X, H) & → & H^{0}(C, H_{C}). \end{aligned}

Let i_{C} : (L+H)_{C} ⟶^{∼} H_{C} be an isomorphism. We can find a pair of sections

s_{1} ∈ H^{0}(X, L + H), s_{2} ∈ H^{0}(X, H)

with i_{C}(s_{1})_{C} = (s_{2})_{C}. Let R_{i} be the zero divisor of s_{i}. Then O(R_{1} −R_{2}) ≃ L and R_{1}, R_{2} intersect C transversally with

R_{1} ∩ C = R_{2} ∩ C ⊂ C \ Q,

as claimed.

Consider the smooth surface π̃ : X̃ → X obtained by blowing up R_{i} ∩ C. The proper preimages R̃_{i} of R_{i} in X̃ don’t intersect the proper preimage C̃ ⊂ X̃ of C. The divisor of π^{∗}L = π^{∗}(R_{1} − R_{2}) doesn’t contain components which are exceptional curves lying over points in C. Hence π^{∗}L is trivial on the open quasi-projective neighborhood X̃ \ supp(π^{∗}(R_{1} − R_{2})) containing C̃. Therefore, the bundle L is induced from Y (as in Lemma 3.6).

Lemma 3.8. — Let K/k be the function field of a surface, C/k a smooth

lemm:cc

curve on a model of K and Q = {q_{0}, …, q_{s}} ⊂ C(k) a finite set of points. Then there exist a model X of K, irreducible divisors D_{j}, H_{j}, H′_{j} on X, with j = 0, …, s, and a positive integer n = n_{Q} such that:

(1) X is smooth and contains C;

(2) D_{j} ∩ C = q_{j} for all j = 1, …, s;

(3) n(D_{j} − D_{0}) restricted to C is a principal divisor;

(4) n(D_{j} − D_{0}) + (H_{j} − H′_{j}) is a principal divisor on X;

(5) the divisors D_{j} are pairwise disjoint;

(6) all intersections between D_{j}, H_{i} and H′_{i} are transversal, pairwise distinct and outside C;

(7) H_{j}, H′_{j} don’t intersect C.

Proof. — Let X be a smooth projective model of K containing C as a smooth curve. Choose divisors D_{j} ⊂ X passing (transversally) through q_{j} (for all j = 0, …, s). Blowing up points in C \ Q we can insure that the (strict transform of) C becomes contractible and that the image of the surface under a contracting morphism is projective (by Lemma 3.6).

Blowing up again (if necessary) and removing components of exceptional divisors, we can insure that the (strict transforms) D_{j} ∩ C = q_{j} (for all j). By Lemma 3.2, there exists an n = n_{Q} such that the restriction of n(D_{j} − D_{0}) to C is a principal divisor. We continue to blow up (outside Q) so that each n(D_{j} − D_{0}) becomes a trivial line bundle on some open neighborhood of C in some model X (using Lemma 3.7).

Throughout, C remains contractible and we write

π : X → Y

for the corresponding blow-down. Now n(D_{j} − D_{0}) is induced from a line bundle on Y (which is projective). In particular, there exist ample classes [H_{j}], [H′_{j}] ∈ Pic(Y) such that

[n(D_{j} − D_{0})] + ([H_{j}] − [H′_{j}])

is a principal divisor on X (here we identified [H_{j}], [H′_{j}] with their full transforms in X). Finally, we can choose representatives H_{j}, H′_{j} ⊂ Y of these classes which are disjoint from π(C), irreducible and satisfy all required transversality assumptions.

More precisely, choose classes [H_{i}] so that

[n(D_{j} − D_{0})] + ([H_{j}]), [n(D_{j} − D_{0})] + ([H′_{j}]), [H_{j}], [H′_{j}]

provide an embedding of Y into a projective space. Consider an embedding of Y into a projective space defined by one of the series [H_{j}], [H′_{j}]. For any finite set of irreducible divisors in Y we find a hyperplane section intersecting the union of these divisors transversally and not containing the given finite set of points in Y. Using induction on j we find representatives of [H_{j}], [H′_{j}] satisfying the lemma.

Lemma 3.9. — Let K/𝔎 be a purely inseparable extension. Then

lemm:purely

– 𝔎 ⊃ k;

– K/𝔎 is a finite extension;

– 𝔎 = k(X′) for some algebraic variety X′.

Definition 3.10. — We write \overline{E}^{K} ⊂ K for the normal closure of a subfield

defn:gener

E ⊂ K (elements in K which are algebraic over E). We say that x ∈ K \ k is generating if \overline{k(x)}^{K} = k(x).

Remark 3.11. — If E ⊂ K is 1-dimensional then for all x ∈ E \k one has

rem:just

\overline{k(x)}^{K} = \overline{E}^{K} (a finite extension of E).

Lemma 3.12. — For any subfield E ⊂ K there is a sequence

lemm:subf

X ⟶^{π_{E}} Y′ ⟶^{ρ_{E}} Y,

where – π_{E} is rational dominant with irreducible generic fiber;

– ρ_{E} is quasi-finite and dominant;

– k(Y′) = \overline{E}^{K} and k(Y) = E.

For generating x ∈ K we write

π_{x} : X → Y

for the morphism from Lemma 3.12, with k(Y) = k(x). For y ∈ K \ k(x) define deg_{x}(y) (the degree of y on the generic fiber of π_{x}) as the degree of the corresponding surjective map from the generic fiber of π_{x} under π_{y}.

Proposition 3.13. — Let K = k(X) be the function field of a smooth

prop:geome

surface, C ⊂ X a smooth irreducible curve and f_{1}, …, f_{s} ∈ K^{∗} rational functions on X, restricting nontrivially to C. Then there exists a model X̃ of K (a blowup of X) such that for every point q in (the strict transform) of C ⊂ X̃ there exists an irreducible divisor D_{q} ⊂ X̃ (possibly the zero divisor) with the property that for all i = 1, …, s the order of D_{q} in the divisor of f_{i} is equal to the order of f_{i} in q.

Proof. — Consider the divisors C and div(f_{i}), i = 1, …, s and a model X̃ of K such that the total preimage D̃ ⊂ X̃ of the union of all these divisors in X̃ has strict normal crossings (resolution of singularities for surfaces). After further blowups we can assume that each irreducible component of D̃ (distinct from C) intersects (the strict transform of) C in at most one point. For each q ∈ C ∩ (D̃ \ C) let D_{q} be this component. For all other q let D_{q} be the zero divisor. These divisors have the required properties.

Lemma 3.14. — Let K = k(X) be the function field of a surface and x, y ∈

lemm:XX

K \ k be such that

deg_{x}(y) = min_{f∈K\\overline{k(x)}^{K}}(deg_{x}(f))

and \overline{k(y)}^{K} = k(y′) for some y′ ∈ K^{∗}. Then y is generating: k(y) = \overline{k(y)}^{K}.

Proof. — If y is not generating then y = z(y′) for some y′ ∈ K and some function z ∈ k(y′)^{∗} of degree ≥ 2. This implies that deg_{x}(y) ≥ 2 deg_{x}(y′), contradicting minimality.

Lemma 3.15. — Let X be a model of K containing a rational curve C and

lemm:xcxc

x ∈ K^{∗} a function such that its restriction x_{C} to C is defined and such that k(C) = k(x_{C}). Then x is generating: \overline{k(x)}^{K} = k(x).

Proof. — The restriction map extends to \overline{k(x)}^{K} and hence is an isomorphism between k(x_{C}) and k(x) = \overline{k(x)}^{K}.

The next proposition characterizes multiplicative groups of fields 𝔎 ⊂ K such that K/𝔎 is a purely inseparable extension. Notice that for a onedimensional field k(C) the subfield 𝔎 is always of the form k(C)^{p^{n}}, for some n ∈ ℕ. Thus for any one-dimensional subfield E ⊂ K there is an r(E) ∈ ℕ such that the intersection of 𝔎^{∗} with E^{∗} consists exactly of r(E)-powers of the elements of E^{∗}. Below we show that this property of intersection with subfields of the special form k(x) = \overline{k(x)}^{K} already characterizes multiplicative groups of such 𝔎 among multiplicative subsets in K^{∗}.

Definition 3.16. — Let 𝔎^{∗} ⊂ K^{∗} be a (multiplicative) subgroup such that

defi:new

for any subfield E = k(x) = \overline{k(x)}^{K} ⊂ K there exists an r = r(E) with the property that 𝔎^{∗} ∩ E^{∗} = (E^{∗})^{r} (r-powers of elements of E^{∗}). For every t ∈ E^{∗} \ k^{∗} we define r(t) = r(E).

Remark 3.17. — Note that r(t) is not defined for t ∈ K^{∗} \ k^{∗} iff \overline{k(t)}^{K} is the function field of a curve of genus ≥ 1.

Definition 3.18. — We will say that y ∈ K^{∗} is a power if there exist an

defi:power

x ∈ K^{∗} and an integer n ≥ 2 such that y = x^{n}.

Proposition 3.19. — Let K = k(X) be the function field of a surface and

prop:KKK

𝔎^{∗} ⊂ K^{∗} a subset such that

(1) 𝔎^{∗} is a multiplicative subgroup of K^{∗};

(2) for every E = k(x) = \overline{k(x)}^{K} ⊂ K there exists an r = r(E) ∈ ℕ with

𝔎^{∗} ∩ E^{∗} = (E^{∗})^{r};

(3) there exists a y ∈ K \ k with r(y) = 1. Then 𝔎 := 𝔎^{∗} ∪ 0 is a field, whose multiplicative group is 𝔎^{∗} and K/𝔎 is a purely inseparable finite extension.

Proof. — Once we know that 𝔎 is a field we can conclude that every x ∈ K^{∗} is either in 𝔎^{∗} or some power of it is in 𝔎^{∗}. Of course, it can only be a power of p so that K/𝔎 is a purely inseparable extension, of finite degree (by Lemma 3.9).

By (3), k ⊂ 𝔎. To conclude that 𝔎 is a field, it suffices to show that for every x ∈ 𝔎 one has x + 1 ∈ 𝔎 (and then use multiplicativity). For every x ∈ 𝔎 \ k with r(x) = 1 we have 𝔎^{∗} ∩ k(x)^{∗} = k(x)^{∗} and

x + κ ∈ 𝔎^{∗}, for all κ ∈ k.

In particular, this holds for y.

Consider x ∈ 𝔎^{∗} with r(x) > 1 or not defined. We claim that for some κ ∈ k

z := \frac{x + y + κ}{y + κ − 1} ∈ 𝔎 and r(z) = 1.

This implies that

z − 1 = (x + 1)/(y + κ − 1) ∈ 𝔎^{∗} and x + 1 ∈ 𝔎^{∗},

(by multiplicativity). We can assume that K/k(C)(y), where k(C) = \overline{k(x)}^{K}, is a finite separable extension. (Otherwise, we can let K be a minimal proper subfield in 𝔎′ ⊂ K containing k(C)(y) and such that K/𝔎′ is purely inseparable and use the intersection of 𝔎 with 𝔎′ instead of 𝔎.)

To prove the claim, choose a model X of K such that both maps

\begin{aligned}π_{x} : X & → & C, k(C) = \overline{k(x)}^{K} \\ π_{y} : X & → & ℙ^{1} = (y : 1) \end{aligned}

are proper morphisms (as in Lemma 3.12). Since x and y are algebraically independent (r(x) > 1), only finitely many components of the fibers of π_{x} are contained in the fibers of π_{y} and there exists a κ ∈ k such that both fibers

π_{y}^{−1}(−κ) and π_{y}^{−1}(1 − κ)

are transversal to the fibers of π_{x}, since we assume that K/k(C)(y) is separable. Note that

div_{0}(y + κ − 1) ⊄ div(x + y + κ),

since y + κ = −1 on div_{0}(y + κ − 1) and x is nonconstant on these fibers (where div_{0} is the divisor of zeroes). It follows in the first case that both

t := (y + κ)/x and z := (x + y + κ)/(y + κ − 1)

are not powers.

Note that t, z are generating elements. Indeed, if we blow up the smooth point q of transversal intersection {y + κ = 0} ∩ {x = 0} then t restricts nontrivially to ℙ_{q}^{1} and similarly

z := (x + y + κ)/(y + κ − 1) = x + 1/(y + κ − 1) + 1

restricts nontrivially to ℙ_{q′}^{1}, where q′ = {x = −1} ∩ {y = 1 − κ}. Note that t ⊂ 𝔎^{∗} and since it is not a power r(t) = 1 and

(1/t) + 1 = (x + y + κ)/(y + κ) ∈ 𝔎.

To show that z ∈ 𝔎 observe that both x, y + κ ∈ 𝔎 so that t ∈ 𝔎. Therefore,

t + 1 = (x + y + κ)/x ∈ 𝔎

and, by (1), x + y + κ ∈ 𝔎. Finally, since (y + κ − 1) ∈ 𝔎 we get z ∈ 𝔎.

Remark 3.20. — If assumption (3) is not satisfied then we can take

(𝔎^{∗})^{1/r(y)} ⋂ K^{∗},

which satisfies all the conditions of the lemma. Thus in general without the assumption (3) we have 𝔎 = (𝔎′)^{r}, where K/𝔎′ is purely inseparable and r ∈ ℕ.

4. Projective structures

sect:proj-str

In this section we explain the connection between fields and axiomatic projective geometry. We follow closely the exposition in [7].

Definition 4.1. — A projective structure is a pair (S, 𝔏) where S is a

defi:proj

(nonempty) set (of points) and 𝔏 a collection of subsets 𝔩 ⊂ S (lines) such that P1 there exist an s ∈ S and an 𝔩 ∈ 𝔏 such that s ∉ 𝔩; P2 for every 𝔩 ∈ 𝔏 there exist at least three distinct s, s′, s′′ ∈ 𝔩; P3 for every pair of distinct s, s′ ∈ S there exists exactly one

𝔩 = 𝔩(s, s′) ∈ 𝔏

such that s, s′ ∈ 𝔩; P4 for every quadruple of pairwise distinct s, s′, t, t′ ∈ S one has

𝔩(s, s′) ∩ 𝔩(t, t′) ≠ ∅ ⇒ 𝔩(s, t) ∩ 𝔩(s′, t′) ≠ ∅.

For s ∈ S and S′ ⊂ S define the join

s ∨ S′ := {s′′ ∈ S | s′′ ∈ 𝔩(s, s′) for some s′ ∈ S′}.

For any finite set of points s_{1}, …, s_{n} define

⟨s_{1}, … s_{n}⟩ := s_{1} ∨ ⟨s_{2} ∨ ⋯ ∨ s_{n}⟩

(this does not depend on the order of the points). Write ⟨S′⟩ for the join of a finite set S′ ⊂ S. A finite set S′ ⊂ S of pairwise distinct points is called independent if for all s′ ∈ S′ one has

s′ ∉ ⟨S′ \ {s′}⟩.

A set of points S′ ⊂ S spans a set of points T ⊂ S if – ⟨S′′⟩ ⊂ T for every finite set S′′ ⊂ S′; – for every t ∈ T there exists a finite set of points S_{t} ⊂ S′ such that

t ∈ ⟨S_{t}⟩. A set T ⊂ S spanned by an independent set S′ of points of cardinality ≥ 1 is called a projective subspace of dimension |S′| − 1.

The axioms imply that projective subspaces of a given projective space S form a lattice and that the dimension function is well defined, i.e.,

dim(T ∪ T′) + dim(T ∩ T′) = dim(T) + dim(T′)

for all pairs of projective subspaces T, T′ ⊂ S. Here we put dim(∅) := −1.

Definition 4.2. — A projective structure (S, 𝔏) satisfies Pappus’ axiom if

defn:pappus

PA for all 2-dimensional subspaces and every configuration of six points and lines in these subspaces as below

the intersections are collinear.

The main theorem of abstract projective geometry is:

Theorem 4.3. — Let (S, 𝔏) be a projective structure of dimension n ≥ 2

thm:abstr

which satisfies Pappus’ axiom. Then there exists a vector space V over a field L and an isomorphism

σ : ℙ_{L}(V) ⟶^{∼} S.

Moreover, for any two such triples (V, L, σ) and (V′, L′, σ′) there is an isomorphism

V/L ⟶^{∼} V′/L′

compatible with σ, σ′ and unique up to homothety v ↦ λv, λ ∈ L^{∗}.

Proof. — See [7], Chapter 6.

Definition 4.4. — A morphism of projective structures

defi:inject

ρ : (S, 𝔏) → (S′, 𝔏′)

is an injection of sets ρ : S ↪ S′ such that ρ(𝔩) ∈ 𝔏′ for all 𝔩 ∈ 𝔏.

Example 4.5. — Let k be a field and ℙ_{k}^{n} the usual projective space over k

exam:Kk

exam:basic

of dimension n ≥ 2. Then ℙ_{k}^{n} carries a projective structure: the set of lines is the set of usual projective lines ℙ_{k}^{1} ⊂ ℙ_{k}^{n}.

Let K/k be an extension of fields (not necessarily finite). Then the set

S := ℙ_{k}(K) = (K \ 0)/k^{∗}

carries a natural (possibly, infinite-dimensional) projective structure. Moreover, multiplication by elements in the group K^{∗}/k^{∗} preserves this structure.

Theorem 4.6. — Let K/L and K′/L′ be field extensions of degree ≥ 3 and

thm:skk

ϕ̄ : S = ℙ_{L}(K) → ℙ_{L′}(K′) = S′

a bijection of sets which is an isomorphism of abelian groups and of projective structures. Then

L ≃ L′ and K ≃ K′.

Proof. — Consider V := K as a vector space over L. By Theorem 4.3, to S there are canonically attached the L-algebra End(V) and GL(V) ⊂ End(V), as the set of elements preserving the collineations of the projective space S (because the action of homotheties on End(V) is trivial). This allows to recover the field K as the subfield of the L-algebra End(V) given by

{0} ∪ {x ∈ GL(V) ⊂ End(V) | x induces a group-translation on S}.

Definition 4.7. — Let K/k be the function field of an algebraic variety X

defn:basic2

of dimension ≥ 2 and S = ℙ_{k}(K) the associated projective structure from Example 4.5. The lines passing through 1 and a generating element of K (see Definition 3.10) and their multiplicative translations by elements in K^{∗}/k^{∗} will be called primary.

Lemma 4.8. — Let K = k(X) be the function field of a surface. For every

lemm:primary-ell

line 𝔩 = 𝔩(1, x) there exists a ℙ^{2} ⊂ ℙ_{k}(K) such that all other lines in this ℙ^{2} are primary.

Proof. — Choose a smooth model X of K and two points q_{1}, q_{2} ∈ X such that x(q_{1}) = 0, x(q_{2}) = 1. Blow up q_{1}, q_{2} and let ℙ_{i}^{1} be the corresponding exceptional curves. Let y ∈ K^{∗} be an element restricting to a generator of k(ℙ_{i}^{1}). The restriction map extends to the normal closure \overline{k(y)} ⊂ K. Hence the normal closure \overline{k(y)} ⊂ K coincides with k(y).

To prove that every line 𝔩 ≠ 𝔩(1, x) ⊂ ℙ^{2} = ℙ_{k}(k ⊕ kx ⊕ ky) is primary we need to show that (y + a + bx)/(y + c + dx) is generating, provided (a, b) ≠ (c, d). If a ≠ c then the restriction of (y +a+bx)/(y +c+dx) to ℙ_{q_{1}}^{1} is equal to (y+a)/(y+c) and hence is a generator of k(ℙ_{q_{1}}^{1}). By the argument of the previous lemma, (y +a+bx)/(y +c+dx) is generating. If a = c, b ≠ d then (y + a + bx)/(y + c + dx) on ℙ_{q_{2}}^{1} coincides with (y + a + b)/(y + c + d) and is also generating since a + b ≠ c + d, by assumption.

Lemma 4.9. — Assume that a set S has two projective structures (S, 𝔏_{1})

lemm:4.10

and (S, 𝔏_{2}), both of dimension ≥ 2, and that for some ℙ_{1}^{2} (in the first projective structure) every line 𝔩_{1} ∈ (𝔏_{1} ∩ ℙ_{1}^{2}), except possibly one line, is also a line in the second structure. Then the set ℙ_{1}^{2} is a projective plane in the second structure (S, 𝔏_{2}), projectively isomorphic to ℙ_{1}^{2} ∈ (S, 𝔏_{1}).

Proof. — Let ℙ̂_{1}^{2} be the set of all lines in ℙ_{1}^{2} and ℙ̂_{1}^{2} \ 𝔩 the set of lines which remain projective lines in ℙ_{2}^{2}. Let 𝔩_{1}, 𝔩_{2}, 𝔩_{3} be three lines from ℙ̂_{1}^{2} \ 𝔩 which don’t have a common intersection point. Then 𝔩_{1}, 𝔩_{2}, 𝔩_{3} lie in the same plane ℙ_{2}^{2}. Since every other line 𝔩′ ∈ ℙ̂_{1}^{2} \ 𝔩 intersects 𝔩_{1}, 𝔩_{2}, 𝔩_{3} then 𝔩′ ⊂ ℙ_{2}^{2}. Thus all lines from ℙ̂_{1}^{2} \ 𝔩 are in ℙ_{2}^{2} which contains all the points of ℙ_{1}^{2}.

They are isomorphic since it is an isomorphism between lines and every point, except possibly one point, is an intersection of two lines in ℙ̂_{1}^{2} \ 𝔩. Since ℙ̂_{2}^{2} coincides with ℙ̂_{1}^{2} outside of one point they coincide.

Corollary 4.10. — Let K/k and K′/k′ be function fields of algebraic

coro:final

surfaces

ϕ̄ : S = ℙ_{k}(K) → S′ = ℙ_{k′}(K′)

an isomorphism of (multiplicative) abelian groups inducing a bijection on the set of primary lines in the corresponding projective structures. Then ϕ̄ is an isomorphism of projective structures and

k ≃ k′ and K ≃ K′.

Proof. — By Lemma 4.8 and Lemma 4.9 ϕ̄ induces an isomorphism of projective structures. It remains to apply Theorem 4.6.

5. Flag maps

sect:af

Notations 5.1. — We fix two distinct prime numbers ℓ and p. Let

nota:pr

– 𝔽 = 𝔽_{q} be a finite field with q = p^{n} and 𝔽^{∗} its multiplicative group;

– Vect_{𝔽} - the set of finite-dimensional 𝔽-vector spaces;

– A a vector space over 𝔽 and ℙ(A) = ℙ_{𝔽}(A) = (A \ 0)/𝔽^{∗} its projectivization;

– M(A) the set of maps from A \ {0} to ℤ_{ℓ};

– for μ ∈ M(A) and B ⊂ A an 𝔽-linear subspace, μ_{B} the restriction of μ to B \ {0}.

Definition 5.2. — An 𝔽-flag on a vector space A ∈ Vect_{𝔽} is a collection

defn:aa

of 𝔽-subspaces (A_{n})_{n=0,...,d} such that – A_{0} = A;

– A_{n} ⊋ A_{n+1}, for all n = 0, …, d − 1. The flag is called complete if d = dim(A).

Definition 5.3. — A map μ ∈ M(A) will be called 𝔽^{∗}-invariant if for all

defn:homo

a ∈ A \ {0} and all κ ∈ 𝔽^{∗} one has

μ(κ · a) = μ(a).

Definition 5.4. — A map μ on A\{0}, for a (possibly infinite-dimensional)

defn:inv

vector space A, will be called an 𝔽-flag map, if – μ is 𝔽^{∗}-invariant;

– every finite-dimensional 𝔽-vector space B ⊂ A has a complete flag (B_{n})_{n=0,...,dim(B)} such that μ_{B} is constant on B_{n} \ B_{n+1}, for all n = 0, …, dim(B) − 1.

The value of μ on B = B_{0} \ B_{1} is called the generic value of μ on B; we denote it by μ^{gen}(B). The 𝔽-subspace B_{1} is called the subspace of nongeneric elements. The set of 𝔽-flag maps will be denoted by Φ_{𝔽}(A).

Example 5.5. — Let K = k(X) be a function field. We can consider it

exam:main-flag

as a vector space over k or over any of the subfields 𝔽 ⊂ k. Let ν be a nonarchimedian valuation on K and χ : Γ_{ν} → ℤ_{ℓ} a homomorphism from the value group of ν (see Section 7). Then χ ∘ ν ∈ Φ_{k}(K).

Definition 5.6. — Let A be an 𝔽-algebra (without zero-divisors). A map

defn:lf

μ ∈ M(A) will be called logarithmic if

μ(a · a′) = μ(a) + μ(a′), for all a, a′ ∈ A \ 0.

The set of such maps will be denoted by L_{𝔽}(A).

Since 𝔽 is torsion, a logarithmic map to ℤ_{ℓ} is 𝔽^{∗}-invariant.

Definition 5.7. — Let A be an 𝔽-vector space. Two maps μ, μ′ ∈ M(A)

c:pair

will be called a c-pair (commuting pair) if for all two-dimensional 𝔽-subspaces B ⊂ A there exist constants λ, λ′, λ′′ ∈ ℤ_{ℓ} (depending on B) with (λ, λ′) ≠ (0, 0) such that for all b ∈ B \ 0 one has

λμ_{B}(b) + λ′μ′_{B}(b) = λ′′.

Theorem 5.8. — Let 𝔽 ⊂ k be a finite field with #𝔽 ≥ 11, and μ, μ′ ∈

thm:cp

L_{𝔽}(K) nonproportional maps forming a c-pair. Then there exists a pair (λ, λ′) ∈ ℤ_{ℓ}^{2} \ (0, 0) such that λμ + λ′μ′ ∈ Φ_{𝔽}(K).

Proof. — This is a special case of the main theorem of [3], where it is proved over general ground fields k. However, the case when k = 𝔽̄_{q} is easier. Following the request of the referee, we now give a complete proof in this special case. The main steps in the proof are:

– characterization of flag maps by their restriction to 2-dimensional 𝔽linear subspaces, for #𝔽 ≥ 11 (see Lemma 5.17);

– reduction to linear spaces over prime fields, resp. 𝔽_{4}, see Lemma 5.19: if μ ∉ Φ_{𝔽′}(A), for a finite field 𝔽′, and μ is 𝔽^{∗}-invariant with respect to a large finite extension 𝔽/𝔽′ then there is a subgroup C ≃ 𝔽_{p}^{2} ⊂ A, (resp. 𝔽_{4}^{2}), so that μ_{C} ∉ Φ_{𝔽_{p}}(C).

– reduction to dimension 3: for any rank two ℤ_{ℓ}-module σ = ⟨μ, μ′⟩ of logarithmic maps generated by a c-pair μ, μ′ ∈ L_{𝔽}(A), not containing a flag map there is a subgroup B = B_{σ} ≃ 𝔽_{p}^{3} ⊂ A (resp. 𝔽_{4}^{3}), such that for any nontrivial μ′′ ∈ σ there is a proper subspace C = C_{μ′′} ⊊ B where μ′′_{C} ∉ Φ_{𝔽_{p}}(C) (this step uses the logarithmic property);

– geometry of collineations on ℙ^{2} = ℙ_{𝔽}(B) over prime fields (resp. 𝔽_{4}): for any σ spanned by a c-pair μ, μ′ on B there is a μ′′ ∈ σ such that μ′′ ∈ Φ_{𝔽}(B) - this shows the existence of the desired flag map on A.

Lemma 5.9. — If A ∈ Vect_{𝔽} and μ ∈ Φ_{𝔽}(A) then there exists a canonical

lemm:linearr

𝔽-flag (A_{n})_{n=0,...,d} such that

μ^{gen}(A_{n}) ≠ μ^{gen}(A_{n+1}),

for all n = 0, …, d − 1.

Proof. — Put A_{0} = A and let A_{n+1} be the additive subgroup of A_{n} spanned by a with μ(a) ≠ μ^{gen}(A_{n}). Since μ is 𝔽^{∗}-invariant, A_{n+1} is an 𝔽-vector space. Indeed, for a, a′ ∈ A_{n+1} and κ, κ′ ∈ 𝔽^{∗} write

a = ∑_{i∈I} b_{i}, a′ = ∑_{j∈J} b′_{j}

with finite I, J. Since

μ(b_{i}) ≠ μ^{gen}(A_{n}), μ(b′_{j}) ≠ μ^{gen}(A_{n}),

for all i ∈ I, j ∈ J, we have

μ(κb_{i}) = μ(b_{i}) ≠ μ^{gen}(A_{n}) and μ(κ′b′_{j}) = μ(b′_{j}) ≠ μ^{gen}(A_{n})

so that κa + κ′a′ ∈ A_{n+1}.

Remark 5.10. — Since a flag map μ is 𝔽^{∗}-invariant, it defines a unique map on (A \ {0})/𝔽^{∗} = ℙ_{𝔽}(A). Conversely, a map μ on ℙ_{𝔽}(A) gives rise to an 𝔽^{∗}-invariant maps on A \ {0}. An 𝔽-flag map on A ∈ Vect_{𝔽} defines a flag by projective subspaces on ℙ_{𝔽}(A). We denote by generic (resp. nongeneric) elements of ℙ_{𝔽}(A) the image of generic (resp. nongeneric) elements from A.

Notations 5.11. — We denote by ℙ̂(A) = ℙ̂_{𝔽}(A) the set of codimension one projective 𝔽-subspaces of ℙ(A).

Definition 5.12. — Assume that A ∈ Vect_{𝔽}, and for all codimension one

defn:duall

𝔽-subspaces B ⊂ A one has μ_{B} ∈ Φ_{𝔽}(B). Define μ̂ by

\begin{matrix}ℙ̂(A) & → & ℤ_{ℓ} \\ B & ↦ & μ̂(ℙ(B)) := μ^{gen}(B). \end{matrix}

Lemma 5.13. — If A ∈ Vect_{𝔽} and μ ∈ Φ_{𝔽}(A) then either μ̂ is constant on

lemm:1pt

ℙ̂(A) or it is constant on the complement to one point.

Proof. — Consider the canonical flag (A_{n})_{n=0,...,d}. If codim(A_{1}) ≥ 2 then for every ℙ(B) ∈ ℙ̂(A) one has μ^{gen}(B) = μ^{gen}(A) and μ̂ is constant. Otherwise, μ^{gen}(B) = μ^{gen}(A), on any B ≠ A_{1} (and differs at ℙ(A_{1}) ∈ ℙ̂(A)).

We need the following elementary

Lemma 5.14. — Let 𝔽 = 𝔽_{q} be a finite field with q ≥ 11 and ℙ^{m} = ℙ_{𝔽}^{m},

lemm:grass

m ≥ 2 a projective space over 𝔽. Then for any four projective hyperplanes and any ten projective subspaces of codimension at least two (all defined over 𝔽) there exists a line (over 𝔽) which is not contained in any of the above hyperplanes and which does not intersect any of the ten codimension two subspaces.

Proof. — One has

#Gr(2, m)(𝔽) ≤ #Gr(2, m + 1)(𝔽)/q^{2}.

The number of 𝔽-lines intersecting a subspace of codimension two in ℙ_{𝔽}^{m} is bounded by #Gr(2, m + 1)(𝔽)/q^{2}. Our claim holds for q ≥ 11.

Lemma 5.15. — Let 𝔽 = 𝔽_{q} be a finite field with q ≥ 11, A ∈ Vect_{𝔽}

lemm:c11

and μ ∈ M(A) an 𝔽^{∗}-invariant map. Assume that there exist 𝔽-subspaces B_{i} ⊂ A, codim(B_{i}) = 1, for i = 1, …, 4 such that

(1) either #{μ^{gen}(B_{i})} ≥ 3 or

(2) μ^{gen}(B_{1}) = μ^{gen}(B_{2}) ≠ μ^{gen}(B_{3}) = μ^{gen}(B_{4}). Then there exists an 𝔽-subspace C ⊂ A, dim_{𝔽}(C) = 2 such that μ_{C} ∉ Φ_{𝔽}(C).

Proof. — By Lemma 5.14, there exists a ℙ^{1} = ℙ(C) ∈ ℙ(A) such that its intersection points with ℙ(B_{i}) are pairwise distinct and generic in the corresponding ℙ(B_{i}) (the nongeneric points of ℙ(B_{i}) are contained in 4 subspaces in codim_{𝔽} ≥ 2, the intersections of B_{i} give rise to 6 more subspaces). Then either μ takes at least three distinct values on ℙ(C) or has distinct values in at least two pairs of points. In both cases μ ∉ Φ_{𝔽}(C).

Corollary 5.16. — Assume that μ_{B} ∈ Φ_{𝔽}(B) for all ℙ(B) ∈ ℙ̂(A) (and

coro:one

#𝔽 ≥ 11). Then μ̂ is constant outside of one point.

Proof. — The map μ̂ takes two different values on ℙ̂(B). By Lemma 5.15, among any three hyperplanes two have the same generic value, so that there can be at most three such values. If there are hyperplanes h_{1}, h_{2}, h_{3} ∈ ℙ̂(A), where μ̂(h_{1}) = μ̂(h_{2}) ≠ μ̂(h_{3}) then for any other h ∈ ℙ̂(A) we have μ̂(h) = μ̂(h_{1}) and μ̂ is constant outside of h_{3}.

Lemma 5.17. — Let A ∈ Vect_{𝔽}, with #𝔽 ≥ 11, and μ ∈ M(A) be an

lemm:1

𝔽^{∗}-invariant map such that for every two-dimensional 𝔽-subspace B ⊂ A, μ_{B} ∈ Φ_{𝔽}(B). Then μ ∈ Φ_{𝔽}(A).

Proof. — Assume the statement holds if dim(A) ≤ n − 1. Then μ̂ is defined and, by Corollary 5.16, either μ̂ is constant on ℙ̂(A) or constant on the complement to one point.

If μ̂ is constant, then the 𝔽-linear envelope of points b ∈ A such that μ(b) ≠ μ̂ has codimension at least two. Indeed, if there is a codimension one subspace B ⊂ A generated by such b then by assumption μ ∈ Φ_{𝔽}(B) and μ^{gen}(B) ≠ μ̂, contradicting the assumption that μ̂ is constant. Otherwise, put A_{1} := B. By the inductive assumption, μ ∈ Φ_{𝔽}(B) and is constant on A \ B. Hence μ ∈ Φ_{𝔽}(A).

Assume that μ̂ is nonconstant and let B ⊂ A be the unique codimension one subspace with differing μ^{gen}(B). Choose an 𝔽-basis b_{1}, …, b_{n−1} in B such that μ(b_{i}) = μ^{gen}(B). Assume that there is a point a ∈ A \ B such that μ(a) ≠ the generic value of μ̂ and let B′ be the codimension one 𝔽subspace spanned b_{1}, …, b_{n−2}, a. Then μ^{gen}(B′) ≠ the generic value of μ̂, contradicting the uniqueness of B. It follows that μ is constant on A \ B.

Remark 5.18. — Let 𝔽/𝔽′ be a finite extension, A ∈ Vect_{𝔽}, considered as an 𝔽′-vector space, and μ ∈ Φ_{𝔽′}(A). If μ is 𝔽^{∗}-invariant, then μ ∈ Φ_{𝔽}(A). Indeed, by Lemma 5.9, the canonical 𝔽′-flag is a flag of 𝔽-vector spaces. We use this observation to reduce our problem to prime fields (resp. 𝔽_{4}).

Lemma 5.19. — Let 𝔽/𝔽′ be a quadratic extension, with #𝔽′ > 2. Let

lemm:444

A be an 𝔽-vector space of dimension 2, considered as an 𝔽′-vector space of dimension 4. Let μ ∈ M(A) be an 𝔽^{∗}-invariant map such that for every 𝔽′-subspace C ⊂ A, dim_{𝔽′}(C) = 2, one has μ_{C} ∈ Φ_{𝔽′}(C). Then μ ∈ Φ_{𝔽}(A).

Proof. — First assume that μ takes only two values on A \ {0}, say 0, 1, and that μ ∉ Φ_{𝔽}(A). Since ℙ_{𝔽}(A) = ℙ_{𝔽}^{1} there exist elements a_{1}, a_{2}, a_{3}, a_{4} ∈ A \ {0} such that the orbits 𝔽^{∗} · a_{i} do not intersect and

0 = μ(a_{1}) = μ(a_{2}) ≠ μ(a_{3}) = μ(a_{4}) = 1.

Then 𝔽^{∗} · a_{i} = Λ_{i} \ {0}, where Λ_{i} is a linear subspace over 𝔽′. The 𝔽′-span Λ_{12} of two nonzero vectors x_{1} ∈ Λ_{1}, x_{2}, ∈ Λ_{2} has μ^{gen}(Λ_{12}) = 0. Hence Λ_{12} contains at most one 𝔽′-subspace ⟨b⟩ of 𝔽′-dimension 1 with generic value 1. The union of the spaces Λ_{12}, for different choices of x_{1}, x_{2}, covers A and

#{b ∈ A | μ(b) = 1} ≤ (q + 1)^{2},

where #𝔽′ = q. Similarly, there are at most (q + 1)^{2} such nongeneric c ∈ A with μ(c) = 0. Since #ℙ^{3}(𝔽′) = q^{3} + q^{2} + q + 1 > 2(q^{2} + 2q + 1), for q > 2, we get a contradiction.

Assume now that μ takes at least 3 distinct values on A \ {0}, say 0, 1, 2, and that there are two vectors a_{1}, a_{2} ⊂ A such that the orbits 𝔽^{∗} · a_{1}, 𝔽^{∗} · a_{2} don’t intersect and 0 = μ(a_{1}) = μ(a_{2}). Such a configuration must exist (take two 𝔽′-spaces of 𝔽′-dimension two spanned by 𝔽^{∗}-orbits; the 𝔽′ span of two generic vectors in these spaces contains elements whose μ-value coincides with the value of μ on one of the two orbits). The modified map, given by

μ̃(a) := \{ \begin{matrix}0 & if μ(a) = 0 \\ 1 & otherwise \end{matrix} ,

satisfies the conditions of the Lemma, and by the above argument μ̃ ∈ Φ_{𝔽}(A). In particular, μ̃ = 0 outside one 𝔽^{∗}-orbit on A \ {0}. Since μ is 𝔽^{∗}-invariant it follows that μ takes two values, and not three as we assumed. Contradiction.

Lemma 5.20. — Let 𝔽′ = 𝔽_{p} (resp. 𝔽_{4}), and 𝔽/𝔽′ be an extension of degree

lemm:p22

divisible by 4. Consider K = k(X) as an 𝔽-vector space and let μ′, μ′′ ∈ L_{𝔽}(K) be a c-pair such that the linear span σ = ⟨μ′, μ′′, 1⟩_{ℤ_{ℓ}} does not contain an Φ_{𝔽}-map. Then there exist an 𝔽′-subspace B ⊂ K with dim_{𝔽′}(B) = 3, two distinct 𝔽′-subspaces C, C′ ⊂ B of dimension 2 and maps μ̃, μ̃′ ∈ σ such that

– μ̃_{C} ∉ Φ_{𝔽′}(C) and μ̃_{C′} is constant;

– μ̃′_{C′} ∉ Φ_{𝔽′}(C′) and μ̃′_{C} is constant; In particular, for every (nonzero) map μ ∈ σ there exists an 𝔽′-subspace C_{μ} ⊂ B, dim_{𝔽′} C_{μ} = 2 with the property that μ_{C_{μ}} ∉ Φ_{𝔽′}(C_{μ}).

Proof. — We will consider K as an 𝔽-vector space as well as an 𝔽′-vector space.

Let μ be an 𝔽^{∗}-invariant map on K. If μ were an 𝔽′-flag map on every twodimensional 𝔽′-subspace of K then, by Lemma 5.19, μ would be an 𝔽-flag map on every 𝔽-subspace B ⊂ K of dim_{𝔽} B = 2. Since #𝔽 ≥ 11 we could apply Lemma 5.17 and conclude that μ ∈ Φ_{𝔽}(K).

Thus, since μ ∉ Φ_{𝔽}(K)-map, there is an 𝔽′-subspace C ⊂ K, dim_{𝔽′}(C) = 2 such that μ_{C} ∉ Φ_{𝔽′}(C). If μ′_{C} is constant, put μ̃′ := μ. Otherwise, using the c-pair property on C we find constants d_{C}, d′_{C}, d′′_{C}, with d′_{C} ≠ 0, such that

d_{C}μ + d′_{C}μ′_{C} = d′′_{C} and put μ̃′ = μ′ − \frac{d′′_{C} − d_{C}μ}{d′_{C}}.

Then μ̃′_{C} = 0. Since the linear combination μ̃′ is not a flag map, there exists a C′, dim_{𝔽′}(C′) = 2, where μ̃′ ∉ Φ_{𝔽′}(C′). If μ_{C′} is constant, put μ̃ := μ. Otherwise, using the c-pair property on C′ we find constants d_{C′}, d′_{C′}, d′′_{C′}, with d′_{C′} ≠ 0, such that

d_{C′}μ + d′_{C′}μ′_{C′} = d′′_{C′} and put μ̃ = μ − \frac{d′′_{C′} − d′_{C′}μ̃′}{d_{C′}} .

Then μ̃_{C′} = 0 and μ̃_{C} ∉ Φ_{𝔽′}(C) (since μ̃′_{C} is constant). Now put

B := C + \frac{c}{c′} · C′,

for some nonzero c ∈ C and c′ ∈ C′. Then dim_{𝔽′}(B) = 3, the maps μ̃_{B}, μ̃′_{B} are linearly independent, and they satisfy the required conditions, by the logarithmic property. For s ≠ 0, we have sμ̃ + s′μ̃′ ∉ Φ_{𝔽′}(C). Otherwise, sμ̃ + s′μ̃′ ∉ Φ_{𝔽′}(\frac{c}{c′} · C′).

A detailed analysis of c-pairs on projective planes as above shows that ⟨μ′, μ′′⟩_{ℤ_{ℓ}} on any such space contains a flag map. This will complete the proof of the main theorem.

Lemma 5.21 (Lemma 4.3.2 in [3]). — Let V ⊂ ℤ_{ℓ}^{2} be such that for any two

lemm:line-point0

pairs of distinct points the affine line through one pair and the affine line through the other have a common point and that this point of intersection is contained in V.

Then V is contained in a line union one point.

Proof. — Otherwise, V contains four points in general position. Embedd ℤ_{ℓ}^{2} into ℙ^{2}(ℚ_{ℓ}), choose coordinates for these four points

(1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1) and (1 : 1 : 1)

and close V for the operation

x, y, z, t ↦ 𝔩(x, y) ∩ 𝔩(z, t), when x ≠ y, z ≠ t, 𝔩(x, y) ≠ 𝔩(z, t).

The closure V̄ of V satisfies the axioms of a projective plane (see Definition 4.1). For example, to verify that any “line” in V̄ contains at least three points it suffices to pick one of the four initial points not on this line and to draw lines through this point and the remaining three points in the initial set.

By the fundamental theorem of projective geometry, V̄ = ℙ^{2}(ℚ). On the other hand, ℙ^{2}(ℚ) is dense in ℙ^{2}(ℚ_{ℓ}). In particular, it cannot be contained in ℤ_{ℓ}^{2}. Contradiction.

Corollary 5.22. — Let B = 𝔽^{3} and μ, μ′ ∈ M(B) be a c-pair of 𝔽^{∗}-

coro:line-point

invariant maps. Then the image of ℙ(B) under map

\begin{aligned}φ & : & ℙ(B) & → & 𝔸^{2}(ℤ_{ℓ}) \\ & & b & ↦ & (μ(b), μ′(b)) \end{aligned}

is contained in a union of an affine line and (possibly) one more point.

Proof. — Observe that the image of every ℙ^{1} ⊂ ℙ(B) is contained in an affine line in ℤ_{ℓ}^{2}. This is simply the geometric interpretation of the condition for μ, μ′ to be a c-pair.

Next, for any two pairs of distinct points (a, b), (a′, b′) in φ(ℙ(B)) the affine lines 𝔩 = 𝔩(a, b), 𝔩′ = 𝔩′(a′, b′) in 𝔸^{2} = ℤ_{ℓ}^{2} through these pairs of points must intersect. (Choose ã, b̃, ã′, b̃′ in the preimages of a, b, a′, b′; the projective lines 𝔩̃, 𝔩̃′ ⊂ ℙ(B) = ℙ^{2} through these points intersect in some q and, by the first observation, φ(q) must lie on both 𝔩 and 𝔩′).

Now it suffices to apply Lemma 5.21.

Assumption 5.23. — We may now assume that – 𝔽 = 𝔽_{p} or 𝔽_{4};

– μ, μ′ ∈ L_{𝔽}(A) is a c-pair of linearly independent maps as in Lemma 5.20,

– B is as in Lemma 5.20, and in particular, there exist two-dimensional C, C′ ⊂ B so that μ_{C} ∉ Φ_{𝔽}(C), μ′_{C′} ∉ Φ_{F} (C′) and μ_{C′} = μ′_{C} = 0.

assu:proof

We can also exclude the following degenerate cases, which contradict our assumption that no linear combination of μ, μ′ is a flag map on B:

(1) φ(ℙ(B)) is contained in a line; this means that μ, μ′ are linearly dependent (modulo constants);

(2) φ(𝔩) is a point, for some 𝔩 ∈ ℙ(B); this implies that φ(𝔩) ∈ φ(𝔩′), for all 𝔩′ ⊂ ℙ(B) and φ(ℙ(B)) is contained in a line, contradiction to (1);

(3) φ(ℙ(B)) is constant outside one line; here the affine map ℤ_{ℓ}^{2} → ℤ_{ℓ} projecting φ(𝔩) to one point gives a nontrivial flag map in the span of μ, μ′.

Lemma 5.24. — Let 𝔩, 𝔩′ ⊂ ℙ^{2} be distinct lines. Let x ∈ ℙ^{2} be a point such

lemm:pi-x

that φ(x) ∉ (φ(𝔩) ∪ φ(𝔩′)). Then there is a natural projective isomorphism π_{x,𝔩′} : 𝔩 → 𝔩′ respecting the level sets of φ. Namely, for every pair y_{1}, y_{2} ∈ 𝔩 with φ(y_{1}) = φ(y_{2}) one has

φ(π_{x,𝔩′}(y_{1})) = φ(π_{x,𝔩′}(y_{2}))

(and vice versa). In particular, if φ(𝔩) ⊂ φ(𝔩′) then φ(𝔩) = φ(ml′).

Proof. — The images φ(𝔩(x, y_{1})) and φ(𝔩(x, y_{2})) span the same affine line L_{x}. We have φ(𝔩′) ⊄ L_{x}. Define π_{x,𝔩′}(y_{i}) := 𝔩(x, y_{i}) ∩ 𝔩′. By Corollary 5.22, φ(π_{x,𝔩′}(y_{i})) are contained in the intersection of φ(𝔩′) and L_{x}, so that φ(π_{x,𝔩′}(y_{1})) = φ(π_{x,𝔩′}(y_{2})).

Corollary 5.25. — Let x, y ∈ ℙ_{𝔽}^{2} be distinct points so that φ(x), φ(y) ∉

coro:same

(φ(𝔩) ∪ φ(𝔩′)) and the line 𝔩(x, y) through x, y passes through the intersection q_{0} := 𝔩 ∩ 𝔩′. Then the composition

π_{x,𝔩} ∘ π_{y,𝔩}^{−1} : 𝔩 → 𝔩

induces a nontrivial translation on 𝔩, with fixed point q_{0}, preserving the level sets of φ. (By symmetry we have a similar translation on 𝔩′.)

π_{x,𝔩′} ∘ π_{y,𝔩′}^{−1} : 𝔩 → 𝔩

In particular, if 𝔽 = 𝔽_{p} (the prime field) then this translation is transitive on 𝔩\(𝔩∩𝔩′) and φ is constant on this complement. If 𝔽 = 𝔽_{4} then the complement to 𝔩 \ (𝔩 ∩ 𝔩′) is a union of two (two point) orbits of this translation and φ is constant on each orbit.

Proof of Theorem 5.8. — We keep the Assumptions 5.23. Put ℙ^{2} = ℙ(B) and 𝔩 = ℙ^{1} = ℙ(C), 𝔩′ = ℙ^{1} = ℙ(C′). For every ℙ^{1} ⊂ ℙ^{2} its image φ(ℙ^{1}) spans a unique affine line.

Assume in addition that there exists a q ∉ 𝔩 such that φ(q) ∉ φ(𝔩′). Let 𝔩′′ := 𝔩(q, q_{0}) be the line through q and q_{0} := 𝔩 ∩ 𝔩′. Then there exist two distinct points x, y ∈ 𝔩′ such that φ(x), φ(y) ∉ (φ(𝔩) ∪ φ(𝔩′′)). This gives a nontrivial translation, preserving the level sets of φ on 𝔩 (and 𝔩′′).

Over the prime field 𝔽_{p}, p > 2, φ restricted to 𝔩 (that is, μ_{𝔩}) is constant on the complement to q_{0}. Contradiction to the assumption that μ_{𝔩} is not a flag map. It follows that for all q ∉ 𝔩, φ(q) ∈ φ(𝔩′). By symmetry, for all q ∉ 𝔩′, φ(q) ∈ φ(𝔩). Thus, for all q ∉ (𝔩 ∪ 𝔩′) we have φ(q) ∈ φ(q_{0}) and φ is constant on every line through q_{0} distinct from 𝔩, 𝔩′. Contradiction to Assumption 5.23 (2).

For 𝔽 = 𝔽_{4} we have

𝔩 = q_{0} ∪ (t_{1} ∪ t_{2}) ∪ (s_{1} ∪ s_{2}),

with

φ(t_{1}) = φ(t_{2}) and φ(s_{1}) = φ(s_{2}).

If φ(q) ∈ φ(𝔩) then φ(𝔩′′) = φ(𝔩), by Lemma 5.24. Then 𝔩 and 𝔩′′ are projectively (and φ-invariantly) equivalent so that

𝔩′′ = q_{0} ∪ (t′′_{1} ∪ t′′_{2}) ∪ (s′′_{1} ∪ s′′_{2}),

with

φ(t′′_{1}) = φ(t′′_{2}) and φ(s′′_{1}) = φ(s′′_{2}).

Since φ(𝔩(t′′_{1}, s_{1})) and φ(𝔩(s′′_{1}, t_{1})) are again both contained in the same affine line L we get translations on 𝔩 on the complement to both s_{1} and t_{1}.

It follows that

φ(q_{0}) = φ(t_{1}) = φ(s_{1})

contradiction to the assumption that μ ∉ Φ(𝔩).

6. Galois groups

Let k be an algebraic closure of a finite field of characteristic ≠ ℓ, K the function field of an algebraic variety X over k, G_{K}^{a} the abelianization of the pro-ℓ-quotient G_{K} of the Galois group G_{K} of a separable closure of K,

sect:gal

G_{K}^{c} = G_{K}/[[G_{K}, G_{K}], G_{K}] ⟶^{pr} G_{K}^{a}

its canonical central extension and pr the natural projection. In our situation, G_{K}^{a} is a torsion-free ℤ_{ℓ}-module.

Definition 6.1. — We say that γ, γ′ ∈ G_{K}^{a} form a commuting pair if for

defn:lift

some (and therefore any) of their preimages γ̃, γ̃′ in G_{K}^{c} one has [γ̃, γ̃′] = 0. A subgroup H of G^{a} is called liftable if any two elements in H form a commuting pair.

Definition 6.2. — A fan Σ_{K} = {σ} on G_{K}^{a} is the set of all those topolog-

defn:fan

ically noncyclic liftable subgroups σ ⊂ G_{K}^{a} which are not properly contained in any other liftable subgroup of G_{K}^{a}.

Remark 6.3. — For function fields K/k of surfaces all groups σ ∈ Σ_{K} are isomorphic to torsion-free primitive ℤ_{ℓ}-submodules σ of rank 2, see Section 9.

Notations 6.4. — Let

nota:kkk

μ_{ℓ^{n}} := \{ \sqrt[ℓ^{n}]{1} \}

and

ℤ_{ℓ}(1) = lim_{n→∞} μ_{ℓ^{n}}.

We often identify ℤ_{ℓ} and ℤ_{ℓ}(1) (since k is algebraically closed). Write

K̂^{∗} := lim_{n→∞} K^{∗}/(K^{∗})^{ℓ^{n}}

for the multiplicative group of (formal) rational functions on X.

Theorem 6.5 (Kummer theory). — The group K^{∗}/k^{∗} is a free ℤ-module.

thm:ga

One has – K^{∗}/(K^{∗})^{ℓ^{n}} = (K^{∗}/k^{∗})/ℓ^{n}, for all n ∈ ℕ ;

– the discrete group K^{∗}/(K^{∗})^{ℓ^{n}} and the compact profinite group G_{K}^{a}/ℓ^{n} are Pontryagin dual to each other, for a μ_{ℓ^{n}}-duality;

– for K^{∗}/k^{∗} ⟶^{∼} ℤ^{I}, one has K^{∗}/(K^{∗})^{ℓ^{n}} ⟶^{∼} (ℤ/ℓ^{n})^{I} and

G_{K}^{a}/ℓ^{n} ⟶^{∼} (ℤ/ℓ^{n}(1))^{I},

hence the duality between K̂^{∗} = { \atop K^{∗}/k^{∗}} and G_{K}^{a} is modelled on that between

{ functionsI → ℤ_{ℓ} tending to 0 at ∞} and ℤ_{ℓ}^{I}.

Lemma 6.6. — Let E/k be the function field of a curve. Then Σ_{E} = ∅.

rem:2

Proof. — By a result of Grothendieck, the pro-ℓ fundamental group (π_{1})ℓ̂ of a curve punctured in finitely many points is free. We have

G_{E}^{a} = lim_{J⊂I^{⟵}} ℤ_{ℓ}^{J}, G_{E}^{c} = lim_{J⊂I^{⟵}} ∧^{2}(ℤ_{ℓ}^{J}),

with the commutation map equal to ∧. This implies that a liftable subgroup of G_{E}^{a} is topologically cyclic.

7. Valuations

sect:val

In this section we recall basic results concerning valuations and valued fields (we follow [4]). Most of this material an adaptation of well-known facts to our context.

Notations 7.1. — A value group, denoted by Γ, is a totally ordered (torsion-

nota:vg

free) abelian group. We use the additive notation ′′+′′ for the group law and ≥ for the order. We have

Γ = Γ^{+} ∪ Γ^{−}, Γ^{+} ∩ Γ^{−} = {0} and γ ≥ γ′ iff γ − γ′ ∈ Γ^{+}.

Then Γ_{∞} = Γ ∪ {∞} is a totally ordered monoid, by the conventions

γ < ∞, γ + ∞ = ∞ + ∞ = ∞, ∀γ ∈ Γ.

Definition 7.2. — A (nonarchimedian) valuation on a field K is a pair

defn:valu

ν = (ν, Γ_{ν}) consisting of a value group Γ_{ν} and a map

ν : K → Γ_{ν,∞}

such that – ν : K^{∗} → Γ_{ν} is a surjective homomorphism;

– ν(κ + κ′) ≥ min(ν(κ), ν(κ′)) for all κ, κ′ ∈ K;

– ν(0) = ∞.

Remark 7.3. — In particular, since Γ_{ν} is nontorsion, ν(ζ) = 0 for every

rem:zeta

element ζ of finite order in K^{∗}.

A valuation is called trivial if Γ = {0}. If K = k(X) is a function field over an algebraic closure k of a finite field then every valuation of K restricts to a trivial valuation on k (every element in k^{∗} is torsion).

FUNCTION FIELDS 31

Lemma 7.4. — Let K = k(X), with k as above, and ν be a nonarchimedian

lemm:zll

valuation on k(X). Then Hom(Γ_{ν}, ℤ_{ℓ}) is a finitely generated ℤ_{ℓ}-module. Proof. — Note that the ℚ-rank of ν is bounded by dim(X) (see [10]). Notations 7.5. — We denote by K_{ν}, 𝔬_{ν}, 𝔪_{ν} and K_{ν} the completion of K

nota:ko

with respect to ν, the ring of ν-integers in K, the maximal ideal of 𝔬_{ν} and the residue field

K_{ν} := 𝔬_{ν}/𝔪_{ν}.

If X (over k) is a model for K then the center 𝔠(ν) of a valuation is the irreducible subvariety defined by the prime ideal 𝔪_{ν} ∩ k[X] (provided ν is nonnegative on k[X]).

It is useful to keep in mind the following exact sequences:

eqn:1 (7.1)  1 → 𝔬_{ν}^{∗} → K^{∗} → Γ_{ν} → 1

and

eqn:2 (7.2)  1 → (1 + 𝔪_{ν}) → 𝔬_{ν}^{∗} → K_{ν}^{∗} → 1.

Notations 7.6. — Write I_{ν}^{a} ⊂ D_{ν}^{a} ⊂ G_{K}^{a} for the images of the inertia and

nota:ine

the decomposition group of the valuation ν in G_{K}^{a}. Notations 7.7. — If χ : Γ_{ν} → ℤ_{ℓ}(1) is a homomorphism then

nota:iner

χ ∘ ν : K^{∗} → ℤ_{ℓ}(1)

defines an element of G_{K}^{a}, called an inertia element of the valuation ν. The group of such elements is I_{ν}^{a} ⊂ G_{K}^{a}. Notations 7.8. — The decomposition group D_{ν}^{a} is by definition equal to the image of G_{K_{ν}}^{a} in G_{K}^{a}. Lemma 7.9. — There is a natural embedding G_{K_{ν}}^{a} ↪ G_{K}^{a} and a (canonical) isomorphism

D_{ν}^{a}/I_{ν}^{a} ≃ G_{K_{ν}}^{a}.

Proof. — See Theorem 19.6 in [6], for example. Definition 7.10. — Let K = k(X) be a function field. Its valuation ν is

– positive-dimensional if tr deg_{k}K_{ν} ≥ 1;

– divisorial if tr deg_{k}K_{ν} = dim(X) − 1.

defn:div

Notations 7.11. — We let V_{K} be the set of all nontrivial (nonarchimedian)

nota:cv

valuations of K and DV_{K} the subset of divisorial valuations. If ν ∈ DV_{K} is realized by a divisor D on a model X of K (see Example 7.13) we sometimes write I_{D}^{a}, resp. D_{D}^{a}, for the corresponding inertia, resp. decomposition group.

Example 7.12. — Let E = k(C) be the function field of a smooth curve.

exam:curve

Every point q ∈ C(k) defines a nontrivial valuation ν_{q} on E (the order of a function f ∈ E^{∗} at q). Conversely, every nontrivial valuation ν on E defines a point q := 𝔠(ν) on C.

Example 7.13. — Let K = k(X) be the function field of a surface.

exam:suff

– Every positive-dimensional valuation is divisorial.

– Every (irreducible) curve C ⊂ X defines a valuation ν_{C} on K with value group ℤ. Conversely, every valuation on K with value group ℤ and nonalgebraically closed residue field defines a curve C on some model X of K.

– Every flag (C, q), (curve, point on its normalization), defines a valuation ν_{C,q} on K with value group ℤ^{2}.

– There exist valuations on K with value group ℚ and center supported in a point (on every model).

Lemma 7.14. — Let K = k(X) be the function field of a surface. If D_{ν}^{a}/I_{ν}^{a}

lemm:surf-valua

is nontrivial then ν is divisorial.

Proof. — The only 1-dimensional valuations on function fields of surfaces are divisorial valuations. For other valuations, the residue field K_{ν} = k is algebraically closed and G_{K_{ν}}^{a} trivial.

8. A dictionary

Write

L_{K} := L_{k}(K) = { homomorphisms K^{∗} → ℤ_{ℓ}(1)}
Φ_{K} := Φ_{k}(K) = { flag maps K → ℤ_{ℓ}(1)}

Proposition 8.1. — One has the following identifications:

G_{K}^{a} = L_{K},
D_{ν}^{a} = {μ ∈ L_{K} | μ trivial on (1 + 𝔪_{ν})},
I_{ν}^{a} = {μ ∈ L_{K} | μ trivial on 𝔬_{ν}^{∗}}.

sect:dict prop:decomp

If two nonproportional μ, μ′ ∈ G_{K}^{a} form a commuting pair then the corresponding maps μ, μ′ ∈ L_{K} form a c-pair (in the sense of Definition 5.7).

Proof. — The first identification is a consequence of Kummer theory 6.5. The second identification can be checked on one-dimensional subfields of K, where it is evident. For this and the third identification we use (7.1) and (7.2). For the last statement, assume that μ, μ′ ∈ L_{K} don’t form a c-pair. Then there is an x ∈ K such that the restrictions of μ, μ′ ∈ L_{K} to the subgroup ⟨1, x⟩ are linearly independent. Therefore, μ, μ′ ∈ G_{K}^{a} define a rank 2 liftable subgroup in G_{k(x)}^{a}. Such subgroups don’t exist since G_{k(x)} is a free pro-ℓ-group.

Example 8.2. — If μ ∈ D_{ν}^{a} and α ∈ I_{ν}^{a} then μ, α form a commuting pair.

exam:cp

Proposition 8.3. — Let K be a field and α ∈ Φ_{K} ∩ L_{K}. Then there exists

prop:may

a unique valuation ν = (ν_{α}, Γ_{ν_{α}}) (up to equivalence) and a homomorphism pr : Γ_{ν_{α}} → ℤ_{ℓ}(1) such that

α(f) = pr(ν_{α}(f))

for all f ∈ K^{∗}. In particular, α ∈ I_{ν}^{a} (under the identification of Proposition 8.1).

Proof. — Let 𝔽 be a finite subfield of k and assume that α(f) ≠ α(f) for some f, f′ ∈ K and consider the line ℙ^{1} = ℙ(𝔽f + 𝔽f′). Since α is a flag map, it is constant outside one point on this ℙ^{1} so that either α(f +f′) = α(f) or = α(f′). This defines a relation: f′ >_{α} f (in the first case) and f >_{α} f′ (otherwise). If α(f) = α(f′) and there exists an f′′ such that α(f) ≠ α(f′′) and f >_{α} f′′ >_{α} f′ then we put f >_{α} f′. Otherwise, we put f =_{α} f′.

It was proved in [3], Section 2.4, that the above definitions are correct and that >_{α} is indeed an order which defines a filtration on the additive group K by subgroups (K_{γ})_{γ∈Γ} such that

– K = ∪_{γ∈Γ}K_{γ} and

– ∩_{γ∈Γ}K_{γ} = ∅, where Γ is the set of equivalence classes with respect to =_{α}. Since α ∈ L_{K} this order is compatible with multiplication in K^{∗}, so that the map K → Γ is a valuation and α factors as K^{∗} → Γ → ℤ_{ℓ} ≃ ℤ_{ℓ}(1). By (7.1), α ∈ I_{ν}^{a}.

Corollary 8.4. — Every (topologically) noncyclic liftable subgroup of

coro:lift

G_{K}^{a} contains an inertia element of some valuation.

Proof. — By Theorem 5.8, every such liftable subgroup contains an Φ-map, which by Proposition 8.3 belongs to some inertia group.

9. Flag maps and valuations

sect:af-val

In this section we give a Galois-theoretic description of inertia and decomposition subgroups of divisorial valuations.

Lemma 9.1. — Let α ∈ Φ_{K} ∩ L_{K}, ν = ν_{α} be the associated valuation and

lemm:am

μ ∈ L_{K}. Assume that α, μ form a c-pair. Then

μ(1 + 𝔪_{ν}) = μ(1).

In particular, the restriction of μ to 𝔬_{ν} is induced from K_{ν}.

Proof. — We have

(1) α(κ) = 0 for all κ ∈ 𝔬_{ν} \ 𝔪_{ν};

(2) α(κ + m) = α(κ) for all κ and m as above;

(3) 𝔪_{ν} is generated by m ∈ 𝔬_{ν} such that α(m) ≠ 0. If m ∈ 𝔪_{ν} is such that α(m) ≠ 0 and κ ∈ 𝔬_{ν} \ 𝔪_{ν} then α is nonconstant on the subgroup A := ⟨κ, m⟩. Then

μ(κ + m) = μ(κ).

Indeed, if μ is nonconstant on A the restriction μ_{A} is proportional to α_{A} (by the c-pair property) and α satisfies (2). In particular, for such m we have μ(1 + m) = μ(1).

If α(m) = 0 then there exists m′, m′′ ∈ 𝔪_{ν} such that m = m′ + m′′ and α(m′) = α(m′′) ≠ 0. Indeed, there exists an m′ ∈ 𝔪_{ν} such that m > m′ > 1 and α(m′) ≠ α(1) = 0. Since α takes only two values on the subgroup ⟨m′, m⟩ ⊂ 𝔪_{ν} we have

α(m′′) = α(−m′ + m) = α(m′).

Therefore,

0 = μ(1 + m′) + μ(1 + m′′) = μ(1 + m + m′m′′).

Put κ = 1 + m + m′m′′ and observe that α(−m′m′′) = 2α(m′) ≠ 0. By the argument above

μ(κ − m′m′′) = μ(κ) = μ(1 + m′ + m′′) = μ(1 + m),

as claimed.

Corollary 9.2. — Inertia elements α ∈ I_{ν}^{a} commute only with elements

coro:divv

μ ∈ D_{ν}^{a}.

Proposition 9.3. — Let K = k(X) be the function field of a surface.

prop:delta

Every σ ∈ Σ_{K} has rk_{ℤ_{ℓ}} σ = 2. Moreover, it defines a unique valuation ν = ν_{σ} of K so that either every element of σ is inertial for ν, or ν is divisorial and there is an element μ ∈ σ which is not inertial for ν, but μ ∈ D_{ν}^{a}.

If distinct σ, σ′ ∈ Σ_{K} have a nonzero intersection then there exists a divisorial valuation ν′′ such that

– σ, σ′ ∈ D_{ν′′}^{a};

– σ ∩ σ′ = I_{ν′′}^{a}.

Conversely, if σ ∈ Σ_{K} is not contained in a D_{ν′′}^{a} for any divisorial valuation ν′′ then for all σ′ ∈ Σ_{K}, σ′ ≠ σ, one has σ ∩ σ′ = 0.

Proof. — We saw that σ ∈ Σ_{K} contains an inertial element α for some valuation ν. Since σ is topologically noncyclic there is a μ ∈ σ, ℤ_{ℓ}-independent on α, and commuting with α. If μ is not inertial, that is, μ ∉ Φ(K), then ν gives a nontrivial element in the (abelianized) Galois group of the residue field K_{ν} of ν. Thus ν is divisorial, K_{ν} is 1-dimensional and every liftable subgroup in G_{K_{ν}}^{a} has ℤ_{ℓ}-rank equal to one. Hence rk_{ℤ_{ℓ}} σ = 2 in this case and, by Corollary 9.2, μ ∈ D_{ν}^{a}. Such a valuation ν is unique, since I_{ν}^{a} ∩ I_{ν′}^{a} = 0 for distinct divisorial ν, ν′.

If σ contains only inertia elements, then there exists a unique valuation ν such that σ ∈ I_{ν}^{a}. Indeed, either 𝔪_{ν} + 𝔪_{ν′} = K or we may assume that 𝔪_{ν} ⊂ 𝔪_{ν′} (and 𝔬_{ν} ⊃ 𝔬_{ν′}). The first case is impossible since the corresponding inertia groups don’t intersect. In the second case, I_{ν}^{a} ⊂ I_{ν′}^{a}, as claimed. Moreover, it follows that rk_{ℤ_{ℓ}} σ = 2, since the ℚ-rank of any valuation on a surface (over 𝔽̄_{q}) is at most two. This gives of ν = ν_{σ} in this case.

If distinct σ, σ′ have a nontrivial intersection, then the subgroup D ⊂ G_{K}^{a} generated by σ, σ′ is not the inertia group of any valuation (the rank of those is ≤ 2, as we have seen above). If the σ ∩σ′ contains a nontrivial inertia element α then D is contained in the decomposition group of this element (all elements of D commute with α) and the corresponding valuation is divisorial. If μ ∈ σ∩σ′ is not an inertia element then there exist inertia elements α ∈ σ and α′ ∈ σ′ corresponding to distinct divisorial valuations ν, ν′. The decomposition groups of distinct divisorial valuations don’t intersect.

Proposition 9.3 allows us to identify intrinsically (in terms of the Galois group) inertia subgroups of divisorial valuations as well as their decomposition groups as follows. Every pair of distinct groups σ, σ′ ∈ Σ_{K} with a nontrivial intersection defines a divisorial valuation ν, whose inertia group

I_{ν}^{a} = σ ∩ σ′.

The corresponding decomposition subgroup is

D_{ν}^{a} = ∪_{σ⊃I_{ν}^{a}}σ.

10. Galois groups of curves

sect:valc

Here we give a Galois-theoretic characterization of subgroups σ ∈ Σ_{K} which are inertia subgroups of rank two valuations of K arising from a flag (C, q), where C is a smooth irreducible curve (on some model of K) and q ∈ C(k) is a point (see Example 7.13). We show that Galois-theoretic data determine the genus of C and all “points” on C, as special liftable subgroups of rank two inside G_{k(C)}^{a}.

Throughout, E = k(C) is the function field of a smooth curve of genus g. We have an exact sequence

0 → E^{∗}/k^{∗} → Div(C) → Pic(C) → 0

(where Div(C) can be identified with the free abelian group generated by points in C(k)). This gives a dual sequence

eqn:crrr (10.1)  0 → ℤ_{ℓ} ⟶^{Δ} M(C(k), ℤ_{ℓ}) → G_{E}^{a} → ℤ_{ℓ}^{2g} → 0,

with the identifications – Hom(Pic(C), ℤ_{ℓ}) = Δ(ℤ_{ℓ}) (since Pic^{0}(C) is torsion);

– M(C(k), ℤ_{ℓ}) = Hom(Div(C), ℤ_{ℓ}) is the ℤ_{ℓ}-linear space of maps from C(k) → ℤ_{ℓ};

– ℤ_{ℓ}^{2g} = Ext^{1}(Pic^{0}(C), ℤ_{ℓ}). Using this model and the results in Section 6, we can interpret

eqn:cm (10.2)  G_{E}^{a} ⊂ M(C(k), ℚ_{ℓ})/constant maps

as the ℤ_{ℓ}-linear subspace of all maps μ : C(k) → ℚ_{ℓ} (modulo constant maps) such that

[μ, f] ∈ ℤ_{ℓ} for all f ∈ E^{∗}/k^{∗}.

Here [·, ·] is the pairing:

eqn:inter (10.3)  \begin{matrix}M(C(k), ℚ_{ℓ}) × E^{∗}/k^{∗} & → & ℚ_{ℓ} \\ (μ, f) & ↦ & [μ, f] := ∑_{q} μ(q)f_{q}, \end{matrix}

where div(f) = ∑_{q} f_{q}q. In detail, let γ ∈ G_{E}^{a} be an element of the Galois group. By Kummer theory, γ is a homomorphism K^{∗}/k^{∗} → ℤ_{ℓ}(1) ≃ ℤ_{ℓ}. Choose a point c_{0} ∈ C(k). For every point c ∈ C(k), there is an m_{c} ∈ ℕ such that the divisor m_{c}(c − c_{0}) is principal. Define a map

\begin{aligned}μ_{γ} : & C(k) & → & ℚ_{ℓ}, \\ & c & ↦ & γ(m_{c}(c − c_{0}))/m_{c}. \end{aligned}

Changing c_{0} we get maps differing by a constant map.

In this interpretation, an element of an inertia subgroup I_{w}^{a} ⊂ G_{E}^{a} corresponds to a “delta”-map (constant outside the point q_{w}). Each I_{w}^{a} has a canonical (topological) generator δ_{w}, given by δ_{w}(f) = ν_{w}(f), for all f ∈ E^{∗}/k^{∗}. The (diagonal) map Δ ∈ M(C(k), ℚ_{ℓ}) from (10.1) is then given by

Δ = ∑_{w∈V_{E}} δ_{w} = ∑_{q_{w}∈C(k)} δ_{q_{w}}.

Definition 10.1. — We say that the support of a subgroup I ⊂ G_{E}^{a} is ≤ s

defn:sup

and write

|supp(I)| ≤ s

if there exist valuations w_{1}, ..., w_{s} ∈ V_{E} such that

I ⊂ ⟨I_{w_{1}}^{a}, ..., I_{w_{s}}^{a}⟩_{ℤ_{ℓ}} ⊂ G_{E}^{a}.

Otherwise, we write |supp(I)| > s.

Lemma 10.2. — Let I ⊂ G_{E}^{a} be a topologically cyclic subgroup such that

lemm:cu

|supp(I)| > s ≥ 2. Then there exist a finite set {f_{j}}_{j∈J} ⊂ E^{∗} and an m ∈ ℕ such that the map

\begin{aligned}ψ : & G_{E}^{a} & → & V := ⊕_{j∈J}ℤ/ℓ^{m} \\ & μ & ↦ & ([μ, f_{j}]_{m})_{j∈J} \end{aligned}

has the following property: for every set {w_{1}, ..., w_{s}} ⊂ V_{E}

ψ(I) ⊄ ⟨ψ(I_{w_{1}}^{a}), ..., ψ(I_{w_{s}}^{a})⟩_{ℤ_{ℓ}}.

Proof. — Let ι ∈ G_{E}^{a} ⊂ M(C(k), ℚ_{ℓ}) be a representative, as in (10.2), of a topological generator of I, where supp(I) > s. There are three possibilities:

(1) ι(C(k)) ⊂ ℚ_{ℓ} is infinite;

(2) there is a b ∈ ι(C(k)) ⊂ ℚ_{ℓ} such that ι^{−1}(b) is infinite and there exist at least s + 1 distinct points q_{s+2}, …, q_{2s+2} ∈ C(k) such that ι(q_{j}) ≠ b for all j = s + 2, …, 2s + 2;

(3) otherwise: ι(C(k)) is finite, there is a b with ι^{−1}(b) infinite and there are at most s distinct points with values differing from b.

In Case (3), |supp(I)| ≤ s.

In Case (1), choose any set Q = {q_{1}, ..., q_{2s+2}} ⊂ C(k) of points with pairwise distinct values. In Case (2) choose distinct q_{1}, ..., q_{s+1} ∈ ι^{−1}(b) and put Q := {q_{1}, ..., q_{2s+2}}. In both cases, if Q′ ⊂ Q is any subset of cardinality |Q′| = s then ι is nonconstant on Q \ Q′. In particular, there exist points q_{s_{1}}, q_{s_{2}} ∈ Q \ Q′ such that

eqn:iota (10.4)  ι(q_{s_{1}}) ≠ ι(q_{s_{2}}).

We may assume that ι(Q) ⊂ ℤ_{ℓ} (replacing ι by a sufficiently high multiple, if necessary). Now we choose an m′′ ∈ ℕ such that all values of ι on Q remain pairwise distinct modulo ℤ/ℓ^{m′′}. Let Div_{Q}^{0}(C) be the abelian group of degree zero divisors on C supported in Q. By Lemma 3.2, there is an n = n_{Q} ∈ ℕ such that nD is principal for every D ∈ Div_{Q}^{0}(C). In particular, for every q_{s_{1}}, q_{s_{2}} ∈ Q there is a function f ∈ E^{∗} such that div(f) = n(q_{s_{1}} −q_{s_{2}}). Write n = ℓ^{m′} n̄, with gcd(n̄, ℓ) = 1, and put m = m′ + m′′.

We have a pairing (Kummer theory)

\begin{matrix}G_{E}^{a} × nDiv_{Q}^{0}(C) & → & ℤ/ℓ^{m} \\ (μ, f) & ↦ & [μ, f]_{m}. \end{matrix}

Notice that [I_{w}^{a}, f] = 0 for all w with q_{w} ∉ Q and all f ∈ E^{∗} supported in Q. Further, for every Q′ ⊂ Q with |Q′| = s and points q_{s_{1}}, q_{s_{2}} ∈ Q \ Q′ as in (10.4) there is an f ∈ E^{∗} with divisor div(f) = n(q_{s_{1}} − q_{s_{2}}) such that

[ι, f] = n · (ι(q_{s_{1}}) − ι(q_{s_{2}})) ≠ 0 mod ℓ^{m}

and

[I_{w′}^{a}, f] = 0

for all I_{w′}^{a} of q′ ∈ Q′. Let {f_{j}}_{j∈J} be a basis for ℓ^{m} · Div_{Q}^{0}(C), with f_{j} ∈ E^{∗}. The map

\begin{matrix}ψ : G_{E}^{a} & → & ⊕_{j∈J}ℤ/ℓ^{m} \\ μ & ↦ & ([μ, f_{j}]_{m})_{j∈J} \end{matrix}

satisfies the required properties.

The next step is an intrinsic definition of inertia subgroups

I_{w}^{a} ⊂ D_{ν}^{a}/I_{ν}^{a} = G_{k(C)}^{a}.

We have a projection

π_{ν} : G_{K}^{a} → G_{K}^{a}/I_{ν}^{a}

and an inclusion

G_{K_{ν}}^{a} = D_{ν}^{a}/I_{ν}^{a} ↪ G_{K}^{a}/I_{ν}^{a}

Proposition 10.3. — Let ν be a divisorial valuation of K. A subgroup

prop:intri

I ⊂ D_{ν}^{a}/I_{ν}^{a}

is the inertia subgroup of a divisorial valuation of k(C) = K_{ν} iff for every homomorphism

ψ : G_{K}^{a}/I_{ν}^{a} → V

onto a finite abelian group V there exists a divisorial valuation ν_{ψ} such that

ψ(I) = ψ ∘ π_{ν}(I_{ν_{ψ}}^{a}).

Proof. — Let C be the smooth model for K_{ν} = k(C),

I = I_{w}^{a} ⊂ D_{ν}^{a}/I_{ν}^{a}

the inertia subgroup of a divisorial valuation of k(C) corresponding to a point q = q_{w} ∈ C(k) and

ψ : G_{K}^{a}/I_{ν}^{a} → V

a homomorphism onto a finite abelian group. Since G_{K}^{a} is a pro-ℓ-group, we may assume that

V = ⊕_{j∈J}ℤ/ℓ^{n_{j}},

for some n_{j} ∈ ℕ. Let n = max_{j}(n_{j}). By Kummer theory,

Hom(G_{K}^{a}, ℤ/ℓ^{n}) = K^{∗}/(K^{∗})^{ℓ^{n}}

so that ψ determines elements

f̄_{j} ∈ K^{∗}/(K^{∗})^{ℓ^{n}}

(for all j ∈ J). Choose functions f_{j} projecting to f̄_{j}. They define a finite set of divisors D_{ij} on X. Moreover, f_{j} are not simultaneously constant on C (otherwise, ψ(G_{k(C)}^{a}) = ψ(I_{k(C)}^{a})). Changing the model X̃ → X, if necessary, we may assume that

– C is smooth (and irreducible); – there exists exactly one irreducible component D in the full preimage of

∪D_{ij} which intersects C in q. Moreover, this intersection is transversal (see Section 3). Then the image of I_{D}^{a} under ψ is equal to the image of I_{w}^{a}.

Conversely, we need to show that if I ≠ I_{w}^{a} (for some w ∈ DV_{K_{ν}}), then there exists a homomorphism

ψ : G_{K}^{a}/I_{ν}^{a} → V

onto a finite abelian group V such that for all ν′ ∈ DV_{K} one has

ψ(I) ≠ ψ ∘ π_{ν}(I_{ν′}^{a}).

We consider two cases

(1) there exist two points q, q′ ∈ C(k) such that I ⊂ ⟨I_{w}^{a}, I_{w′}^{a}⟩;

(2) otherwise.

Case 1. There exists a rational map π : X → ℙ^{1} such that its restriction

π : C → ℙ^{1}

is surjective, unramified at q, q′ and π(q) ≠ π(q′). Under the induced map of Galois groups

π_{∗}(I) ⊂ ⟨I_{π(w)}^{a}, I_{π(w′)}^{a}⟩

but is not contained in either I_{π(q)}^{a} or I_{π(q′)}^{a}. Thus there exist a finite abelian group V and a map ψ : G_{k(ℙ^{1})}^{a} → V such that ψ(I) ∉ ψ(I_{w′′}^{a}) for any q′′ ∈ ℙ^{1}. It follows that

ψ ∘ π_{∗}(I) ∉ ψ ∘ π_{∗}(I_{ν}^{a})

for any ν ∈ DV_{K}.

Case 2. By Lemma 10.2, there exist a finite set of functions f̄_{j} ∈ k(C), with support in a finite set Q = {q_{0}, ..., q_{s}} ⊂ C(k), and an m ∈ ℕ such that the homomorphism

\begin{matrix}ψ̄ : G_{k(C)}^{a} & → & V = ⊕_{j∈J}ℤ/ℓ^{m} \\ μ & ↦ & ([μ, f̄_{j}]_{m})_{j∈J} \end{matrix}

has the property that for all w, w′ ∈ DV_{k(C)}

ψ(I) ∉ ⟨ψ(I_{w}^{a}), ψ(I_{w′}^{a})⟩_{ℤ_{ℓ}}.

Next we choose a model for X and C as in Lemma 3.8. In particular, there exist functions g_{j} with divisor

div(g_{j}) = n · (D_{j} − D_{0}) + (H_{j} − H′_{j})

such that all the divisors are irreducible, with transversal intersections and div(g_{j})|C = n(q_{j} − q_{0}). These functions g_{j} define a homomorphism

ψ : G_{K}^{a}/I_{ν}^{a} → V.

If D is a divisor on X then ψ ∘ π_{ν}(I_{D}^{a}) = 0 unless D = D_{j} for some j. In this case ψ ∘ π_{ν}(I_{D_{j}}^{a}) = ψ(I_{w_{j}}^{a}).

Let ν′ ∈ DV_{K} and 𝔠(ν′) ⊂ X be its center on X. There are three cases:

– 𝔠(ν′) ⊄ D_{j} for any j: then ψ ∘ π_{ν}(I_{ν′}^{a}) = 0;

– 𝔠(ν′) ∈ D_{j}^{0}, where D_{j}^{0} = D_{j} \ (∪_{j′≠j}D_{j} ∩ D_{j′}): then

ψ ∘ π_{ν}(I_{ν′}^{a}) ⊂ ψ(I_{w_{j}}^{a});

– 𝔠(ν′) ∈ D_{j} ∩ D_{j′} for some j, j′: then

ψ ∘ π_{ν}(I_{ν′}^{a}) ⊂ ⟨ψ(I_{w_{j}}^{a}), ψ(I_{w_{j′}}^{a})⟩_{ℤ_{ℓ}}.

All three possibilities contradict our assumptions.

Lemma 10.4. — Let E = k(C) be the function field of a curve. Then

lemm:gen-type

g(C) ≥ 1 iff there exists a non-zero homomorphism from G_{E}^{a} to a finite (abelian) group which maps all inertia elements to 0.

Proof. — Indeed, every curve of genus ≥ 1 over a finite field of characteristic p has unramified coverings of degree ℓ. These coverings define maps of Galois groups, which are trivial on all inertia elements. If C is rational then G_{E}^{a}, and hence its image under every homomorphism (onto any finite group), is generated by inertia elements (see the exact sequence (10.1)).

Remark 10.5. — Combining this with Proposition 10.3 we can decide in

rem:gen-tt

purely Galois-theoretic terms which divisorial valuations of K correspond to nonrational (irreducible) curves C on some model X of K. We call such valuations nonrational.

11. Valuations on surfaces

sect:recc

The next stage of the recognition process leads us to the following problem: How to characterize subgroups { \atop k(C)}^{∗} ⊂ K̂^{∗}? In this section we recall a geometric argument (used in algebraic K-theory) characterizing pairs of functions f, g ∈ K^{∗} which are contained in k(C)^{∗} ⊂ K^{∗}, for some curve C (such curves correspond to projections X → C).

Let K = k(X) be the function field of a smooth surface X over k and ν a divisorial valuation of K. We have a well-defined (bilinear, with respect to multiplication) residue map

eqn:ress (11.1)  \begin{matrix}K^{∗} × K^{∗} & → & K_{ν}^{∗}/k^{∗} \\ f, g & ↦ & f^{ν(g)}/g^{ν(f)}. \end{matrix}

On a smooth model X of K, where ν = ν_{D} for some divisor D ⊂ X, we can define

eqn:res-1 (11.2)  ϱ_{ν} = ϱ_{D} : K^{∗} × K^{∗} → K_{ν}^{∗}/k^{∗}

as follows:

– ϱ_{ν}(f, g) = 1 if both f, g are invertible on D;

– ϱ_{ν}(f, g) = f_{D}^{m} if f is invertible (f_{D} is the restriction to D) and g has multiplicity m along D;

– ϱ_{ν}(f, g) = (f^{m_{g}}/g^{m_{f}})_{D} in the general case, when f, g have multiplici-

ties m_{f}, m_{g}, respectively. The definition does not depend on the choice of the model.

The following is a standard result in K-theory. We include a proof since we will need its ℓ-adic version.

Lemma 11.1. — For f, g ∈ K^{∗}

lemm:fg

ϱ_{ν}(f, g) = 1 ∀ν ∈ DV_{K} ⟺ f, g ∈ E = k(C) ⊂ K for some curveC.

Proof. — (⇐) On an appropriate model X we have ν = ν_{D} for a divisor D ⊂ X and π : X → C is regular and flat with irreducible generic fiber (and f, g ∈ k(C)^{∗}). By definition, ϱ_{ν}(f, g) = 1 if D is not in a fiber of π. If D is in a fiber then there is a t ∈ k(C)^{∗}, ν_{D}(t) ≠ 0 such that both ft^{m_{f}} , gt^{m_{g}} are regular and constant on D (for some m_{f}, m_{g} ∈ ℕ) so that ϱ_{ν}(f, g) = 1.

(⇒) Assume that ϱ_{ν}(f, g) = 1 for every ν ∈ DV_{K}. Every nonconstant function f defines a unique map (with irreducible generic fiber)

π_{f} : X → C_{f}

which corresponds to the algebraic closure of k(f) in K (we will say that f is induced from C_{f}). We claim that π_{f} = π_{g}.

Since f is induced from C_{f}, we have

div(f) = ∑_{q∈Q} a_{q}D_{q},

where Q ⊂ C_{f}(k) is finite and D_{q} = π^{−1}(q). Then D_{q}^{2} = 0 and D_{q} is either a multiple of a fiber of π_{g} or it has an irreducible component D ⊂ D_{q} which dominates C_{g} (under π_{g}). In the second case, the restriction of g to D_{q} is a nonconstant element in k(D_{q}). Then ν_{D}(f) ≠ 0, while ν_{D}(g) = 0. Hence ϱ_{D}(f, g) ≠ 0 since it coincides with g_{D}^{−ν_{D}(f)} ≠ 1, a contradiction. Therefore, all D_{q} are contained in the finitely many fibers S of π_{g}. That means div(f) does not intersect the fibers R_{t}, t ∈ C_{g}, t ∉ S which implies that f is constant on such R_{t}. Hence f belongs to the normal closure of k(C_{g}) in K, and in fact f ∈ k(C_{g}) since k(C_{g}) is algebraically closed in K, by construction. Thus f is induced from C_{g} and hence C_{f} = C_{g} and π_{f} = π_{g}.

12. ℓ-adic analysis: generalities

sect:ella

Hypothetically, surjective homomorphisms G_{K}^{a} → G_{k(C)}^{a} (or dually, inclu-

sions { \atop k(C)^{∗}} ⊂ K̂^{∗}) are characterized as follows: assume, we have a commutative diagram

\xymatrix{G_{K}^{c} \ar[d] \ar[r] & A^{c} \ar[d] \\ G_{K}^{a} \ar[r]_{ψ} & A}

where the abelian group A is a rank two torsion-free ℤ_{ℓ}-module and A^{c} is its free central extension. Then there exists a unique field k(C) ⊂ K and a factorization of ψ:

G_{K}^{a} → G_{k(C)}^{a} → A.

Here we solve a dual problem. We distinguish, Galois-theoretically, a certain subgroup inside K̂^{∗} which contains K^{∗}/k^{∗}. The main result is the Galoistheoretic determination of pairs f̂, ĝ of elements of this subgroup which are contained in the completion Ê^{∗} of the same one-dimensional field.

In detail, to every f ∈ K^{∗} one associates its divisor D = D_{f} on X. Conversely, D (uniquely) determines the image of f ∈ K^{∗}/k^{∗}. Recall that the Galois group G_{K}^{a} determines K̂^{∗}, a group substantially bigger than K^{∗}/k^{∗}. The goal is to detect the ℓ-adic subspace K^{∗}/k^{∗} ⊗ ℤ_{ℓ} ⊂ K̂^{∗}.

We start with the theory of divisors with ℤ_{ℓ}-adic coefficients associated to elements in K̂^{∗}. Such an element is, in general, represented by a divisor with infinite support on X, with rapidly decreasing coefficients (in the ℓ-adic topology on Z_{ℓ}). The Galois datum (G_{K}^{a}, Σ_{K}) allows us to distinguish between rational and nonrational irreducible divisors (via the corresponding valuations) and to characterize intrinsically a subspace FS(K) ⊂ K̂^{∗} (of divisors with finite nonrational support, see 12.2 and 12.3), containing K^{∗}/k^{∗} ⊗ ℤ_{ℓ}.

In order to further shrink FS(K) using Galois data we use the fact that for any nontrivial f ∈ K^{∗} there are many other g ∈ K̂^{∗} with ρ̂(f, g) = 0 (where ρ̂(f, g) = 0 is the ℓ-adic generalization of ρ(f, g) = 0). Those are arbitrary elements g ∈ Ê^{∗}, E = \overline{k(f)}^{K}. However, for a sufficiently generic element f̂ ∈ K̂^{∗} the element g with ρ(f, g) = 0 is equal to f^{a}, a ∈ ℤ_{ℓ}.

Thus the property that for f ∈ K̂^{∗} the set of g ∈ K̂^{∗} with ρ̂(f, g) = 0 contains many elements different from f^{a}, a ∈ ℤ_{ℓ} can be used to select a smaller subgroup FS_{X}(K) ⊂ FS(K), containing K^{∗}/k^{∗}. Elements in FS_{X}(K) have finite support on every model X. We show in Section 14 that ρ(f, g) = 0, f, g ∈ FS_{X}(K) implies that f, g ⊂ Ê^{∗}, E = k̄(x)^{K} for some x ∈ K^{∗}.

We have an exact sequence

eqn:seqq (12.1)  0 → K^{∗}/k^{∗} ⟶^{ρ_{X}} Div(X) ⟶^{φ} Pic(X) → 0,

where Div(X) is the group of (Weil or Cartier) divisors of X. We identify an f ∈ K^{∗}/k^{∗} with its image under ρ_{X}. Let

D̂iv(X) := {D = ∑_{m∈M} â_{m}D_{m}}, resp. D̂iv_{nr}(X) ⊂ D̂iv(X),

be the group of divisors (resp. nonrational divisors) with rapidly decreasing coefficients:

– M is a countable set;

– for all r ∈ ℤ the set

{m | |â_{m}|_{ℓ} ≤ r}

is finite;

– for D ∈ D̂iv_{nr}(X), all D_{m} are nonrational. Clearly, the group of finite ℓ-adic divisors

Div(X)_{ℓ} := Div(X) ⊗_{ℤ} ℤ_{ℓ} ⊂ D̂iv(X).

Every element

f̂ ∈ K̂^{∗} = lim_{n→∞} K^{∗}/(K^{∗})^{ℓ^{n}}

has a representation

f̂ = (f_{n})_{n∈ℕ} or f = f_{0}f_{1}^{ℓ}f_{2}^{ℓ^{2}} ⋯,

with f_{n} ∈ K^{∗}. We have homomorphisms

ρ̂_{X} : K̂^{∗} → D̂iv(X),
f̂ ↦ div(f̂) := ∑_{n∈ℕ} ℓ^{n} · div(f_{n}) = ∑_{m} â_{m}D_{m},
ρ̂_{X,nr} : K̂^{∗} → D̂iv(X) ⟶^{pr} D̂iv_{nr}(X),

where D_{m} ⊂ X are irreducible divisors,

â_{m} = ∑_{n∈ℕ}a_{nm}ℓ^{n} ∈ ℤ_{ℓ},

with a_{nm} ∈ ℤ, and

div(f_{n}) = ∑_{m} a_{nm}D_{m}.

Here div(f_{n}) is the Cartier divisor of f_{n} and ∑_{m} a_{nm}D_{m} is its image in the group of Weil divisors.

Note that the map ρ̂_{X} has a kernel

T_{ℓ}(X) = lim_{⟵} Tor^{1}(ℤ/ℓ, Pic^{0}(X)[ℓ]),

where Pic^{0}(X)[ℓ] ⊂ Pic^{0}(X) is the ℓ-power torsion subgroup. The group T_{ℓ}(X) is isomorpic to ℤ_{ℓ}^{g}.

If T_{ℓ}(X) ≠ 0 then either X contains only finitely many rational curves, or X, modulo purely inseparable covers, is a rational pencil over a curve C of genus g(C) ≥ 1.

Every ν ∈ DV_{K} gives rise to a homomorphism

ν : K̂^{∗} → ℤ_{ℓ}

and a residue map

ϱ̂_{ν} : K̂^{∗} × K̂^{∗} → K̂_{ν}.

On a smooth model X, where ν = ν_{D} for some divisor D ⊂ X, ν(f̂) is the ℓ-adic coefficient at D of div(f̂), while ϱ̂_{ν} is the natural generalization of (11.1).

Definition 12.1. — We say that distinct f̂, ĝ ∈ K̂^{∗} commute if ϱ̂_{ν}(f̂, ĝ) =

defi:commu

0, for all divisorial ν. We say that they have disjoint support if for all divisorial valuations ν ∈ DV_{K}

ν(f̂) · ν(ĝ) = 0.

We say that f̂ ∈ K̂^{∗} has nontrivial commutators if there exist ĝ ∈ K̂^{∗} with disjoint support (from f̂) which commute with f̂.

Notations 12.2. — We put

nota:supp

supp_{K}(f̂) := { ν ∈ DV_{K} | f̂ nontrivial on I_{ν}^{a} };
supp_{X}(f̂) := {  D_{m}  | â_{m} ≠ 0 }.

Definition 12.3. — We say that f̂ has finite nonrational support if the set

rem:alt

of nonrational ν ∈ supp_{K}(f̂) is finite (see Lemma 10.4 for the definition and Galois-theoretic characterization of nonrational valuations). Let

FS(K) ⊂ K̂^{∗}

be the set of such elements.

Definition 12.4. — We say that f̂ has finite support on the model X if

defn:fs

supp_{X}(f̂) is finite. Put

FS_{X}(K) := {f̂ ∈ K̂^{∗} | ρ_{X}(f̂) ∈ Div(X)_{ℓ}}.

Lemma 12.5. — The definition of FS_{X}(K) does not depend on the choice

lemm:indep

of a smooth model X.

Proof. — For any two smooth models X′, X′′ we can find a smooth model X dominating both. The difference between the sets of irreducible divisors Div(X′), resp. Div(X′′), and Div(X) is finite and consists only of rational curves.

Corollary 12.6. — Let K be the function field of a surface X containing

coro:almost

only finitely many rational curves. Then

FS(K) = FS_{X}(K).

This gives an intrinsic, Galois-theoretic description of FS_{X}(K) in this case. We proceed to give such a description in general. Note that for f̂ ∈ FS(K), its nonrational component ρ̂_{X,nr}(f̂) is independent of the model X. More precisely, for any birational morphism X′ → X we can identify D̂iv_{nr}(X′) = D̂iv_{nr}(X). Under this identification

ρ_{X′,nr}(f̂) = ρ_{X,nr}(f̂).

Let F(K) be the set of all f ∈ K^{∗}/k^{∗} such that ρ_{X,nr}(f) ≠ 0 and for every rational divisorial valuation ν such that ν(f) = 0 and some (equivalently, every) model X of K, where ν = ν_{C} for a rational curve C ⊂ X, either

– f_{C} = 1 ∈ k(C)^{∗}/k^{∗} or

– ρ_{C}(f_{C}) ≠ 0 mod ℓ. Geometrically, this condition means that if C is not a component of the divisor of f then there is a point in C ∩ div(f) whose multiplicity is prime to ℓ.

Lemma 12.7. — The set F(K) generates K^{∗}/k^{∗}. Moreover, for every pair

lemm:imp

of commuting elements f̂, ĝ ∈ FS(K) with disjoint support such that there exists an f ∈ F(K) with

f = f̂ mod (K^{∗})^{ℓ},

one has f̂ ∈ FS_{X}(K) and ĝ ∈ FS_{X}(K), for every model X of K.

Proof. — Let y ∈ K^{∗} be a function such that the generic fiber of the corresponding map π_{y} : X → ℙ_{y}^{1}, from some model X of K, is an irreducible nonrational curve. (Notice that such y generate K^{∗}.) Using such y we construct F(K) as follows.

For generic quadratic, coprime polynomials P, Q ∈ k[y], the preimage in X of (0 ∪ ∞) ⊂ ℙ^{1} under the composition of π_{y} with the map

\begin{aligned}ϕ : & ℙ_{y}^{1} & → & ℙ^{1} \\ & y & ↦ & f(y) := P (y)/Q(y) \end{aligned}

contains at least 4 irreducible smooth fibers of π_{y}. If f were nonconstant on a rational curve C (on some model X of K) and f_{C} were an ℓ-th power then the local ramification indices of f and hence of y were divisible by ℓ. Thus we would have a map π_{y} : C → ℙ_{y}^{1} with all local ramification indices over 4 points divisible by ℓ, and by Hurwitz’ theorem, g(C) > 0, which contradicts the rationality of C. It follows that f ∈ F(K). Clearly, the elements from F(K) generate k(y)^{∗}.

Next, write

\begin{aligned}ρ_{X}(f̂) & = & ∑_{i∈I} n_{i}D_{i} + ℓ ∑_{j=1}^{∞} n_{j}C_{j}, \\ ρ_{X}(ĝ) & = & ∑_{i∈I′} n′_{i}D′_{i} + ℓ ∑_{j=1}^{∞} n′_{j}C′_{j}, \end{aligned}

where I, I′ are finite sets and the second sum is an infinite series over distinct rational curves C_{j}, C′_{j} ⊂ X. By assumption, the sets {D_{i}}_{i∈I}, {C_{j}}_{j∈ℕ}, {D′_{i}}_{i∈I′}, {C′_{j}}_{j∈ℕ} are disjoint.

By assumption, ρ_{ν′_{j}}(f̂, ĝ) = 0, for all ν′_{j} corresponding to C′_{j}. Since C′_{j} are rational, this residue equals the residue of f on C′_{j}, which is nonzero mod ℓ, contradiction. Thus, if (f̂, ĝ) = 0, then supp_{X}(ĝ) is finite. The restriction of g′ to any irreducible component of the divisor of f̂ is identically zero. This implies that g′ is a product of ℓ-adic powers of elements belonging to the same field k(y) as f. Thus all rational curves in the support of f̂ also belong to the fibers of y. There are finitely many such curves since some fibers contain nonrational curves.

We have an exact sequence

0 → K̂^{∗} ⟶^{ρ̂_{X}} D̂iv(X) ⟶^{φ_{ℓ}} Pic(X)_{ℓ} → 0,

where we denote by Pic(X)_{ℓ} the quotient group D̂iv(X)/K̂^{∗}. Write

D̂iv(X)^{0} ⊂ D̂iv(X)

for the group generated by ρ̂_{X}(K̂^{∗}) and identify an f̂ ∈ K̂^{∗} with its image.

Lemma 12.8. — Let X/k be a smooth projective surface, M a finite set and

lemm:DD

D = ∑_{m∈M} a_{m}D_{m} ∈ Div(X)_{ℓ} := Div(X) ⊗_{ℤ} ℤ_{ℓ}, a_{m} ∈ ℤ_{ℓ}

a divisor such that φ_{ℓ}(D) = 0. Then there exist a finite set I, functions f_{i} ∈ K^{∗} and numbers a_{i} ∈ ℤ_{ℓ}, linearly independent over ℤ, such that for all i ∈ I

supp_{X}(f_{i}) ⊂ supp_{X}(D)

and

D = ∑ a_{i}b_{i}div(f_{i}), b_{i} ∈ ℚ.

If NS(X) = Pic(X) then we can take all b_{i} = 1.

Proof. — We have a diagram

Ker(φ) → ⊕_{m∈M}ℤD_{m} ⟶^{φ} Λ ⊂ Pic(X) → 0
↓ ↓ ↓
Ker(φ_{ℓ}) → ⊕_{m∈M}ℤ_{ℓ}D_{m} ⟶^{φ_{ℓ}} Λ_{ℓ} ⊂ Pic(X)_{ℓ} → 0.

The group Λ is finitely generated and its image in NS(X) = Pic(X)/Pic^{0}(X) has the same rank as Λ (since Pic^{0}(X) is a torsion group):

rk_{ℤ}Λ = rk_{ℤ_{ℓ}}Λ_{ℓ} and rk_{ℤ}Ker(φ) = rk_{ℤ_{ℓ}}Ker(φ_{ℓ}).

In particular, Ker(φ_{ℓ}) has a basis {D_{i}}_{i∈I} (over ℤ_{ℓ}), where each D_{i} is a ℤintegral linear combinations of D_{m} (with m ∈ M) and is also in Ker(φ). It follows that D_{i} = div(f_{i}) for some function f_{i} ∈ K^{∗} with support in D. Finally, if φ_{ℓ}(D) = 0, we can find a representation

D = ∑_{i} a_{i}div(f_{i}^{b_{i}}),

with b_{i} ∈= ℚ and a_{i} ∈ ℤ_{ℓ}, linearly independent over ℤ (passing to a subset of I, if necessary).

Corollary 12.9. — There is an exact sequence

1 → K^{∗}/k^{∗} ⊗ ℤ_{ℓ} → FS_{X}(K) → Pic^{0}(X)[ℓ] → 1

where Pic^{0}(X)[ℓ] ⊂ Pic^{0}(X) is the ℓ-power torsion subgroup.

Proof. — If suffices to recall that elements in Pic^{0}(X)[ℓ], are represented, in our description, by elements f^{1/ℓ^{n}}, for some n ∈ ℕ, which define unramified ℓ-power cyclic covering of a model X.

Lemma 12.7 and Lemma 12.8 allow us to define FS_{X}(K) intrinsically. Namely, for every

f̄ ∈ FS(K)/ℓ = K^{∗}/ℓ = K̂^{∗}/ℓ

denote by Ff̄ ⊂ FS(K) the group ℤ_{ℓ}-generated by (f̂/f̂′)^{1/ℓ}, where f̂and f̂′ are elements which have nontrivial commutators and which both reduce to f̄ modulo ℓ. Then define

FS_{0}(K) = ∩_{f̄∈K^{∗}/ℓ}Ff̄.

Note that for all f̄ ∈ F(K)/ℓ and every model X of K one has

Ff̄ ⊂ FS_{X}(K).

Lemma 12.8 implies that, conversely

FS_{X}(K) ⊂ Ff̄,

for every f̄. In particular, for every f ∈ F(K), with f̄ = f mod ℓ, both sets coincide. Therefore,

eqn:f0 (12.2)  FS_{0}(K) = FS_{X}(K),

for all models X.

Moreover, notice that elements in K^{∗}/k^{∗} ⊗ ℤ_{ℓ} ⊂ FS_{X}(K) are Galoistheoretically characterized as elements whose projection to Pic^{0}(X) is trivial. As a group, K^{∗}/k^{∗} ⊗ ℤ_{ℓ} is generated by elements whose ν-values (for ν ∈ DV_{K}) are not all divisible by ℓ.

13. ℓ-adic analysis: curves

sect:curves-ell

In this section we begin the process recognition of the lattice K^{∗}/k^{∗} ⊂ K̂^{∗}. We solve an analogous problem for the function field of a rational curve. This result will play an essential role in the analysis of surfaces.

Proposition 13.1. — Let k̃ be the closure of a finite field, char(k̃) ≠ ℓ, C

prop:cc

a curve over k̃ of genus g with function field E = k̃(C) and

Ψ : G_{k(ℙ^{1})}^{a} → G_{E}^{a}

an isomorphism of Galois groups inducing an isomorphism on inertia groups of divisorial valuations, that is, a bijection on the set of such groups and isomorphisms of corresponding groups. Let

Ψ^{∗} : { \atop k(ℙ^{1})^{∗}} → Ê^{∗}

be the dual isomorphism. Then E = k̃(ℙ^{1}) and there is a constant a ∈ ℤ_{ℓ}^{∗} such that Ψ^{∗}(k(ℙ^{1})^{∗}/k^{∗}) = a · E^{∗}/k̃^{∗}.

Proof. — Recalling the exact sequence (10.1), we have a commuting diagram

\xymatrix{0 \ar[r] & ℤ_{ℓ}(Δ_{C(k̃)}) \ar[r] & M(C(k̃)) \ar[r] & G_{E}^{a} \ar[r] \ar[d] & ℤ_{ℓ}^{2g} \ar[r] & 0 \\ 0 \ar[r] & ℤ_{ℓ}(Δ_{ℙ^{1}(k)}) \ar[r] & M(ℙ^{1}(k)) \ar[r] & G_{k(ℙ^{1})}^{a} \ar[r] & 0}

Since Ψ is an isomorphism on inertia groups I_{w}^{a}, for each w, the sets C(k̃) and ℙ^{1}(k) coincide and we get a unique isomorphism of ℤ_{ℓ}-modules (of maps to ℤ_{ℓ})

M(C(k̃)) = M(ℙ^{1}(k̃)).

In particular, we find that g = 0 and E = k̃(ℙ^{1}). Further, we have an induced isomorphism

ℤ_{ℓ}( ∑_{w∈V_{E}} δ_{w}) = ℤ_{ℓ}( ∑_{w′∈V_{k(ℙ^{1})}} δ_{w′})

so that

( ∑_{w∈V_{E}} δ_{w}) = a( ∑_{w′∈V_{k(ℙ^{1})}} δ_{w′})

for some a ∈ ℤ_{ℓ}^{∗}. This implies that δ_{w} = aδ_{w′}, for all w ∈ V_{E} and the corresponding w′ ∈ V_{ℙ^{1}}. In particular, for the dual groups we have

E^{∗}/k̃^{∗} = (K^{∗}/k^{∗})^{a},

where a ∈ ℤ_{ℓ}^{∗}.

14. ℓ-adic analysis: surfaces

sect:surr

Let K = k(X) be a function field of a smooth surface X over k. We will need an ℓ-adic version of Lemma 11.1 .

Proposition 14.1. — Let f̂, ĝ ∈ FS_{0}(K) be such that

prop:gff

– ϱ_{ν}(f̂, ĝ) = 0 for every ν ∈ DV_{K};

– supp_{K}(f̂) ∩ supp_{K}(ĝ) = ∅. Then there is a 1-dimensional field E = k(C) ⊂ K such that f̂, ĝ ∈ Ê^{∗}.

Proof. — By Lemma 12.8,

f̂ = ∏_{i∈I} f_{i}^{a_{i}}, resp. ĝ = ∏_{j∈J} g_{j}^{b_{j}},

where – I, J are finite sets;

– f_{i}, g_{j} ∈ K^{∗} for all i, j;

– a_{i} ∈ ℚ_{ℓ} (resp. b_{j} ∈ ℚ_{ℓ}) are linearly independent over ℚ. Fix a valuation ν and choose a (smooth) model X so that ν = ν_{D} for some divisor D ⊂ X. Then

ϱ_{ν}(f̂, ĝ) = ∏ ϱ_{D}(f_{i}, g_{j})

and we can compute it using only those pairs f_{i}, g_{j} which have D in their support. In particular,

f̂^{m_{g}}/ĝ^{m_{f}} = ∏(f_{i}^{a_{i}m_{j}}/g_{j}^{b_{j}m_{i}}),

where m_{j} (resp. m_{i}) is the order of g_{j} (resp. f_{i}) on D. This order vanishes unless D ∈ supp(f̂) ∪ supp(ĝ). By assumption, if D ∈ supp(f̂) then D ∉ supp(ĝ) (and n_{j} = 0) so that

ϱ_{D}(f̂, ĝ) ∈ k̂(D)^{∗}.

Since the nonzero numbers a_{i} are linearly independent over ℚ the equality ∑ a_{i}m_{i} = 0 implies that m_{i} = 0 (for all i) and that g_{D} ∈ k^{∗}.

Similarly, g_{D} = ∏(g_{j})_{D}^{b_{j}}, where b_{j} are linearly independent over ℤ, and g_{D} ∈ k^{∗} implies that (g_{j})_{D} ∈ k^{∗} (for all j ∈ J). It follows that

ϱ_{ν}(f_{i}, g_{j}) = 0

for all f_{i}, g_{j} and every valuation ν = ν_{D}. By Lemma 11.1, all f_{i}, g_{j} belong to the same 1-dimensional field E ⊂ K and hence f̂, ĝ ∈ Ê^{∗}.

Remark 14.2. — For every f ∈ K^{∗} the element g = (f + a)(f + b) where

rem:sati

a ≠ b and ab ≠ 0, satisfies the conditions of Proposition 14.1.

Proposition 14.3. — Let 𝔎^{∗} ⊂ FS_{0}(K) ⊂ K̂^{∗} be a subset with the fol-

prop:surfa

lowing properties:

– 𝔎^{∗} is closed under multiplication;

– 𝔎^{∗} ∩ Ê^{∗} = a_{E} · E^{∗}/k^{∗} for every 1-dimensional subfield E = k(x) ⊂ K, with a_{E} ∈ ℤ_{ℓ}^{∗};

– there exists a ν_{0} ∈ DV_{K} such that

{[δ_{0}, f̂] | f̂ ∈ 𝔎^{∗}} ≃ ℤ

for a topological generator δ_{0} of I_{ν_{0}}^{a}. Then 𝔎^{∗} ⊂ K^{∗}/k^{∗} ⊗ ℤ_{(ℓ)}.

Proof. — For x ∈ K \ k let E = k(x) be the corresponding 1-dimensional field. By assumption, there exists an a_{E} ∈ ℤ_{ℓ} such that

𝔎^{∗} ∩ Ê^{∗} = a_{E} · E^{∗}/k^{∗}.

If some (any) topological generator δ_{0} of I_{ν_{0}}^{a} is not identically zero on Ê^{∗} then there exists a (smooth) model X, where ν_{0} is realized by a divisor D_{0}, together with a morphism

X → ℙ^{1} = ℙ_{E}^{1}

such that D_{0} dominates ℙ^{1}. It follows that

a_{E} ∈ ℚ ∩ ℤ_{ℓ}^{∗} = ℤ_{(ℓ)}.

It remains to observe that every x ∈ K^{∗} can be written as a product

x = x′ · x′′

such that δ_{0} is nontrivial on both E′ = k(x′) and E′′ = k(x′′).

Corollary 14.4. — After a choice of δ_{0}, for every 1-dimensional E ⊂ K

coro:pm

and every f ∈ E^{∗}/k^{∗} we can Galois-theoretically distinguish its poles from its zeroes.

The last essential step is a Galois-theoretic characterization of the partial projective structure on 𝔎^{∗}/k^{∗}, more precisely, the characterization of generating elements and primary lines in 𝔎^{∗}/k^{∗} (see Definition 3.10 and Definition 4.7).

Lemma 14.5. — Let x ∈ K^{∗} be a generating element, E := k(x) and

lemm:unnn

r = r(x) ∈ ℕ the smallest positive integer such that x^{r} ∈ 𝔎^{∗}. Then

– r = p^{m} for some m ∈ ℕ (with p = char(k));

– (E^{∗}/k^{∗}) ∩ (𝔎^{∗}/k^{∗}) = (E^{p^{m}})^{∗}/k^{∗};

– (pointwise) p^{m}-th powers of primary lines in E^{∗}/k^{∗} coincide with primary lines in (E^{p^{m}})^{∗}/k^{∗}.

Proof. — The first property follows since K/𝔎 is a finite purely inseparable extension, by Propositions 3.19 and 14.3. Next, we claim that a generator y ∈ 𝔎 is a p^{m}-th power of a generator of K (for some m depending on y). Indeed, E := \overline{k(y)}^{K} ⊂ K is a finite and purely inseparable extension of k(y), E := k(x) (for some x ∈ K). Thus

y = (ax^{p^{m}} + b)/(cx^{p^{m}} + d) = ((a′x + b′)/(c′x + d′))^{p^{m}}

for some m ∈ ℤ, a, b, c, d ∈ k and their p^{m}-th roots a′, b′, c′, d′ ∈ k (since k is algebraically closed).

In particular, a generator y ∈ 𝔎^{∗} is in E^{∗} ∩ 𝔎^{∗} (and is the minimal positive power of a generator in E contained in E^{∗} ∩ 𝔎^{∗}). This implies the third property: the generators of E^{p^{m}} are p^{m}-th powers of the generators of E.

Corollary 14.6 (Definition). — Assume that y, y′ are primitive elements

coro:pro-uni

in (E^{p^{m}})^{∗} ⊂ 𝔎^{∗} such that

– y, y′ have support in 2 points;

– the pole of y coincides with the pole of y′.

Then (the images of) y, y′ in 𝔎^{∗}/k^{∗} are contained in a primary line passing through (the images of) 1, y, y′.

Proof. — Definition 10.1 and Lemma 10.2 give a Galois-theoretic characterization of the notion “support in 2 points”. By Corollary 14.4 we can Galoistheoretically distinguish zeroes and poles of y ∈ 𝔎^{∗}/k^{∗}. It remains to apply Lemma 14.5.

15. Proof

sect:proof

In this section we prove our main theorem: if

(G_{K}^{a}, Σ_{K}) = (G_{L}^{a}, Σ_{L}),

where L is a function field over an algebraic closure of a finite field of characteristic ≠ ℓ, then K is a purely inseparable extension of L.

Step 1. We have a nondegenerate pairing

G_{K}^{a} × K̂^{∗} → ℤ_{ℓ}(1).

This implies that K̂^{∗} = L̂^{∗}.

Step 2. We have Σ_{K}^{div} = Σ_{L}^{div} and we identify intrinsically the inertia and decomposition groups of divisorial valuations:

I_{ν}^{a} ⊂ D_{ν}^{a} ⊂ G_{K}^{a} :

every liftable subgroup σ ∈ Σ_{K}^{div} ⊂ Σ_{K} contains an inertia element of a divisorial valuation (which is also contained in at least one other σ′ ∈ Σ_{K}). The corresponding decomposition group is the “centralizer” of the (topologically) cyclic inertia group (the set of all elements which “commute” with inertia). This identifies DV_{K} = DV_{L}.

Step 3. For every ν ∈ DV_{K} we characterize intrinsically

I_{w}^{a} ⊂ D_{ν}^{a}/I_{ν}^{a}

(see Proposition 10.3).

Step 4. We distinguish divisorial valuations with nonrational centers (see Lemma 10.4 and Remark 10.5).

Step 5. For f̂ ∈ K̂^{∗} we have two notions of support: supp_{K}(f̂) (intrinsic) and supp_{X}(f̂) (depending on a model X) and two notions of finiteness: f̂ is nontrivial on at most finitely many nonrational divisorial valuations ν, resp. f̂ has finite divisorial support on a model. We defined FS(K) ⊂ K̂^{∗} as the set of elements satisfying the first notion of finiteness. If some (any) model X of K contains only finitely many rational curves, both notions of finiteness of support coincide and one obtains an intrinsic Galois-theoretic characterization of K^{∗}/k^{∗} ⊗ ℤ_{ℓ} ⊂ K̂^{∗}, as elements in FS(K). In general, it may happen that some g ∈ L^{∗}/l^{∗} has an “infinite rational tail” on some (every) model X of K:

ρ_{X}(g) = ρ_{X,nr}(g) + ∑_{j≥1} n_{j}C_{j},

where C_{j} are irreducible rational curves on X. In Lemma 12.7 we show that many elements of L^{∗}/l^{∗} ⊂ FS(L) = FS(K) have finite support on every model X of K, and vice versa. In particular, we prove that

FS_{0}(K) = K^{∗}/k^{∗} ⊗ ℤ_{ℓ} = L^{∗}/l^{∗} ⊗ ℤ_{ℓ}

(up to an ℓ-torsion group related to Pic^{0}(X), for some model X of K), where FS_{0}(K) ⊂ FS(K) ⊂ K̂^{∗} has an intrinsic Galois-theoretic description.

Step 6. For every pair of elements f̂, ĝ ∈ FS_{0}(K) satisfying

– supp_{K}(f̂) ∩ supp_{K}(ĝ) = ∅;

– ϱ_{ν}(f̂, ĝ) = 0 for all ν ∈ DV_{K} there exists a subfield E = k(C) ⊂ K such that f̂, ĝ ∈ Ê^{∗} (Proposition 14.1).

Step 7. Proposition 13.1identifies E^{∗}/k^{∗} inside Ê^{∗}, up to conformal equivalence, for all one-dimensional E = k(x), which are integrally closed in K.

Step 8. Proposition 14.3identifies 𝔎^{∗} := K^{∗}/k^{∗}∩L^{∗}/l^{∗} (as a multiplicative group) with a multiplicative subgroup of K^{∗}/k^{∗} ⊗ ℤ_{(ℓ)}.

Step 9. By Proposition 3.19, 𝔎^{∗} is isomorphic to K_{1}^{∗}/k^{∗}, and L_{1}^{∗}/l^{∗}, where K_{1}/K and L_{1}/L are finite purely inseparable extensions. Therefore, 𝔎^{∗} carries two structures of an abstract projective space compatible with the multiplicative structure (see Example 4.5).

Step 10. By Theorem 4.6 the field is uniquely determined by the partial projective structure on 𝔎^{∗} consisting of primary lines (see Lemma 4.8 and Lemma 4.9).

Step 11. Lemma 14.5 and Corollary 14.6 give a Galois-theoretic characterization of generating elements and primary lines in 𝔎^{∗}. These define a (unique) partial projective structure on 𝔎^{∗} (in particular, the projective structures induced by ℙ(K_{1}) and ℙ(L_{1}) coincide). In particular, the fields K_{1} and L_{1} both contain k and are isomorphic.

Step 12. If follows that K and L are finite purely inseparable extensions of the same field. This concludes the proof of Theorem 1.

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