Higher Dimensional Class Field Theory: the Variety Case

Home , Basic Number Theory, Global field, Ramification group

Higher Dimensional Class Field Theory: The variety case

Linda M. Gruendken

A Dissertation

Mathematics

Presented to the Faculties of the University of Pennsylvania in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

2011

Florian Pop Supervisor of Dissertation

Jonathan Block Graduate Group Chairperson

Dissertation Committee Florian Pop, Samuel D. Schack Professor of Algebra, Mathematics David Harbater, Christopher H. Browne Distinguished Professor in the School of Arts and Sciences, Mathematics Robin Pemantle, Merriam Term Professor of Mathematics HIGHER DIMENSIONAL CLASS FIELD THEORY: THE VARIETY CASE

2011

LINDA MEIKE GRUENDKEN Acknowledgments

I would like to thank my advisor, Florian Pop, for suggesting this topic, and for his continued support throughout both the research and writing phases of this thesis.

I am also very grateful to David Harbater, Rachel Pries, Scott Corry, Andrew

Obus, Adam Topaz for their helpful discussions in various phases of completion of this thesis. I would also like to thank Ching-Li Chai and Ted Chinburg for many interesting discussions in the early stages of the UPenn Ph.D. program.

Finally, I would like to thank my mother and my father for their love and support.

iii ABSTRACT

Higher Dimensional Class Field Theory: The variety case

Linda M. Gruendken

Prof. Dr. Florian Pop, Advisor

n Let k be a finite field, and suppose that the arithmetical variety X ⊂ Pk is an open subset in projective space. Suppose that CX is the Wiesend idèleclass

ab ab group of X, π1 (X) the abelianised fundamental group, and ρX : CX −→ π1 (X) the Wiesend reciprocity map. We use the Artin-Schreier-Witt and Kummer Theory of affine k-algebras to prove a full reciprocity law for X. We find necessary and sufficent conditions for a subgroup H < CX to be a norm subgroup: H is a norm subgroup if and only if it is open and its induced covering datum is geometrically bounded. We show that ρX is injective and has dense image. We obtain a one- to-one correspondence of open geometrically bounded subgroups of CX with open

ab 00 subgroups of π1 (X). Furthermore, we show that for an ´etalecover X −→ X with maximal abelian subcover X0 −→ X, the reciprocity morphism induces an

0 isomorphism CX /NCX00 ' Gal(X /X).

iv Contents

1 Introduction 1

2 Preliminaries 5

2.1 Basic Facts and Generalities ...... 5

2.2 Fundamental groups ...... 10

2.3 Covering Data ...... 17

3 The Wiesend Id`eleClass Group 34

3.1 Deﬁnitions and some Functiorial Properties ...... 34

3.2 The Reciprocity Homomorphism ...... 44

3.3 The Base Cases of the Induction Argument: Reciprocity for Regular

Schemes of Dimension One and Zero ...... 49

4 From the Main Theorem to the Key Lemma 55

4.1 Introducing induced covering data ...... 56

4.2 Geometrically bounded covering data ...... 60

v 4.3 Realisable subgroups of the class groups ...... 68

4.4 The Key Lemma ...... 73

5 Wildly ramiﬁed Covering Data 78

5.1 A Review of Artin-Schreier-Witt Theory ...... 80

5.1.1 Artin-Schreier Theory ...... 80

5.1.2 Artin-Schreier-Witt Theory ...... 85

5.2 Covering data of Prime Index ...... 94

n 5.2.1 Aﬃne n-space Ak ...... 94

n 5.2.2 The Case of Open Subsets of Pk ...... 102

5.3 Covering data of Prime Power Index ...... 113

6 Tamely ramiﬁed covering data revisited 122

6.1 A review of Kummer Theory ...... 123

6.2 Tamely ramiﬁed covering data of cyclic factor group ...... 129

vi Chapter 1

Introduction

Class Field Theory is one of the major achievements in the number theory of the

first half of the 20h century. Among other things, Artin reciprocity showed that the unramified extensions of a global field can be described by an abelian object only depending on intrinsic data of the field: the Class Group.

In the language of Grothendieck’s algebraic geometry, the theorems of classical global class ﬁeld theory [12, Ch. VI] can be reformulated as theorems about one- dimensional arithmetical schemes, whose function ﬁelds are precisely the global

ﬁelds. (A summary of results in convenient notation is presented in Section 3.3).

The class ﬁeld theory for such schemes X then turns into the question of describing the unramiﬁed abelian covers of the schemes, i.e. describing the fundamental group

ab π1 (X). It is therefore natural to ask for a generalisation of class ﬁeld theory to arithmetical schemes of higher dimensions.

1 Several attempts at a Higher Class Field Theory have already been made, with diﬀerent generalisations of the class group to higher dimensional schemes: Katz-

Lang [4] described the maximal abelian cover of a projective regular arithmetic scheme and Serre [15] gave a description of the abelian covers of schemes over Fp in terms of generalised Jacobians. Finally, Parshin and Kato, followed by several others, proposed getting a higher dimensional Artin reciprocity map using algebraic K-Theory and cohomology theories. Although promising, these approaches become quite technical, and the heavy machinery involved makes the results very complicated and diﬃcult to apply in concrete situations.

It was G. Wiesend [21] who had the idea to reduce the higher dimensional class

field theory to the well developed and known class field theory for arithmetical curves. He defines an ”idèleclass group” CX in terms of the arithmetical curves and closed points contained in X, and gets a canonical homomorphism

ab ρX : CX −→ π1 (X) in the hope of establishing properties similar to that of the Artin reciprocity map.

Wiesend’s work was supplemented by work by M. Kerz and A. Schmidt, where more details of Wiesend’s approach were given [5]. Most notably, in the so-called ﬂat case of an arithmetical scheme X over Spec Z (c.f. Deﬁnition 2.1.2), they were able to prove surjectivity of the canonical homomorphism, and they provided a concrete description of the norm subgroups in CX .

The focus of this thesis is on the still-open regular variety case, where X is a

2 regular arithmetical variety over some ﬁnite ﬁeld k of characteristic p.

In this case, Kerz and Schmidt have already proved a higher dimensional reci-

ab,t procity law for the abelianised tame fundamental group π1 (X), which classifies all the tame étaleabelian covers of X. A generalised Artin map can be defined between

t ab,t ρX,t : CX −→ π1 (X) ,

t where CX denotes the tame class group, as defined by Wiesend [21]. As in the flat case, the proof crucially relies on finiteness theorems for the geometric part of the tame fundamental group [4].

These finiteness results are known to be false for the full fundamental group due to the presence of wild ramification: For any affine variety of dimension ≥ 1 over a

finite field, the p-part of the fundamental group is infinitely generated.

n Let k be a ﬁnite ﬁeld, and X ⊂ Pk an open subvariety, then we prove:

n Theorem 1.0.1. Let X ∈ Pk be an open subvariety, and let

ab ρX : CX −→ π1 (X) be the Wiesend recprocity morphism. Then the following hold:

1) There exists a one-to-one correspondence between open and geometrically bounded

ab −1 subgroups of CX and open subgroups N of π1 (X); it is given by ρX (N) 7→ N. The

ab reciprocity morphism ρX is a continuous injection with dense image in π1 (X).

2) A subgroup of CX is a norm subgroup iﬀ it is open, of ﬁnite index and geometrically bounded.

3 3) If X00 −→ X is an ´etaleconnected cover, X0 −→ X the maximal abelian subcover, then NCX00 = NCX0 and the reciprocity map gives rise to an isomorphism

' 0 CX /NCX00 −→ Gal(X /X).

This thesis is organised as follows: We begin by introducing notation and re- viewing basic facts from the theories of arithmetical schemes, fundamental groups and covering data. We then review Wiesend’s definition of the idèleclass group and the reciprocity homomorphism, giving the right definitions as to guarantee commutativity of all relevant diagrams. (Unfortunately, in all the published works so far, the given morphisms do not make all the diagrams commute as stated.) We also review the results of classical class field theory as the ”base” case of our induction argument in the proof of the Main Theorem 4.3.4. All of Chapter 4 is devoted to assembling and refining the necessary tools for the proof of the Main Theorem

4.3.4, and in the process, the proof of the Main Theorem 4.3.4 is reduced to the Key

Lemma 4.4.4. Theorem 1.0.1 is shown as a corollary to the Main Theorem 4.3.4.

n The Key Lemma is shown for open subvarieties X ⊂ Pk in two steps. In Chapter

5, we analyse the behaviour of index-pm wildly ramiﬁed covering data on X to show

Part 1). The second part of the Key Lemma, which was already known from the results of Wiesend, Kerz and Schmidt ([21], [5]), is reproven in Chapter 6 without making explicit use of geometric ﬁniteness results.

4 Chapter 2

Preliminaries

2.1 Basic Facts and Generalities

In this thesis, we shall be concerned with arithmetical schemes X over Spec Z:

Deﬁnition 2.1.1. X is said to be an arithmetical scheme over Z if it is integral, separated and of ﬁnite type over Spec Z.

We assume that all schemes are arithmetical, unless otherwise stated, and distinguish two cases:

Definition 2.1.2. If the structural morphism X −→ Spec Z has open image, X will be called flat. If this is not the case, then the image of X −→ Spec Z is a closed point p ∈ Spec Z, and X is a variety over the residue field of the point k = k(p) = Fp.

We now collect some general facts and tools to be used in later chapters:

5 Fix a field k, and let K ⊃ k be a field over k. Then let degtr K denotes the transcendence degree of a field K. For a scheme X, let dim X denote the Krull dimension of a scheme X as a topological space (cf. [9, Definition 2.5.1]).

Definition 2.1.3. Let X be an arithmetical scheme with structural morphism f : X −→ Spec Z, and let f(X) be the closure of the image of f in Spec Z. Denote the function field of X by K(X). Then the Kronecker dimension d of X is defined as

dim X = degtr K(X) + dim(f(X)) .

Remark 2.1.4. For example, both Spec Q and Spec Z have Kronecker dimension one, while Spec Fp has Kronecker dimension zero. An arithmetical variety of Kro- necker dimension one is given by Spec Fp[t].

Deﬁnition 2.1.5. A curve is an arithmetical scheme of Kronecker dimension one.

If X is an arithmetical scheme of arbitrary dimension, a curve in X is a closed integral subscheme of Kronecker dimension one.

Under this definition, the curves are precisely those arithmetical schemes whose function field is a global field.

Definition 2.1.6. Let X be any scheme. An étalecover of X is a finite étalecover

Y −→ X, and a pro-´etalecover of X is the projective limit of ´etalecovers of X.

n Let k be an arbitrary ﬁeld, and consider a subvariety X ⊂ Pk . If C ⊂ X is a

n curve, i.e. a one-dimensional integral closed subscheme of X, then C ⊂ Pk is quasi-

6 projective. We recall from [9, Section 7.3.2] and [2, Section 1.7] the deﬁnitions of the genus and degree of a quasi-projective curve:

Definition 2.1.7. Let C be a geometrically connected projective curve. Then the arithmetic genus ga(C) is defined as ga(C) = 1 − χk(OX ), where χk(OX ) denotes the Euler-Poincarécharacteristic of the structural sheaf OX . Let C be a quasi- projective curve, with regular compactification C. Then the arithmetic genus is defined by setting ga(C) := ga(C).

Let C be a geometrically connected projective variety over k which is also smooth. Then the arithmetic genus is equal to the geometric genus gC .

n Definition 2.1.8. If k is an arbitrary field, and X ⊂ Pk is a projective variety of dimension d, the degree of X over k is defined as the leading coefficient of the

Hilbert polynomial pX (t), multiplied by d!. It is denoted by degk X.

n Fact 2.1.9. Let k be an arbitrary ﬁeld, X ⊂ Pk be a subvariety. For every positive integer d, there exists a number g = g(n, d) such that for all curves C ⊂ X of degree

≤ d, we have gC ≤ g. In particular, this holds for all regular curves C = Ce ⊂ X.

2 Note: The converse of this is not true: If C ⊂ Pk is the completion of

n 2 2 V (x − y ) ⊂ Ak ⊂ Pk, then C has degree n, but gC = 0 as C is rational.

Fact 2.1.10. Let X be any reduced scheme of ﬁnite type over the perfect ﬁeld k.

Then there exists a dense open subscheme which is aﬃne and smooth.

7 Proposition 2.1.11 (Chebotarev Density Theorem). Let Y −→ X be a gener- ically Galois cover of connected normal schemes over Z. Let Σ be a subset of

G = G(Y |X) that is invariant under conjugation, i.e. gΣg−1 = Σ for all g ∈ G. Set

S = {x ∈ X : F robx ∈ Σ}. Then the Dirichlet density δ(S) is deﬁned and equal to

δ(S) = |Σ|/|G|.

Proof. See [16].

Lemma 2.1.12 (Completely Split Covers). Let X be a connected, normal scheme of finite type over Z. If f : Y −→ X is a finite étalecover in which all closed points of X split completely, then this cover is trival. If Y is connected, then f is an isomorphism.

Proof. Let Y 0 be the Galois closure of Y −→ X, then a closed point x ∈ X is completely split in Y if it is completely split in the cover Y 0, so without loss of generality we may assume that Y −→ X is Galois with Galois group G. Since x ∈ X splits completely if and only if F robx = 1 for all Frobenius elements above x, this follows directly from the Chebotarev Density Theorem 2.1.11. Let

S = {x : F robx = 1}, and δ(S) the Chebotarev density of S. Then

1 = δ(S) = 1/|G| , whence it folllows that |G| = 1.

As being completely split is equivalent to saying that the pullback Y × x −→ x to every closed point x ∈ |X| is the trivial cover, we make the following deﬁnition:

8 Deﬁnition 2.1.13. An ´etalecover f : Y −→ X is called locally trivial if it is completely split.

Lemma 2.1.14 (Approximation Lemma). Let Z be a regular arithmetical curve,

X a quasi-projective arithmetical scheme, and let X → Z be a smooth morphism in of arithmetical schemes. Let Y −→ X be a ﬁnite cover of arithmetical schemes, and let x1, . . . , xn be closed points of X with pairwise diﬀerent images in Z. Then there exists a curve C ⊂ X such that the points xi are contained in the regular locus

Creg of C, and such that C × Y is irreducble.

Proof. See [5].

The following proposition will be essential for dealing with covering data and trivialising morphisms in Chapters 3-5:

Proposition 2.1.15. Let X be a regular, pure-dimensional, excellent scheme,

X0 ⊂ X a dense open subscheme, Y 0 −→ X0 an étalecover and Y the normal- ization of X in k(Y 0). Suppose that for every curve C on X with C0 = C ∩ X0 6= ∅, the étalecover Y 0 × Ce0 −→ X × Ce0 extends to an étalecover of Ce. Then Y −→ X is étale.

Proof. See [5, Proposition 2.3] for a proof.

9 2.2 Fundamental groups

In this section, we give a brief survey of relevant results in the theory of fundamental groups, all of which are taken from [19, Section 5.5].

Let X be a connected scheme. Then the ﬁnite ´etalecovers of X, together with morphisms of schemes over X form a category of X-schemes, which we denote by

EtX .

Now let Ω be an algebraically closed field, and fixs ¯ : Spec Ω −→ X, a geometric point of X. If Y −→ X is an element of EtX , consider the geometric fiber Y ×X

Spec Ω, and let F ibs¯(Y ) denote its underlying set. Any morphism Y1 −→ Y2 in

EtX induces a morphism Y1 ×X Spec Ω −→ Y2 ×X Spec Ω. Applying the forgetful functor, we get an induced set-theoretic map F ibs¯(Y1) −→ F ibs¯(Y2).

Thus, F ibs¯(·) deﬁnes a set-valued functor on the categeroy EtX , which we call the ﬁber functor at the geometric point s¯.

We recall that an automorphism of a functor F is a morphism of F −→ F which has a two-sided inverse (cf. [20]), and deﬁne the fundamental group as follows:

Deﬁnition 2.2.1. The fundamental group of X with geometric basepoints ¯ is the automorphism group of the ﬁber functor F ibs¯ associated tos ¯, and is denoted by

π1(X, s¯).

Lemma 2.2.2. The fundamental group is proﬁnite and acts continuously on F ibs¯(X).

Proof. See [19, Theorem 5.4.2].

10 Proposition 2.2.3. The functor F ibs¯ induces an equivalence of the category of

finite étalecovers of X with the category of finite continuous π1(X, s¯)-Sets. Under this correspondence, connected covers correspond to sets with transitive action, and

Galois covers to ﬁnite quotients of π1(X, s¯)

Proof. Cf. [19, Thm. 5.4.3]).

Now lets, ¯ s¯0 : Spec Ω −→ X be two geometric points of X.

Deﬁnition 2.2.4. A path froms ¯ tos ¯0 is an isomorphism of ﬁber functors

' F ibs → F ibs0 .

Whenever such a path p exists, the fundamental groups are (non-canonically)

0 isomorphic as proﬁnite groups via conjugation by p: π1(Y, s¯) ' π1(Y, s¯ ) [19, 5.5.2].

Proposition 2.2.5. If X is a connected connected scheme and s are s¯0 two geometric points of X, then there always exists a path p from s¯ and s¯0.

Proof. See [19, Cor. 5.5.2].

Thus, for a connected scheme X, the fundamental group π1(X) is well-deﬁned up to conjugation by an element of itself. In particular, the maximal abelian quotient

ab π1 (X) does not depend on the choice of geometric basepoint [19, Remark 5.5.3].

ab Deﬁnition 2.2.6. The maximal abelian quotient π1 (X) is called the abelianised fundamental group of X.

Now let X, Y be schemes with geometric pointss ¯0 ands ¯.

11 Deﬁnition 2.2.7. In the situation above, if f : Y −→ X is a morphism of schemes such that f ◦ s¯ =s ¯0, then the morphism is said to be compatible withs ¯ ands ¯0.

Spec Ω s¯ / Y G GG GG f s¯0 GG GG# X Proposition 2.2.8. In the situation above, a morphism that is compatible with s¯

0 0 and s¯ induces a morphism on fundamental groups fe : π1(Y, s¯) −→ π1(X, s¯ ).

Proof. See the remarks following Remark 5.5.3 in [19].

If f : Y −→ X is a finite morphism that is not compatible with the given geometric points, then we can find another geometric points ¯00 of Y so thats ¯0 and s¯00 are compatible. Indeed, let x be the image point ofs ¯0 in X. If y is any element of f −1(x), k(y) is a finite extension of k(x) by assumption. Since Ω is algebraically closed, we have k(y) ⊂ Ω, and may define a geomtric points ¯00 of Y by declaring its image to be y.

The fundamental groups of X with respect to the two points are isomorphic via conjugation by some path p, so we obtain a morphism on the fundamental groups induced by f via composition with the isomorphism:

−1 p · p 00 0 fe : π1(X, s¯) ' π1(X, s¯ ) −→ π1(Y, s¯ )

Remark 2.2.9. In the situation above, whether basepoints are already compatible

−1 0 or not, the preimage fe (N) of any normal subgroup N of π1(X, s¯ ) is always well

12 deﬁned. We shall thus drop the geometric points from our notation in later chapters, when dealing with covering data.

Remark 2.2.10. While the fundamental group functor π1(X, s¯) is not representable in EtX , the category of finte étalecovers of X, it is pro-representable and thus representable in the larger category of profinite limits of finite étalecovers of X.

The corresponding universal element is also called a universal cover of X, and there is a one-to-one correspondence between universal covers and a system of compatible geometric basepoints for the collection of all pro-´etalecovers of X.

In particular, ﬁxing a universal cover with geometric basepoint xe amounts to choosing a system of compatible basepoints for every pro-´etalecover Y −→ X.

The following proposition summarizes some further properties of fundamental groups:

Proposition 2.2.11. Let X be a connected scheme. Fix a universal cover Xe of X, let xe denote its geometric basepoint, and let f : Y −→ X be a ﬁnite connected ´etale subcover. Then

1. The induced morphism fe : π1(Y ) −→ π1(X) is injective with open image

NY := fe(π1(Y )), and the association Z 7→ fe(π1(Z)) is one-to-one if we re-

strict to connected ´etalesubcovers Z of Xe.

2. f is a trivial cover if and only if fe is an isomorphism.

3. f is Galois if and only if NY is normal in π1(X). If this is the case, the Galois

13 group G = Aut(Y/X) is isomorphic to the group π1(X)/NY .

4. If g : W −→ X is another ´etalecover, and let w, y be the geometric basepoints

of W , respectively Y that are combaptible with xe. Let Z ⊂ Y ×X Y be the

connected component of Y × Z containing the geometric basepoint y × w, then

Z / W g f Y / X

corresponds to the open subgroup NY ∩ NZ . If f and g are Galois over X with

groups Gf and Gg, then so is W −→ X, and it has Galois group G, where

G < Gf × Gg is a subgroup projecting surjectively onto Gf and Gg.

5. There exists a minimal ´etalecover Z −→ Y such that Z −→ X is Galois. NZ

is then the smallest normal subgroup of π1(X) contained in NY .

Proof. Apply Proposition 2.2.3 and Propositions 5.5.4-6 of [19].

Recall from Definition 2.1.6 that a pro-étalecover f : Y −→ X is the inverse limit of finite étalecovers. Analogously to above, we then have the following proposition:

Proposition 2.2.12. Let X be a connected scheme. Fix a universal cover Xe of X, let xe denote the associated geometric basepoint, and let f : Y −→ X be a pro-´etale subcover. Then

14 1. The induced morphism fe : π1(Y ) −→ π1(X) is injective with closed image

NY := fe(π1(Y )), and the association Z 7→ fe(π1(Z)) is again one-to-one if we

restrict to connected ´etalesubcovers Z of Xe.

2. f is a trivial cover if and only if fe is an isomorphism.

3. f is Galois if and only if NY is normal in π1(X). If this is the case, the Galois

group G = Aut(Y/X) is isomorphic to the group π1(X)/NY .

4. If g : W −→ X is another pro-´etalecover, let w, y denote the geometric

basepoints of W , respectively Y which are compatible with xe. Let Z ⊂ W ×X Y

be the irreducible component which contains the geometric basepoint y × w.

Then we have a commutative diagam

Z / W g f Y / X

and the cover f × g : Z −→ X corresponds to the open subgroup NY ∩ NW . If

f and g are Galois over X with groups Gf and Gg, then so is f × g, say with

Galois group G, then G < Gf × Gg is a subgroup projecting surjectively onto

Gf and Gg.

5. There exists a minimal ´etalecover Z −→ Y such that Z −→ X is Galois. NZ

is then the smallest normal subgroup of π1(X) contained in NY .

Notation 2.2.13. Lastly, we establish and summarise some notation for later chapters:

15 1. We have already introduced the tilde notation for morphisms between funda-

f mental groups which are induced by morphisms of schemes: If f : Y −→ X

is a morphism, then fe : π1(Y ) −→ π1(X) denotes the induced map on the

fundamental groups.

2. If X is an arithmetical scheme, then the closed points in an arithmetical

scheme X are those points with ﬁnite residue ﬁeld, and the set of closed

points is denoted by |X|. We let ix : x ,→ X denote the inclusion morphism.

3. Recall that a closed integral subscheme C,→ X of dimension one is called a

curve in X. We denote the normalisation of a curve by Ce, and note that the

normalisation might lie outside the scheme X.

We let iC : Ce −→ C,→ X denote the composition of the normalisation

morphism with the inclusion of the curve in X.

4. Let iC : Ce −→ C,→ X be the composition of the normalisation morphism

with the inclusion of a curve into X as deﬁned above, then we denote by

eiC : π1(Ce) −→ π1(X) the induced morphism on the fundamental groups.

Similary, for the inclusion ix : x ,→ X of a closed point, the induced morphism

π1(x) −→ π1(X) is denoted by eix.

5. Now let f : Y −→ X be a morphism of arithmetical schemes. If y is a closed

point in Y , then x = f(y) is also a closed point. If C ⊂ X is a curve, then the

closure f(C) of the image of C is either a closed point or a curve D. We let

16 f|y = fe×eix and fe|C = f ×eiC denote the ﬁber products of f with the maps

induced by ix and iC , respectively.

Let D ⊂ C ×X Y be the irreducble component determined by the ﬁxed uni-

versal cover. Then we have a commutative diagram:

i eD / π1(De) π1(Y )

fe|C fe i eC / π1(Ce) π1(X)

Similarly, let y ∈ x×X Y be the closed point determined by the univesal cover,

then we get the commutative diagram:

eiy π1(y) / π1(Y )

fe|x fe eix π1(x) / π1(X)

2.3 Covering Data

Let X be an arithmetical scheme, and recall from Notation 2.2.13.2 that a curve C of X is an integral, closed, one-dimensional subscheme of X, not necessarily regular.

Recall also that Ce denotes the normalisation of a curve C.

In this section, we consider collections of open normal subgroups

D = (NC ,Nx)C,x, where C and x range over the curves and closed points of X, respectively. For each C, respectively x, NC / π1(Ce) and Nx / π1(x) are taken to be normal subgroups of the fundamental groups of the normalisation of C and the

17 fundamental group of x, respectively. Since the subgroups NC , Nx are open and normal, they correspond to ﬁnite Galois covers of Ce and x, respectively.

We notice that whenever C is any curve containing the closed point x, the ﬁbered product gives commutative diagrams

x × Ce / Ce

iC i x x / X in which x × Ce is ﬁnite. If x is a regular point of C, then the left vertical morphism is an isomorphism.

Now let xe be a point of x × Ce, then we get an induced diagram of fundamental groups

/ π1(xe) π1(Ce)

iC∗ ix∗ π1(x) / π1(X) .

Deﬁnition 2.3.1. A collection D = (NC ,Nx)C,x of open normal subgroups

NC < π1(Ce), Nx < π1(x) is a covering datum if the NC and Nx satisfy the following compatibility condition:

(*) For every curve C ⊂ X, x ∈ |X|, and any xe lying above x in x × Ce, the

preimages of NC and Nx under the the canonical morphisms above agree as

subgroups of π1(xe).

Deﬁnition 2.3.2. Let f : Y −→ X, and recall the notations for induced morphisms on the fundamental groups set in 2.2.13.1 and 2.2.13.4. If D is a covering datum

18 on X, then the pullback of D via f is the covering datum on Y deﬁned as follows:

−1 1. If y is a closed point in Y , x = f(y), we let Ny = fey (Nx).

−1 2. For a curve C ⊂ Y , we let NC = feC (Nf(C)), where f(C) is the closure of the

image of C in X, i.e. either a point or a curve.

Deﬁnition 2.3.3. Let X be an arithmetical scheme, and let D be a covering datum on X.

We say that D is a covering datum of index m on X if we have [π1(Ce):

NC ], [π1(x): Nx] ≤ m for all points x and curves C in X, and if we have equality for at least one curve or point.

If we do not necessarily have equality, we say instead that D is of index bounded by m, or of bounded index.

We say that D is a covering datum of cyclic index l if the associated covers

D D YC −→ C and x −→ x all have cyclic Galois groups Z/sZ where s|l.

This terminology is slightly diﬀerent from that in [5], for reasons explained in

Section 4.3.

Deﬁnition 2.3.4. Let X be an arithmetical scheme, and let D,D0 be two covering data on X. Let C ⊂ X denote a curve, and let x ∈ |X| denote a closed point of

D D D0 D0 X. Let fC : YC −→ C, fC : YC −→ C denote the covers of C deﬁned by D

0 D D D0 D0 and D . Also let fx : x −→, fC : x −→ x denote the cover of x deﬁned by D, respectively D0.

19 0 D D0 We say that D is a subdatum of D if fC is a subcover of fC for all curves

D D0 C ⊂ X, and if fx is a subcover of fx for all x ∈ |X|.

Deﬁnition 2.3.5. A covering datum D is called trivial if Nx = π1(x) for all closed points x ∈ |X|, and N(C) = π1(Ce) for all curves C ⊂ X. We say that D is weakly trivial if Nx = π1(x) for all closed points x ∈ |X|.

Deﬁnition 2.3.6. Let D be a covering datum on a scheme X. Given an ´etalecover

Y = YN −→ X corresponding to the open subgroup N ⊂ π1(X), we say that:

1. f trivialises D if the pullback of D to Y is the trivial covering datum.

2. f weakly trivialises D if the pullback of D to Y is weakly trivial.

3. f weakly realises the covering datum if N(x) = Nx for all closed points

x ∈ |X|.

4. f realises D if N(x) = Nx for all closed points x ∈ |X| and N(C) = NC for

all cuvers C ⊂ X.

We call f a (weak) trivialisation, respectively realisation, of D.

i 5. If the pullback i∗(D) to an open subset U ,→ X has one of the above properties, we say that D has that property over U.

Let f : XN −→ X be a Galois ´etalecover corresponding to the open normal subgroup N / π1(X), and recall Notations 2.2.13.1 and 2.2.13.4. For every

20 curve C ⊂ X with normalisation Ce and every closed point x ∈ |X|, we deﬁne the

−1 −1 pullbacks N(C) := eiC (N), N(x) = eix (N). They correspond to the ´etalecovers fC : XN × Ce −→ Ce and fx : XN × x −→ x, respectively. Then for any xe ∈ Ce × x, the diagram

/ π1(xe) π1(Ce)

π1(x) / π1(X) commutes since it is induced from the commutative diagram

/ xe Ce

x / X

In particular, the pullbacks of N(C) and N(x) must agree in π1(xe) for any x and

C, i.e. the datum (NC ,Nx)C,x is a covering datum. Thus, the Galois ´etalecover

N f : XN −→ X induces a covering datum of X, which we shall denote by D .

D Notation 2.3.7. Given a covering datum D on X, we let YC −→ Ce, y −→ x be the

´etalecovers of connected schemes corresponding to NC / π1(Ce) and Nx / π1(x).

Then given a morphism Y −→ X, we have diagrams

D / / Y × Ce × YC Y × Ce Y × x × y Y × x

D / y / x YC Ce 0 D Notation 2.3.8. Now let Z, Z be the connected components of Y ×Ce and Y ×Ce×YC , respectively, and let z, z0 be points in the ﬁnite sets Y × x and Y × x × y containing

21 the relevant geometric base points. Then we get the following diagrams of connected schemes:

q qx Z0 C / Z z0 / z

pC fC px fx g gx D C / y / x YC Ce Following the remarks on the correspondence between connected pro-´etalecovers and closed subgroups of the fundamental group of a scheme X, we note that the

0 0 covers Z −→ Ce and z −→ x correspond to the subgroups NC ∩N(C) and N(x)∩Nx of π1(Ce) and π1(x), respectively. We also note that we can identify N(C) with

D feC (π1(Z)), NC with geC (π1(YC )) and similarly for closed points.

It is now possible to express the properties 2.3.6.1 through 2.3.6.4 of f in re- lation to D as inclusions/equalities of subgroups in π1(Ce) and π1(x), but also as

D inclusions/equalities of subgroups in π1(YC ), π1(YC ) and π1(y), π1(Y × x). Full characterisations are given as follows:

Corollary 2.3.9. Let D be a covering datum on a scheme X.

1. If f : Y −→ X is a pro-´etalecover, then TFAE:

(a) f trivialises D.

(b) Nx ⊇ N(x) in π1(x) for all closed points x ∈ |X|, and NC ⊇ N(C) in

π1(Ce) for all curves C ⊂ X.

22 −1 (c) We have fex (Nx) = π1(Y × x) for all closed points x ∈ |X| and

−1 feC (NC ) = π1(Y × Ce) for all curves C ⊂ X.

(d) The covers pC and px are trivial for any closed points x ∈ |X| and any

curve C ⊂ X.

2. Similarly, TFAE:

(a) f is a locally trivial cover (cf. Deﬁnition 2.1.13).

(b) f weakly trivialises D.

−1 (d) We have fex (Nx) = π1(y) for all closed points x ∈ |X|.

(e) The covers px are trivial for all closed points x ∈ |X|.

3. For a pro-´etalecover f : Y −→ X, TFAE:

(a) f realises D.

−1 −1 (b) We have fex (Nx) = π1(Y × x), gex (N(x)) = π1(y) for all closed points

−1 −1 D x ∈ |X| and feC (NC ) = π1(Y × Ce), geC (N(C)) = π1(YC ) for all curves

C ⊂ X.

px, qx are trivial for any closed points x ∈ |X|.

4. Analogously to 3), TFAE:

23 (a) f weakly realises D.

−1 −1 (b) We have fex (Nx) = π1(Y × x), gex (N(x)) = π1(y) for all closed points

x ∈ |X|.

Lemma 2.3.10. Let X be an arithmetical scheme, and let D be a covering datum on

X. An ´etalecover Y −→ X which weakly trivialises (weakly realises) D trivialises

(realises) the covering datum.

Proof. We use the notations introduced in 2.3.8, and make repeated use of the equivalence of conditions listed in Cor 2.3.9.2 and 2.3.9.3: f weakly trivialises D if and only if the morphisms px are trivial covers of arithmetical schemes. So now consider the covers p . If x is a point of C×x, then the morphism p induced by base C e e xe changing from x to xe is again trivial. Thus, pC is locally trivial. By Lemma 2.1.12, a locally trivial ﬁnite cover of arithmetical schemes is trivial, so pC is a trivial cover for any curve C ⊂ X, as claimed.

Similarly, if f weakly realises D, then we know that all the morphisms px and qx are trivial for all closed points. As above, this implies that pC and qC are locally trivial and thus trivial.

Definition 2.3.11. Let X be an arithmetical scheme, and let D be a covering datum on X. Then D is called effective if it is realised by a pro-étale cover Y −→ X. If this is the case, the associated cover YN is called the realisation of the covering datum

24 D. If the realisation of D is a ﬁnite cover, we say that D has a ﬁnite realisation.

Remark 2.3.12. Note that if D has a ﬁnite realisation f : XN −→ X, then there exists a curve C ⊂ X such that the canonical morphism iC : Ce −→ X

(cf. Deﬁnition 2.2.13) and the induced morphism eiC : π1(Ce) −→ π1(X) induces an isomorphism

π1(Ce)/N(C) ' π1(X)/N .

In particular, the degree of the realisation f is equal to the index of the covering datum.

Proof. XN corresponds to the open normal subgroup N / π1(X). We deﬁne

−1 −1 N(C) := eiC (N) for any curve C ⊂ X, and likewise N(x) := eix (N) for closed points x. Then we have natural inclusions

π1(Ce)/N(C) ,→ π1(X)/N and π1(xe)/N(x) ,→ π1(X)/N

for all x and C. As N(x) = Nx and N(C) = NC , this implies that the index of D is bounded by deg(f). Now let C be an irreducible curve as guaranteed by

Lemma 2.1.14, then fC : Y × Ce −→ Ce has the same Galois group as f. Therefore

π1(Ce)/N(C) ,→ π1(X)/N

is an isomorphism, and we have deg(f) = [π1(Ce): NC ]. Thus, D must have index exactly equal to deg(f), as claimed.

Proposition 2.3.13. Let X be a normal, arithmetical scheme and D = (NC ,Nx)(C,x) a covering datum on X.

25 1. Then D has at most one ﬁnite realisation.

2. Let Y1, Y2 be two ´etalecovers of X weakly realising D over an open subset U.

Then Y1 = Y2.

i 3. If there exists an open subscheme U ,→ X such that the pullback i∗(D) can

be weakly realised by an étalecover Y −→ U, then D is effective with finite

realisation. The realisation of D is given by Y 0, the normalisation of X in

K(Y ).

Proof. (cf. [5, Lemma 3.1])

1. Let XN1 , XN2 be two realisations of D, and let Ni/, i = 1, 2 be the corre-

sponding open normal subgroups. Then N1 ∩ N2 correspons to a completely

split cover of Y1, so by Lemma 2.1.12, it is an isomorphim. It follows that

N2 ⊂ N1, and thus N2 = N1 by symmetry.

2. Let N1, N2 be the open subgroups of π1(X) associated to the two realisa-

−1 tions Y1, Y2, and let Ni(x) := eix (Ni) denote their pullbacks to π1(x) for

any closed point x. Let Y = Y1 × Y2. If N1(x) = N2(x) for all x ∈ |U|, then

U × Y −→ U × Y1 is completely split, and thus an isomorphism by

Lemma 2.1.12. For V an irreducible component of Y1 × U, this means that

V × Y1 −→ V is an isomorphism of connected schemes. In particular, the

Galois closure of this cover is just V × Y1 itself, and is connected. Now if

0 Y → Y1 were not an isomorphism, then the Galois closure Y → Y1 would

26 have nontrivial Galois group as well. Since V × Y1 is connected, the Galois

groups of the two vertical covers in

V × Y 0 / Y 0

V / Y1

are isomorphic. Thus this would imply that V × Y1 has nontrivial Galois

groups as well, contrary to assumption.

3. If Y 0 is the normalisation of X in Y , Y 0 −→ X is ﬁnite, and we have

Y = Y 0 × U. As D is a covering datum on all of X, all covers that i∗(D)

0 induces on curves C = C ∩ U of U extend to ´etalecovers YC of the full

curve C in X. By Proposition 2.1.15, Y 0 −→ X is an ´etalecover; we let

N / π1(X) be the corresponding open normal subgroup. Recall the notations

set in 2.2.13.5), and let N(C) := i−1(N), N(x) := i−1(N) denote the pullback eCe ex

to π1(Ce), respectively π1(x).

Now let C be a curve on X with C ∩ U 6= ∅. Then the preimages i−1(N(C)) exe

−1 and i (NC ) in π1(x) agree for every point x of Ce lying over U. Applying the exe e e

argument in 1) to the scheme Ce, we see that the normal subgroups N(C) and

NC of π1(Ce) coincide, and so N(x) = Nx for every regular point x of C. By

Lemma 2.1.14, every point is contained in the regular locus of a curve meeting

U, so we get N(x) = Nx for all closed points x ∈ |X|. By Lemma 2.3.10, we

0 conclude that Y = YN is a realisation of D.

27 Remark 2.3.14. In general, a realisation of a covering datum is automatically ﬁnite if the covering datum is of bounded index and tame. It is always ´etaleby Lemma

2.1.15. Since the p-part of the fundamental group is not finitely generated, effective covering data which are note tame have realisations which are not necessarily finite

´etalecovers, but only pro-´etale.

Theorem 2.3.15. Let X be a regular arithmetical scheme, and let D = (NC ,Nx)C,x be a covering datum on X that is trivialised by a finite cover f : Y −→ X. Then D is effective with a finite realisation.

Corollary 2.3.16. Let X be a regular arithmetical scheme, and let D = (NC ,Nx)C,x

i be a covering datum on X. If there exists an open subscheme U ,→ X such that the pullback i∗(D) can be weakly trivialised by a finite cover Y −→ U, then D is effective with a finite realisation.

Proof. This follows directly from Theorem 2.3.15 and Lemma 2.3.13.2.

Proof of Theorem 2.3.15. We first show that it suffices to prove the Theorem under the additional assumption that f is étale.

Claim 2.3.17. If the covering datum D is trivialised by a finite cover f : Y −→ X, then there also exists a finite étalecover f 0 : Y 0 −→ X trivialising D.

0 00 0 0 Proof. Let X ⊂ X be an open dense subset such that f|X0 : Y = X ×X Y −→ X is ´etale.(Such an X0 exists by the purity of the branch locus, cf. [9, Section 8.3, Ex.

28 0 2.15].) Then the restriction D|X0 of D to X is trivialised by f|X0 , and thus eﬀective by the results of the previous paragraphs. In particular, there exists a subgroup

0 0 N < π1(X ) giving a weak realisation of D over the open subscheme X , which must be a full realisation by Proposition 2.3.13. Since the realisation of a covering datum

D trivialises D by deﬁnition, taking Y 00 to be the cover corresponding to N proves the claim.

Returning to the proof of Theorem 2.3.15, by Proposition 2.3.13, it suﬃces to

ﬁnd a subgroup N of π1(X) which gives a weak realisation over a dense open subset

−1 U, i.e. an open normal subgroup N such that eix (N) = Nx for all x ∈ U. In particular, we can replace X by any open dense subset. Using Lemmas 2.1.10 and 2.3.13, we may thus assume without loss of generality that X is quasi-projective, and that there exists a smooth morphism X −→ S to a curve S. Replacing Y −→ X by its

Galois hull, we may also assume that Y −→ X is Galois, say with group G.

By Lemma 2.1.14, there exists a curve C ⊂ X such that D = C×Y is irreducible and such that fC : D −→ C is again a Galois ´etalecover with group G. By the

Chebotarev Density Theorem 2.1.11, there are inﬁnitely many n-tuples (x1, . . . , xn)

of points in C such that G is the union of the conjugacy classes [F robxi ]. Since the regular locus of C is of codimension one, it is ﬁnite, so by replacing those xi which

reg are not contained in the regular locus of C, we may assume that xi ∈ C for all i.

Since Y trivialises D, we have feC (π1(De)) ≤ NC as subgroups of π1(Ce). Let

29 N = eiC (NC ) be the image of NC in π1(X), then we shall show that N gives a weak realization of D over an open subset of X. More precisely, if we denote

−1 N(x) = eix (N) for any x ∈ |X|, then we show that N(x) = Nx for all x which g : X → Z maps to points which are distinct from the images of S = {x1, . . . , xn} under g. The set U of such x is open in X since the set of images of S is closed, and g is continuous in the Zariski topology. For x ∈ U, the Approximation Lemma

0 2.1.14 yields a curve C ⊂ X containing x, x1, . . . xn as regular points, and such that

C0 × Y is irreducible.

Lemma 2.3.18. Let C be a curve in X such that C × Y is irreducible, let

N = eiC (NC ) be the image of NC in π1(X), and N(z) = eiz(N) the pullback to

0 π1(z). Then N(z) = Nz for all regular points z of C .

−1 Proof. We ﬁrst show that if we deﬁne N(C) := eiC (N), then N(C) = NC . Indeed, eiC is easily seen to be injective by applying the second criterion of

[19, Corollary 5.5.8] to the canonical morphism j = eiC : Ce −→ X:

First we consider the case where C is normal, i.e. Ce is already contained in X, and also assume that X is aﬃne. If j is a closed immersion of aﬃne schemes cor-

∼ responding to a surjective morphism B A = B/I, and D −→ Spec A is a ﬁnite

´etalecover, then D is also aﬃne, say D = Spec E. We thus have to show that if

E = A[f1, . . . , fn] = A[x1, . . . , xn]/(g1, . . . , gn) is a finite étalealgebra over A, where the gi are polynomials with coefficients in A, then E is the tensor product F ⊗ B of some finite étalealgebra F . But taking hi ∈ B[x1, . . . , xn] to be any lifts of the gi,

30 then we can take F = B[x1, . . . , xn]/(h1, . . . , hn). Clearly, F is finite over B, and it is flat since Spec F −→ Spec B is finite and surjective [9, Remark 4.3.11]. Last, it can easily be seen that if F were not étaleover B, then E would have to be ramified over B/I as well (e.g. by applying the criterion of [9, Example 4.3.21]).

For the not necessarily aﬃne case of a regular arithmetical scheme X, take

U ⊂ X to be a dense aﬃne open subset. Then the previous argument gives a ﬁnite

´etalecover V of U such that we have a commutative diagram

DU = C ∩ U × V / V

C ∩ U / U / X

Taking Y to be the normalisation of X in the function ﬁeld K(V ) of V over

K(U) = K(X), we have that V = Y × U. Note that D −→ C is normal since

C is locally Noetherian, as well as assumed to be normal, and the cover is ﬁnite

´etale.Thus D is equal to the normalisation of C in k(DU ). Now compare this with

D0 = C × Y , a normal cover of C since the ´etalenessof Y −→ X is preserved under

0 0 0 base change to C. D is birational to DU and to D , which implies D = D ; then, the claim follows.

Now to the case where C ⊂ X may be non-regular, so that Ce is not necessarily a subscheme of X. We have iC : Ce −→ C −→ X, and are given a cover De of Ce.

Since Ce and C are birational, they have the same function field. So let D be the normalisation of C in k(D), and apply the first part to D −→ C. Then D = C × Y for some finite étalecover Y , and since De = D×Ce, the claim follows by associativity

31 of the ﬁbre product.

−1 In conclusion, we have that N(C) = eiC (eiC (NC )) = NC . Now if z is a regular point of C, then there is a unique point π1(ze) of Ce lying above z ∈ C, and we have ∼ a natural isomorphism π1(ze) = π1(z) ﬁtting into the following diagram:

/ π1(ze) π1(Ce) w; ∼ ww = ww ww ww π1(z)

A diagram chase comparing N(z) to Nz now easily proves the claim.

Returning to the proof of Theorem 2.3.15, we have N(xi) = Nxi for all i. Let

0 −1 0 N(C ) = eiC0 (N) denote the preimage under the natural map eiC0 : π1(Cf) → π1(X),

0 M its image in G, and M the image of N in G. If xei is the unique point of Ce lying above xi, we have a diagram:

0 π (C) / G o π (C0) NC / M,M o NC0 1 e O 1 f Fb O ; Hc H v: F x HH vv FF xx HH vv FF xx HH vv FF xx H vv F xx π (x ) N 1 ei xi

Noting that the image of Nxi in G must be the same for both triangles, we

0 have, in particular, that F robxi ∈ M iﬀ F robxi ∈ M for all i. Since the xi were

picked such that the {F robxi } are all the conjugacy classes of G, this implies that

M = M 0, and the diagram commutes. Since C0 was irreducible in Y −→ X, we have

0 0 0 Nx = N (x) by Lemma 2.3.18. Now since N(x) and N (x) both map to M = M when composing the canonical morphism eix : π1x −→ π1(X) with the projection

32 π1(X) −→ G, they must be equal. Thus we have Nx = N(x) as claimed. This

ﬁnishes the proof of the theorem.

33 Chapter 3

The Wiesend Id`eleClass Group

In this chapter, we let X be an arithmetical scheme, and define the Wiesend idèle class group for both the flat and variety case (cf. Definition 2.1.2). We define the reciprocity homomorphism and establish some functorial properties.

3.1 Deﬁnitions and some Functiorial Properties

Let X be an arithmetical scheme. As before, we let |X| denote the set of closed points of X and let C ⊂ X be a curve. If X is in the ﬂat case, then we have two possibilities: If the image of the structural morphism C −→ Spec Z is a point p ∈ Spec Z then C is called vertical. Otherwise, C −→ Spec Z has dense image.

Then the regular compactification P (C) of C is isomorphic to some order of Spec R, where R ⊂ OK is an order inside the ring of integers of a number field K at some element f. If C is regular, then the regular compactification P (C) of C is isomorphic

34 to Spec OK .

Deﬁnition 3.1.1. Let C ⊂ X be a curve inside an arithmetical scheme X. If the structural morphism of C factors through Spec Fp for some prime p, C is called a vertical curve. Otherwise, C is called horizontal.

Note that if X is a variety, then all curves are vertical. For a vertical curve, we denote by C∞ the ﬁnite set of (normalized) discrete valuations of K(C) without a center on Ce. If C is horizontal, we let C∞ be the ﬁnite set of discrete valuation of

K(C) corresponding to the points without a center on Ce together with the ﬁnite set of archimedean places of K(C).

Now recall (e.g. from [9, Remark 8.3.19]) the following:

Fact 3.1.2. Let C be any integral curve. There is a one-to-one correspondence between the set of discrete valuations on its function K(C) having center on C and the set of closed points in the normalisation Ce of C.

It is given by associating to each discrete valuation its unique center; its inverse is given by associating to a closed point x the valuation νx it deﬁnes.

For a scheme X, let the id`elegroup (after Wiesend) be the abelian topological group group deﬁned by

M M M × JX = Z.x ⊕ K(C)ν x∈|X| C⊂X ν∈C∞ with the direct sum topology. Note that this is a countable direct sum of locally compact abelian groups (cf. [21, Section 7, Remarks after 1st Deﬁnition]). A typical

35 element is of the form ((nx.x)x, (tC,ν)(C,ν)), where the indices x, C, ν run over the set of closed points, of curves contained in X and valuations in C∞, respectively, and where at most ﬁnitely many components are non-zero.

Remark 3.1.3. If X is an arithmetical scheme with id`elegroup JX , then JX is

Hausdorﬀ but not necessarily locally compact. If C ⊂ X is a horizontal curve

arch on X, let C∞ denote the set of archimedean places. If C ⊂ X is vertical, set

arch C∞ = ∅. Then the subgroup

1 M M × JX = K(C)ν arch C⊂X ν∈C∞ is the connected componenet of the identity.

Deﬁnition 3.1.4. Let f : X −→ Y be a morphism of arithmetical schemes. We deﬁne the induced morphism fJ : JX −→ JY on ideal class groups as follows:

1) The image y = f(x) of a closed point x ∈ X is a closed point, and the extension of residue ﬁelds k(x)/k(y) is ﬁnite. So let fJ (1.x) = [k(x): k(y)] . y.

2) For C ⊂ X a curve, we either have f(C) = y a closed point, or f(C) is dense in a curve D ⊂ Y .

2a) In the first case, since closed points of integral varieties of finite type over Z have finite residue field, C is a variety over some Fpn = k(y). For ν a valuation on

K(C), let k(ν) = Oν/mν denote the residue ﬁeld, a ﬁnite extension of k(y). Now f

× gives rise to an embedding k(y) ,→ k(ν), so for t ∈ K(C)ν , we can deﬁne the image under fJ as fJ (t) = ν(t)[k(ν): k(y)].y.

36 2b) In the second case: Let D be the closure of f(C) in Y , then f gives rise to the ﬁnite extension K(C)/K(D), and ν restricts to a valuation ω on K(D). If ω

× does not have a center in De, then JY has a factor K(D)ω , so and we can deﬁne

fJ : K(C)ν −→ K(D)ω by the norm NK(C)ν /K(D)ω .

2c) Lastly, if ω has a center z on De, then JY does not contain the summand

× K(D)ω , but instead the discrete summand Z.z, so we deﬁne

× fJ (t) = ν(t)[k(ν): k(z)] . z for all t ∈ K(C)ν .

Finally, define fJ : JX −→ JY by setting the image of fJ equal to the sum of the images of the components, as they were defined above. Then J is a continuous homomorphism by definition.

Remark 3.1.5. The identity morphism X −→ X induces the identity JX −→ JX .

Moreover, the composition of two induced morphisms is the induced morphism of the composition. Thus we get a functor J from the category of arithmetical schemes over Z to the category of Wiesend id`elegroups with induced morphism fJ .

We shall use the functorial properties of the induced morphisms repeatedly.

Among other things, they imply that if the diagram

f 2 C / D

f 1 f 3 X / Y f 4

37 of schemes over S is commutative, then so is the diagram

2 fJ JC / JD

1 3 fJ fJ / JX 4 JY fJ of induced morphisms on the id`elegroups.

For a curve C ⊂ X with normalisaton Ce, the composition iC of the canonical maps

Ce −→ C −→ X thus induces a morphism

(i ) : J −→ J (3.1.1) C J Ce X

There also exists a natural inclusion map j : K(C)× −→ J given by component- C Ce wise inclusion:

t 7→ ((νx(t). x), (iν(t))ν) ,

where νx is the discrete valuation associated to the closed point x ∈ C, and iν : K(C) −→ K(C)ν is the inclusion of the function ﬁeld in its completion at

ν. Composing jC with iC∗, and taking the direct sum over all curves C ⊂ X, we get a map

X × j = ΣC (iC∗ ◦ jC ): K(C) ,→ JX . (3.1.2) C⊂X

Deﬁnition 3.1.6. Let X be an arithmetical scheme of Kronecker dimension dim(X) ≥ 2, or a regular curve. Then the Wiesend id`eleclass group of X is

38 deﬁned to be the quotient of JX by the image of j:   M M M × X CX = JX / im(j) =  Z.x ⊕ K(C)ν  / im(iC∗ ◦ jC ) x∈|X| C⊂X ν∈C∞ C⊂X together with the quotient topology. For an arithmetical scheme of dimension zero, i.e. a point x, we deﬁne analogously Cx = Jx = Z.x .

Remark 3.1.7. For an arithmetical scheme X with class group CX , let DX denote the connected component of the identity. Then DX is equal to the closure of the

1 image of the subgroup JX in CX deﬁned in 3.1.3 (also see [21, Section 7]).

Proposition 3.1.8. For a morphism f : X −→ Y , the induced map fJ on the ideal class groups descends to a continuous homomorphism f∗ : CX −→ CY on the class groups.

Proof. To show this, we have to prove that im(j) is contained in the kernel of the canonical map f J : JX −→ JY −→ CY . Since im(j) = ΣC⊂X im(iC∗ ◦ jC ), this

× amounts to showing that each (iC∗ ◦ jC )(K(C) ) is is contained in the kernel of f J .

There are several cases to consider:

1) f(C) = y is a closed point of Y . Then for t ∈ K(C)×, we have

(f | ◦ j )(t) = f | (((i (t)) , (ν (t)) ) (3.1.3) J C C J C ν ν∈C∞ x x∈Ce X X = ν(t)[k(ν): k(y)].y + νx(t)[k(x): k(y)].y ν∈C∞ x∈Ce X = ( ν(t)[k(ν): k(y)]).y

ν∈VK(C)

= (deg(div(t)).y = 0 (3.1.4)

39 where VK(C) denotes the set of places on K(C). We obtain a canonical diagram

j i ∗ × C / J C / K(C) Ce JX

(f|C )J fJ can. 0 / Z.y / JY , where the ﬁrst square commutes by the above computation, and the second square commutes as the middle vertical map is just f ◦ i restricted to J . Thus, the J C∗ Ce

× outer rectangle also commutes, and the image of K(C) is in the kernel of f J , as required.

2) f(C) is dense inside a curve D of Y . There exists a unique fe : Ce −→ De induced by f such that the following diagrams commute:

Ce / C / X

∃!h f|C f De / D / Y. × × Since the maps jC : K(C) −→ JC and jD : K(D) −→ JD factor through

J −→ J and J −→ J , respectively, we get diagrams Ce C De D

j × C / J / / K(C) Ce JC JX

NK(C)/K(D) hJ (f|C )J fJ j × D / J / / K(D) De JD JY . We show that the left square is commutative:

Indeed, we show that hJ ◦ jC = jD ◦ NK(C)/K(D) agree on every direct summand of J . We have a direct summand for every discrete valuation ω on K(D). If ω De does not have a center on D, then the ω-summand is K(D)×, and if ω = ν for some e ω ye

40 center y ∈ D, then the ω-summand is .y. Let p and p denote the corresponding e e Z e ω ye projections.

× × Recall that for a valuation ν ∈ VK(C) iν : K(C) −→ K(C)ν denotes the

× × × canonical inclusion t ∈ K(C) . For t ∈ K(C) , we let s = NK(C)/K(D)(t) ∈ K(D) , then

j (t) = ((ν (t)) ), (i (t)) ) , C x x∈|Ce| ν ν∈C∞

j (s) = ((ν (s)) ), (i (s)) ) . D y y∈|De| ω ω∈D∞

1) First consider the case of an ω-summand with ω ∈ D∞. First note that a ν-summand gets mapped into the ω-summand iﬀ ν is a valuation in C∞ and

ν|K(D) = ω. As ω does not have a center on De, any ν on K(C) restricting to ω on

K(D) does not have a center on Ce. In particular, any valuation restricting to ω is automatically already in C∞.

× Furthermore, hJ restricted to K(C)ν is the local norm

× × × NK(C)ν /K(D)ω : K(C)ν −→ K(D)ω . Thus in K(D)ω , we have

(pω ◦ jD ◦ NK(C)/K(D)(t) = (pω ◦ jD)(NK(C)/K(D)(t))

= iω(NK(C)/K(D)(t))

= NK(C)/K(D)(t)

× for all t ∈ K(C)ν , and

X X (pω ◦ hJ ◦ jC )(t) = hJ ((iν(t)) = NK(C)ν /K(D)ω (t)

ν:ν|K(D)=ω ν:ν|K(D)=ω

41 × for all t ∈ K(C)ν . Therefore, showing that hJ ◦jC and jD ◦NK(C)/K(D) agree on

K(D)ω amounts to the well-known fact that global norms are the product of local norms (for a proof, see e.g. [12, Ch. III]):

X × NK(C)/K(D)(t) = NK(C)ν /K(D)ω (t) for all t ∈ K(C)ν .

ν:ν|K(D)=ω 2) Now consider ω-summands for ω = ν with center y ∈ D. Then K(C)× maps ye e e ν into the ω-summand Z.ye iﬀ we have ν|K(D) = ω. Furthermore, we get a contribution from the discrete xe-summand Z.xe if and only if fe(xe) = ye, which is equivalent to the condition ν | = ν . We have xe K(D) ye

(p ◦ j ◦ N )(t) = (p ◦ j )(N (t)) = v (N (t)) . y ω D K(C)/K(D ye D K(C)/K(D) ye K(C)/K(D) e

and

X X (p ◦ F ◦ j )(t) = ν(t)[k(ν): k(y)] . y + ν (t)[k(x): k(y)] . y ye e D e e ye e e e ν∈C∞ : ν| =ν x:f(x)=y K(D) ye e e e   X X = ν(t)[k(ν): k(y)] + ν (t)[k(x): k(y)] . y  e ye e e  e ν∈C∞ : ν|K(D)=νy ν : x∈C, ν | =ν e xe e e xe K(D) ye   X =  ν(t)[k(ν): k(ye)] . ye ν : ν| =ν K(D) ye

Thus, showing that Fe◦jC and jD ◦NK(C)/K(D) agree on the ye-summand amounts to showing that   X v (N (t)) = ν(t)[k(ν): k(y)] ye K(C)/K(D)  e  ν : ν| =ν K(D) ye

42 which is again a fact of basic number theory of global ﬁelds.

Deﬁnition 3.1.9. Let f : X −→ Y be a morphism of regular arithmetical schemes, and let f∗ : π1(X) −→ π1(Y ) be the induced morphism on fundamental groups.

Then we denote the image subgroup f∗(CX ) inside CY by NCX , the and call it the f-norm subgroup.

We shall also make use of the following proposition:

Proposition 3.1.10. Let X be a regular arithmetical scheme, and let JX , CX denote the Wiesend id`elegroup, respectively the Wiesend id`eleclass group of X.

Then J is generated by the images of J under the canonical maps i , and C X Ce C∗ X is generated by the images of C under i . Ce C∗

Proof. The proof easily reduces to the following

Claim 3.1.11. If X is a regular arithmetical scheme, then for every point x ∈ X, there exists a curve C ⊂ X such that x is a regular point on C.

. Indeed, let x be a point of X. Since being regular is an open condition, by shrinking X if necessary, we may reduce to a regular open aﬃne U = Spec A contained in X, where x corresponds to a prime ideal p ⊂ A. Let Ap = OX,x be the local ring at x. As x is a regular point of X, we can choose (t1, . . . , td), all elements of Ap, such that they form a system of local parameters at x. (Replacing ti by a multiple by an element of A − p if necessary, we may assume that ti ∈ A for all i.)

By [9, Theorem 2.5.15], B := Ap/(t1, . . . , td−1) is regular, local and of dimension 1,

43 and therefore integral; the closure of its generic point in Spec A is thus an integral curve C.

2 Now x is a regular point of Spec B iﬀ tr 6∈ (t1, . . . , tr−1) in Ap (cf. [9, 4.2.15]).

Letting mp = (td) denote the image of mp under the natural surjection A −→ B,

2 then if td were contained in (t1, . . . , td) , {t1, . . . , td} would not be an independent

2 set of generators of mp modulo mp , i.e. not a coordinate system of Ap. Thus x is a regular point of Spec B.

3.2 The Reciprocity Homomorphism

ab In this section, we show the existence of a homomorphism ρX : CX −→ π1 (X), which will be called the reciprocity homomorphism. This morphism is a generalisation of the Artin map from classical class ﬁeld theory: For X a regular curve, ρX will coincide with the Artin map as deﬁned in [12, Section VI.7]. The details of this are shown in the next section.

We shall construct the homomorphism ρX by ﬁrst deﬁning a map ψ at the level

ab of the id`elegroup: ψ : JX −→ π1 (X), and then show that ψ factors through CX .

We deﬁne as follows:

1) For a closed point x ∈ X, set ψ(1.x) = F robx, the Frobenius element of

ab π1 (X) associated to x.

2) For a curve C ⊂ X, recall that Ce denotes the normalisation of C, and iC the canonical homomorphism iC : Ce −→ X (cf. 2.2.13). Also recall from 3.1.1 that

44 (i ) : J −→ J denotes the induced morphism on id`elegroups . C J Je X

If ν a valuation of K(C) without a center on Ce, we let ψ = (iC )J ◦ ρC , where

ρ : J −→ πab(C) is the reciprocity homomorphism associated to the normal C Ce 1 e curve Ce (cf. [12, Ch. ]).

P × Then ψ is trivial on the image of j : C K(C) −→ JX , so it descends to a

ab homomorphism ρX : CX −→ π1 (X).

Deﬁnition 3.2.1. Let X be a regular arithmetical scheme. Then the continuous

ab homomorphism ρX : CX −→ π1 (X) deﬁned above is called the reciprocity homomorphism of X.

Let f : X −→ Y , and recall from Section 2.2 that fe denotes the induced morphism on fundamental groups. Recall also that the abelianised fundamental

ab ab groups π1 (X), π1 (Y ) are well-deﬁned without reference to geometric basepoints.

¯ ab ab Notation 3.2.2. Given f, fe as above, we let f : π1 (X) −→ π1 (Y ) denote the induced morphism on the abelianised fundamental groups.

Proposition 3.2.3. The recprocity morphism is compatible with induced homo- morphisms on the class and fundamental groups. I.e. if we let f : X −→ Y is a morphism of regular arithmetical schemes, then the diagram

f∗ CX / CY

ρX ρY f¯ ab / ab π1 (X) π1 (Y ) commutes.

45 Proof. It is most convenient to show this at the level of the id`elegroups, i.e. to show that the diagram

fJ JX / JY

ρX ρY f¯ ab / ab π1 (X) π1 (Y ) commutes. We then check commutativity componentwise on every summand CX as follows:

1) If x is a closed point, then so is y = f(x). Then k(y) ' Fq and k(x) ' Fqn

ab are ﬁnite ﬁelds, π1 (x) = π1(x)), Cx = Z. x and similarly for y. The reciprocity morphisms are

ρx : Z. x −→ π1(x)

ρy : Z. y −→ π1(y) .

ab Here, ρx : Z. x −→ π1 (x) maps 1. x 7→ F robx, where F robx = F robFqn denotes the

Frobenius associated to the ﬁnite ﬁeld k(x). Similarly, F roby = F robFq , so we have

n F robx = F roby in π1(y); thus the upper square in the following diagram commutes.

Z.x / Z.y 1.x / n.y

n π1(x) / π1(y) F robx / F robx = F roby

ab / ab / n π1 (X) π1 (Y ) F robx F robx = F roby

By a slight abuse of notation, we have let F roby also denote the image of F roby

ab under the canonical homomorphism π1(x) −→ π1 (X). We shall keep this notation

46 also in the following paragraphs. Commutativity of the lower square follows from

ab functoriality of π1 and π1 (cf. Section 2.2 or [19, Ch. 5]).

2) Now let C ⊂ X be a curve, and recall the canonical morphism iC : Ce −→ C −→ X (cf. 2.2.13). Recall also the notation for induced morphisms on the idèleand idèleclass groups (cf. Definition 3.1.4, Proposition 3.1.8 and (3.1.1)).

2.1) f(C) is a closed point y. Then for any valuation ν ∈ C∞, the induced map P fJ on the id`elegroups maps K(C)ν ⊂ JX into the y-summand Z.y ⊂ JY . By the functorial properties of induced maps on the abelianised fundamental group, the square

(f| ) ab C ∗ / π1 (Ce) π1(y)

iC∗ iy f ab ∗ / ab π1 (X) π1 (Y ) commutes. Moreover, by the definition of the Artin map on idèleclass group with restricted ramification (cf. [12, Section]), the square

× / K(C)ν Z.y

ρ (.) C ρy=F roby (f| ) ab C ∗ / π1 (Ce) π1(y) commutes for any valuation ν.

2.2) f(C) is dense inside a curve D ⊂ Y . Then the associated function ﬁeld

K(C) is an extension of K(D), and ω := ν|K(D) is a valuation of K(D) for all valuations ν on K(C). We distinguish two cases:

47 2.2.1) If ω does not have a center on De, then ω ∈ D∞, and fJ restricted to

× × × K(C)ν is given by the norm N = NK(C)ν /K(D)ω : K(C) −→ K(D)ω . We show that the following diagram commutes:

× N / × K(C)ν K(D)ω

ρC ρD (f| ) ab C ∗ / ab π1 (C) π1 (D)

n × Indeed, let t = π u be an element of K(C)ν and let f = [k(y): k(x)]. Then

0 fn 0 fn N(t) = u π , where u is a unit of OK(D), so ρD(N(t)) = F robk(ω), where F robk(ω) is the generator of Gal(Fp/k(ω)) inside Gal(Fp/Fp).

n ab ab On the other hand, ρC (t) = F robk(ω). Viewing π1 (C) and π1 (D) as the Galois groups of the maximal unramiﬁed outside C∞ (resp. D∞) extension of K(C) (resp.

K(D)), (f|C )∗ is just the restriction map to K(C) (resp. K(D)), under which

f f F robk(ν) gets mapped to F robk(ω). So (f|C )∗(F robk(ν)) = F robk(ω) and the above diagram commutes, as claimed.

2.2.2) Now assume that ω has a center on the normalisation De, say y. Then ω is the valuation νy associated to y, and k(ν) is a ﬁnite extension of k(ω) = k(y).

We need to show commutativity of

f × J / K(C)ν Z.y

ρC ρD (f| ) ab C ∗ / π1 (C) π1(y) .

48 × n Indeed, let π be a uniformiser of K(C)ν. For t ∈ K(C) , write t = uπ , where u is a unit of K(C)ν. We have fJ (t) = ν(t)[k(ν): k(y)]. y, which gets mapped to

n[k(ν):k(y)] F roby in π1(y).

n [k(ν):k(y)] On the other hand, we have ρC (t) = F robk(ν) , so since F robk(ν) = F roby , the diagram commutes as claimed.

3.3 The Base Cases of the Induction Argument:

Reciprocity for Regular Schemes of Dimen-

sion One and Zero

In this section, we summarise the reciprocity laws for arithmetical schemes of di- mensons one and zero. These will form the base case for induction on dimension in the proof of the Key Lemma in Chapter 5.

In dimension one, this amounts to a reformulation of results of classical class

field theory with restricted ramification in the language of schemes, which we give below. We begin with the dimension zero case, where reciprocity is essentially clear from the definitions.

An arithmetical scheme of dimension zero is equal to Spec k for some ﬁnite ﬁeld k. If x is a closed point in some higher-dimensional arithemtic scheme X, then

49 the residue field k(x) is a finite field, and x ' Spec k(x). The fundamental group of this scheme is abelian, and isomorphic to Zb, the procyclic group generated by the Frobenius automorphism F robx of the finite field k(x). The class group of x is equal to Cx = Jx = Z. x, and the reciprocity morphism ρx : Cx −→ π1(x) maps

1. x 7→ F robx. We get the following

Proposition 3.3.1. Let x be a zero-dimensional arithmetical scheme. Then ρx induces an exact sequence

ρx ˆ 0 −→ Cx −→ π1(x) −→ Z/Z −→ 0 .

The following corollary summarises the key statements for a zero-dimensional class ﬁeld theory:

Corollary 3.3.2. Let x be an arithmetical scheme of dimension zero and Cx ' Z x the id`eleclass group of X.

1. There exists a one-to-one correspondence between open subgroups of Cx and

−1 open subgroups N of the fundamental group π1(x); it is given by ρx (N) ← N.

The inverse correspondence is given by H 7→ ρx(H), where the bar denotes

the topological closure. If H < Cx corresponds to N < π1(x), then

CC /H ' π1(x)/N .

2. The norm subgroups of Cx are precisely the open subgroups of ﬁnite index.

50 3. If y −→ x is an ´etaleconnected cover, then the reciprocity map gives rise to

an isomorphism

' Cx/NCy −→ Gal(y/x).

Proof. The claims follow directly from Proposition 3.3.1; we show the details of

3.3.2.1: Open subgroups of Cx are of the form nZ. x for non-negative integers n,

n and open subgroups of π1(x) are of the form < F robx >, where the bar denotes the topological closure. Then clearly, the association

n −1 N = < F robx > 7→ ρx (N) = nZ. x

gives a one-to-one correspondence between open subgroups of Cx and π1(x).

The inverse correspondence is given by associating to a subgroup H = nZ. x ≤

n Cx the topological closure of the image subgroup: N = ρx(H) = < F robx >.

Now let C be a regular arithmetical curve. Then the required results are given by the classical class field theory of (number or function) field extensions with restricted ramification, as derived in [13, Chapter VIII].

There are two distinct cases: Either C is a curve over Fp for some prime p, or

C is open in Spec OK for some number ﬁeld K.

We first consider the case where C is a regular curve over Fp, which is the main focus of this thesis. Then K(C) is a function field, i.e. a finite extension of Fp(t).

C is either proper or aﬃne. Let P (C) be the regular compactiﬁcation of C, and let S = P (C)\C be the complement of C. Then S is non-empty if and only if C is

51 aﬃne. The elements of S are closed points, i.e. of codimension one in C. We can identify S with the set of valuations without a center on C:

Proposition 3.3.3. There exists a contravariant equivalence of categories of connected étalecovers of C to finite, unramified-outside-S extensions of K(C).

Proof. We give the contravariant functor which deﬁnes the equivalence of categories:

It is given by associating to a connected étalecover Y −→ C its function field K(Y ), which is unramified over C by definition.

The inverse functor is given by associating to a ﬁnite extension L of K(C) the normalisation CL of C in L. Indeed, the normalisation morphism CL −→ C is

ﬁnite, and if L is unramiﬁed at the primes contained in C, then CL −→ C will be

´etale.

Remark 3.3.4. Now recall that the the maximal unramified-outside-of-S extension of K(C) is defined as the union of all finite field extensions of K(C) which are unramified at primes not in S. Let it be denoted by KS, then Proposition 3.3.3 implies that π1(C) ' Gal(KS/K(C))

Remark 3.3.5. As before, the choice of a geometric basepoint for π1(C) amounts to

ﬁxing a separable closure of K(C).

× We have JC = ⊕Z.x ⊕ν∈S K(C)ν , and thus CC ' CS(K) is the S-id`eleclass group of K(C) as deﬁned in [13, VIII.3]. Then [13, Thm. 8.3.14] yields the following:

Theorem 3.3.6. If C is a regular curve over a ﬁnite ﬁeld, then there exists a

52 canonical exact sequence

ρ C ab ˆ 0 −→ CC −→ π1 (C) −→ Z/Z −→ 0 .

0 ab,0 If CC and π1 (C) denote the degree-0 parts of the class and fundemental group, respectively, then we obtain surjectivity of the reciprocity morphism restricted to

0 CC , and thus an isomorphism

0 ab,0 CC ' π1 (C) .

In the ﬂat case, Theorem [13, 8.3.12] implies the following:

Theorem 3.3.7. If C is a ﬂat arithmetical curve, let DC denote the connected component of the identity in the Wiesend id`eleclass group CC . Then there exists a canonical exact sequence

ρ C ab 0 −→ DC −→ CC −→ π1 (C) −→ 0 .

Corollary 3.3.8. Let C be a regular arithmetical curve.

1. There exists a one-to-one correspondence between open subgroups of CC and

ab −1 open subgroups N of π1 (C); it is given by ρC (N) ← N. The inverse cor-

respondence is given by H 7→ ρC (H), where the bar denotes the topological

closure.

ab If H < CC corresponds to N < π1 (C), then

ab CC /H ' π1 (C)/N .

53 2. The norm subgroups of CC are precisely the open subgroups of ﬁnite index.

3. If D −→ C is an ´etaleconnected cover, D0 −→ C the maximal abelian sub-

cover, then NCD = NCD0 and the reciprocity map gives rise to an isomor-

phism

' 0 CC /NCD −→ Gal(D /C).

Proof. In the flat case, the first statement is immediate from the surjection of the reciprocity homomorphism; here, taking the topological closure is a trivial opera- tion. In the function field case, the first statement follows directly from the exact sequence and the topology of Zˆ/Z.

For the second statement, we note that for an ´etalecover f : D −→ C of regular curves, the induced morphism f∗ : CD −→ CC is identical to the natural morphism CS(K(D)) −→ CS(K(C)) of S-class groups which is induced by the norm of the classical id`eleclass group, NK(D)/K(C) : IK(D) −→ IK(C). Then, Corollaries

[13, 8.3.13] and [13, 8.3.15] show that the norm subgroups are precisely the open subgroups of ﬁnite index.

Lastly, we observe that the norms subgroup NCD correspond to a unique abelian

´etalecover of C by 3.3.8.2, which proves the third statement.

54 Chapter 4

From the Main Theorem to the

Key Lemma

In this chapter, we introduce the main result of this thesis, the Main Theorem

4.3.4, which will yield the desired generalisation of known reciprocity laws to all

n open subvarieties of Pk .

We begin begin by introducing induced covering data, the link between covering data considerations and subgroups of the Wiesend id`eleclass group. We show that any open subgroup H < CX of ﬁnite index gives rise to an induced covering datum

DH , and call H finitely realisable if DH is effective with a finite realisaion. We obtain a one-to-one correspondence between finitely realisable subgroups and open subgroups of the fundamental group.

n The central result is Theorem 4.3.4, which concerns open subvarieties X of Pk .

55 It identifies the realisable open subgroups of finite index CX as those which are geometrically bounded (cf. Definition 4.2.4). As a corollary, we obtain a one-to-one correspondence between open, geometrically bounded subgroups of finite index and open subgroups of the fundamental group, as well as a description of the kernel

ab and image of the reciprocity morphism ρX : CX −→ π1 (X). We also obtain a description of the norm subgroups of CX .

In the last section, we show that the proof of the Main Theorem for an open

n subvariety X ⊂ Pk can be reduced to proving the Key Lemma. The Key Lemma

n for open subvarieties X ⊂ Pk will be shown in Chapters 5 and 6.

4.1 Introducing induced covering data

In this section, we provide the link between covering data considerations and subgroups of the Wiesend idèleclass group: Any open subgroup of finite index H < CX gives rise to an induced covering datum DH . We give a proof that if the induced covering datum DH is effective with realisation XN , the image of N in the abelianised fundamental group is the image of the subgroup H.

Construction 4.1.1. Let X be a regular arithmetical scheme, and let H < CX

−1 be an open subgroup of the Wiesend id`eleclass group. Let HC := iC∗(H) and

H := i−1(H) denote the preimages in C and C ' .x, respectively. By the x x∗ Ce x Z reciprocity law of the base cases (cf. Section 3.3), the images ρ (H ) = N and Ce C C

ab ρx(Hx) = Nx are open subgroups of π1 (Ce), respectively π1(x). Let NC denote the

56 ab subgroup of π1(Ce) corresponding to N C ≤ π1 (Ce).

Deﬁnition 4.1.2. Let X be a regular arithmetical scheme, and let H < CX be an open subgroup of the Wiesend id`eleclass group. Then the induced covering datum

DH is the covering datum DH := (NC ,Nx)C,x deﬁned by the subgroups NC , Nx deﬁned in the construction above.

Let x be a point on the curve C ⊂ X. To show that the induced datum is a covering datum, we must prove that DH satisﬁes the compatibility condition for a covering datum at (x, C).

If xe is a point of x × Ce, then the commutative square

/ xe Ce

x / C induces commutative squares at the level of class groups and fundamental groups:

C / C π (x) / xe Ce 1 e π1(Ce)

Cx / CC π1(x) / π1(C) .

Let H denote the preimage of H in C . Then H is the preimage of H and of xe xe xe C

Hx under the natural morphisms above. By the functoriality of the reciprocity morphism, see the diagram below. Thus, ρ (H ) = N is actually equal to the xe xe xe preimage of Nx and NC in π1(xe). In particular, these two concide whenever x is a point on C, and the compatibility condition for a covering datum is satisﬁed.

57 C o C / C x xe Ce

ρx ρ ρ xe Ce o / π1(x) π1(xe) π1(Ce) .

Thus, the datum DH deﬁnes a covering datum on X.

Deﬁnition 4.1.3. If X is an arithmetical scheme, H < CX an open subgroup, the covering datum DH deﬁned above is called the covering datum induced by H.

Remark 4.1.4. Let X be an arithmetical scheme, and recall the deﬁnititions made in 2.3.3.

1. If H < CX of ﬁnite index, then induced covering datum DH is of index bounded

by m.

2. If H < CX is such that CX /H is cyclic, then DH is a of cyclic index.

In the following, if N < π1(X), we again denote by N its image in the abeliani-

ab sation π1 (X).

Proposition 4.1.5. Let X be a regular arithmetical scheme, and let H < CX be an open subgroup of the ideal class group. If the induced covering datum DH is eﬀective

−1 with realisation YN , then ρX (N) = H, and reciprocity induces an isomorphism

ab CX /H ' π1 (X)/N.

58 0 0 0 Proof. Let HC be the preimage of N in CX , and HC , Hx the preimages of H in

π1(Ce), π1(x) in the commutative diagram

ab / ab π (Ce) π1 (X) 1 O O ρ Ce ρX i C C∗ / Ce CX where the arrows in the top rows are induced by functoriality of the fundamental group descended to the abelianisation. By the class ﬁeld theory for arithmetic curves, we then have H0 = H for all curves C ⊂ X. Since the images of C C C Ce

0 generate CX , this implies that H = H .

To prove the second part of the Proposition, let C ⊂ X be any curve and recall again the canonical induced morphism eiC : π1(Ce) −→ π1(X). If N / π1(X) is an

−1 open subgroup, let NC := eiC (N) denote its preimage in π1(Ce). By Lemma 2.1.14, there exists a curve C ⊂ X such that the injection π1(Ce)/NC ,→ π1(X)/N is an isomorphism. Let N N C denote the images of N, respectively NC in the abelianised

−1 fundamental groups, and recall from above that ρX (N) = H. Then we obtain a diagram of injections

ρ X ab CX /H / π (N) O 1 O ' ? ρ C /H Ce / ab Ce C π1 (Ce)/N C

From classical class ﬁeld theory for curves (cf. Corollary 3.3.8), we have that

59 the Artin map ρ induces an isomorphism Ce

ρ : C /H ' πab(C) . Ce Ce C 1 e

Thus, all the arrows in the diagram must be isomorphisms, as claimed.

4.2 Geometrically bounded covering data

In this section, we define the notion of a bounded covering datum on an arithmetical variety X over the finite field k. The condition for boundedness shall make use of the degree of a curve; for this end, we shall first set the discussion in projective space:

Notations/Remark 4.2.1. If not otherwise explicitly stated, we will from now work in the following context:

i 1. We consider U ,→ X, an open, dense quasi-projective and regular subset

n n n (which exists by Fact 2.1.10), say U ⊂ Pk . Let Ak ⊂ Pk be the canonical

n embedding. Then replacing U by U ∩ Ak if necessary, we may assume that

n n n n U,→ Ak ⊂ Pk . Let U ⊂ Pk be the closure of U in Pk .

2. We consider only those curves C ⊂ U which are regular curves on U, i.e.

n curves such that the regular compactiﬁcation P (C) is contained in U ⊂ Pk .

(Note that since k is perfect, regularity and smoothness are equivalent condi-

n tions for curves over k.) Then P (C) = C, the topological closure of C in Pk .

60 3. Note that for curves C ⊂ U as above, the arithmetical genus ga(P (C)) is the

the same as the geometric genus gC of P (C).

4. Given a positive integer d, we also let g(n, d) denote the integer from

Fact 2.1.9.

5. In the above context, let D0 be a covering datum on X, and consider the

∗ 0 pullback D := i (D ), a covering datum on U. For every NC ∈ D, consider the

D D ´etalecover fC : YC −→ C, thus deg fC = [π1(C): NC ]. Notice that YC −→ C

is smooth, and let g D denote the geometric genus of (the completion of) C. YC

Let k be the ﬁeld of constants of Y D. Finally, let deg R denote the C C kC C

degree of the ramiﬁcation divisor of the ramiﬁcation locus of fC over kC .

Then Hurwitz’s formula gives (cf. [2, Cor. IV.2.4] and [9, Section. 7.4]):

1 g = (deg f )(g − 1) + 1 + (deg R ) (4.2.1) YC C C 2 kC C 1 = [π (C): N ](g − 1) + 1 + (deg R ) 1 C C 2 kC C

D D where RC is the branch locus of fC inside the compactiﬁcation Y C of YC .

6. Recall also that a covering datum D on U is of bounded index if there exists

a positive integer m such that [π1(Ce): NC ] ≤ m for all curves C ⊂ X (see

D Deﬁnition 2.3.3). If this is the case, the maximal genus of a cover YC −→ C

in the covering datum is solely determined by deg R , the degree of the kC C

ramification divisors RC over the field of definition kC of fC :

1 g ≤ m(g − 1) + 1 + (deg R ) YC C 2 kC C

61 i n Deﬁnition 4.2.2. Let U ,→ Pk be a quasi-projective arithmetical variety, D a covering datum on U. Then D is called geometrically bounded if for every positive integer d, there exists a positive integer δ = δ(d) such that the following holds:

For all curves C as in Notation 4.2.1 that satisfy deg C ≤ d, the cover kC

D Y −→ C deﬁned by D satisﬁes g D ≤ δ(d) C YC

Remark 4.2.3. Contrary to the deﬁnition of covering data of bounded index, this

n is a condition only involving the regular curves C in U with C ⊂ U ⊂ Pk regular; no assumption is made about the behaviour of the covering datum at curves with non-regular compactiﬁcations in U.

Deﬁnition 4.2.4. Let X be an arithmetical variety, and let H < CX be an open subgroup of the Wiesend id`eleclass group. We say that the subgroup H is geometrically bounded if the induced covering datum DH on X is geometrically bounded.

Recall from 4.2.1 that we let C denote the (regular) closure of C in U, and that

C is open in C. Since fC is ´etaleover C, the complement C\C = {x1, . . . , xn} contains the branch locus RC of fC .

The degree of RC can then be given in terms of the higher ramiﬁcation groups

i above the points xj: If Gy is the i-th ramiﬁcation group at y, a point lying over some xj, we have

n ∞ ! X X X i deg RC = |Gy| − 1 [k(y): kC ] . j=1 −1 i=0 y∈fC (xj )

62 D Deﬁnition 4.2.5. In notations as in (4.2.1), let YC −→ C be a cover of regular

D curves, and let y ∈ Y C be a point. The ramiﬁcation number my of y is deﬁned to

my be the biggest integer my such that the my-th ramiﬁcation group G is non-zero.

D −1 If YC −→ C is assumed to be Galois, we have my = mxj for all y ∈ fC (xj).

i P∞ i Then since Gy is a subgroup of GC for all j and i, the sum i=0 | Gy| − 1 is bounded above by mj · (|GC | − 1). Note also that the set SC given by

−1 SC = {y : y ∈ fC (xj) for some j} is ﬁnite and contains the ramiﬁcation locus.

So let M = maxy∈SC [k(y): k], and m = maxj mxj , then

deg R ≤ n · m · M · (|G | − 1) . (4.2.2) kC C C

Here, n = n(C) is the number of points in C\C, a constant depending on the

n curve C once the embedding i : U,→ Pk is ﬁxed. Similarly, we have M = M(C) is a constant depending only on C once i is ﬁxed.

The proposition below shows that for all regular curves, we may replace n(C),

M(C) by upper bounds which only depends on the degree of the curve.

n Proposition 4.2.6. Let U ⊂ Pk be quasi-projective arithmetical variety over the

ﬁnite ﬁeld k and let D be a covering datum on U. For every positive integer d, there exist integers N = N(d), M = M(d) such that

1. n(C) ≤ N(d) whenever C ⊂ U is a regular curve degk C ≤ d.

2. M(C) ≤ M(d) whenever C ⊂ U is a regular curve degk C ≤ d.

63 Proof. 1. We have C\C = U\U ∩ C = {x1, . . . , xn(C)}. Then since by inter-

section theory,

n(C) X degk U\U · degk C = multj(xj)[k(xj): k] ≥ n(C) (4.2.3) j=1

In particular, for any regular curve C ⊂ U of degree degk C ≤ d, we have

n(C) ≤ degk C · deg U\U ≤ d · degk U\U.

2. For all y ∈ SC , we have

[k(y): kC ] = [k(y): k(xj)] · [k(xj): k] ≤ deg fC · [k(xj): k]

for some xj ∈ C\C. Noting that

[k(xj): k] ≤ degk C\C ≤ degk U\U · degk C ≤ d · degk U\U,

for all xj ∈ C\C, we get

M(C) ≤ d · deg fC · degk U\U, (4.2.4)

as claimed.

n Deﬁnition 4.2.7. Let U ⊂ Pk be a quasi-projective arithmetical variety over the

finite field k, and let D be a covering datum on U. Let C ⊂ U be a regular curve whose regular compactification P (C) is contained in U (cf. Notations 4.2.1). Recall

D that kC denotes the ﬁeld of constants of YC −→ C, thus a ﬁnite extension of k.

64 Suppose that C has degree deg C ≤ d. The set of ramiﬁcation numbers of D is kC

D deﬁned to be the set of the ramiﬁcation numbers m(y,C) of points y ∈ Y C above points of C\C.

The ramiﬁcation numbers of D are uniformly bounded if there exists a positive integer M such that m(y,C) ≤ M for all pairs (C, y) of curves C ⊂ U and closed

D D points y ∈ Y C \YC .

Then we get the following:

Proposition 4.2.8. Let D be covering datum of bounded index h on a quasi- projective arithmetical scheme U. Then D is geometrically bounded if and only if the ramiﬁcation numbers of D are uniformly bounded.

Proof. Let C ⊂ U be a regular curve of degree deg ≤ d, and let Y D −→ C be the kC C cover induced by D. Putting together Equation (4.2.2) and Proposition 4.2.6, we have

deg R ≤ N(d) · m · M(d) · (|G | − 1) ≤ N(d) · m · M(d) · (h − 1) (4.2.5) kC C C C C

where mC = maxy {mC,y} is the maximum of all the ramiﬁcation numbers of points above C\C. Clearly, if the ramiﬁcation numbers of D are not uniformly bounded,

D n→∞ there exist curves Cn with at least one point yn ∈ YCn such that m(Cn,yn) −→ ∞ .

Then

1 n→∞ gY D = deg fC (gC − 1) + 1 + degk RC −→ ∞ . Cn 2 C

65 Conversely, if there exists m such that m(C,y) ≤ m for all pairs of curves C and

D closed points y lying above C\C in YC −→ C, then

deg R ≤ N(d) · m · M(d) · (|G | − 1) . kC C C and thus

1 gY D = deg fC (gC − 1) + 1 + degk RC Cn 2 C N(d) m M(d)(h − 1) ≤ h(g − 1) + 1 + C 2 whenever C is a curve with deg C ≤ d. kC

n Now consider a Galois ´etalecover Y −→ U. As above, let U ⊂ Pk denote the closure of U in projective space, and C the closure of a regular curve C ⊂ U.

N If Y = UN corresponds to the open normal subgroup N / π1(U), let D be the corresponding covering datum.

n Proposition 4.2.9. Let U ⊂ Pk be a regular quasi-projective arithmetical variety, and let f : Y −→ U be the ´etalecover corresponding to the open normal subgroup

N N / π1(U). Then the induced covering datum D is geometrically bounded.

Proof. Let Y be the normalisation of U in K(Y ), then f extends to a finite cover f : Y −→ U. As the cover is étaleover U, the ramification locus R is contained in

U\U ([9, Section 8.3, Ex. 2.15]). Now let C ⊂ U be a regular curve of degreee ≤ d,

C its compactification in U and RC the ramification locus of fC : Y × C −→ C. As f is étaleover C, we have R ⊆ Y \Y . Noting that deg R = [k : k] deg R , C C k C C kC C

66 we get

degk RC = (degk C) · (degk R) ≤ d [kC : k] (degk R) . for any curve of degree deg C ≤ d. Then kC

1 g = (deg f )(g − 1) + 1 + deg R YC C C 2 kC C deg R ≤ (deg f)(g(n, d) − 1) + 1 + k C 2[kC : k] d ≤ (deg f)(g(n, d) − 1) + 1 + (deg R) 2 k is bounded by constants depending only on invariants of U, invariants of the given

étalecover Y −→ U and the maximal degree d of the curve over the field of definition kC of the pullback Y × C −→ C, as claimed.

Corollary 4.2.10. Let U be as in Proposition 4.2.9, D a covering datum on U. If

D is eﬀective with a ﬁnite realisation, then D is geometrically bounded.

N Proof. Let XN −→ X be the ﬁnite realisation of D, then D = D. The covering datum DN is geometrically bounded by the proposition.

Now we deﬁne the notion of a geometrically bounded covering data for any regular arithmetical scheme, and obtain the analogous statement to Corollary 4.2.10.

Definition 4.2.11. Let X be a regular scheme over a finite field, and D a covering datum on X. Then D is geometrically bounded on X if there exists a quasi-projective

i regular open subscheme U ,→ X such that the pullback i∗(D) is geometrically bounded.

67 Remark 4.2.12. If the Main Theorem 4.3.4 holds for the scheme X, i. e. if any geometrically bounded covering datum on X is ﬁnitely realisable, then the above condition is equivalent to requiring that the pullback i∗(D) to any open quasi- projective subscheme U ⊆ X is geometrically bounded. Indeed, if D is an induced

i0 covering datum that is effective with finite realisation, let U 0 ,→ X be another quasi-projective regular open subscheme. Apply Lemma 2.3.13.3 to U 0 and U ∩ U 0, then the pullback i0∗(D) is also finitely realisable, and thus geometrically bounded by Corollary 4.2.10.

Corollary 4.2.13. Let X ∈ Sch(Fp) be any regular arithmetical variety, D a covering datum on X. If D is eﬀective with a ﬁnite realisation, then D is geometrically bounded.

i Proof. Let U ,→ X be a quasi-projective regular open subscheme, and let Y −→ X be the étalecover realising D. Then i∗(D) is effective with finite realisation Y × U, so i∗(D) is geometrically bounded by Corollary 4.2.10.

4.3 Realisable subgroups of the class groups

In this section, we define the (finitely) realisable open subgroups of the Wiesend class group CX of an arithmetical scheme X as those open subgroups whose induced covering datum is (finitely) realisable. We shall show that there exists a one-to-one

68 ab correspondence of these subgroups with open subgroups of ﬁnite index in π1 (X), and provide an explicit description of these groups as the bounded open subgroups of ﬁnite index of CX , which will be proven in the following chapters. This explicit description and its proof is the essential feature of the Main Theorem, and is the main result of this thesis.

Definition 4.3.1. An open subgroup H < CX of finite index is called finitely realisable with realisation N if the induced covering datum is effective with finite realisation YN .

The following lemma shows that ﬁnite realisations of an induced covering datum are unique. In particular, a covering datum induced by an open subgroup of the class group has at most one ﬁnite realisation.

Lemma 4.3.2. Let X be a regular arithmetical scheme, and let N 1, N 2 be two open

ab subgroups of π1 (X). Then the following are equivalent:

1) N 1 ⊂ N 2

−1 −1 2) ρX (N 1) ⊂ ρX (N 2)

−1 −1 3) (ρX ◦ iC∗) (N 1) ⊂ (ρX ◦ iC∗) (N 2) for all curves C ⊂ X

−1 −1 4) (ρX ◦ ix∗) (N 1) ⊂ (ρX ◦ ix∗) (N 2) for all closed points x ∈ |X| . In particular, an induced covering datum has at most one ﬁnite realisation.

Proof. (Taken from [5].) The implications 1) ⇒ 2) ⇒ 3) ⇒ 4) are clear since every closed point is a regular point on some curve. If Condition 4) holds, then in

69 particular, N1,x ⊂ N2,x for all closed points x, so the cover associated to N1/N1 ∩N2 is completely split. Hence by Lemma 2.1.12, it is trivial, which implies 1).

Proposition 4.3.3. Let X be a regular arithmetical scheme. Then the map

−1 N 7→ ρX (N) deﬁnes a one-to-one correspondece between ﬁnitely realisable open

ab subgroups of CX and open subgroups N of the abelianised fundamental group π1 (X).

Proof. A correspondence is one-to-one if and only if it has a well-deﬁned, two-sided inverse correspondence.

We deﬁne an inverse correspondence as follows: If H < CX is an open subgroup of the class group which is ﬁnitely realisable, say by fN : XN −→ X, where N /π1(X),

ab map H 7→ N, where N is the image of N in π1 (X). Then, if DH is finitely realisable with finite realisation N, the realisation is unique by Lemma 4.3.2. Thus, the inverse correspondence is well-defined. By Proposition 4.1.5, we then also have

−1 ρX (N) = H.

ab −1 Conversely, if N is an open normal subgroup of π1 (X), ρX (N) is an open subgroup of CX . Let N be the preimage of N in π1(X). Let HC , Hx be the preimages of ρ−1(N) in C and C , respectively, then by reciprocity for curves and X Ce x

−1 −1 points, we have ρ (N(C)) = HC for all curves C ⊂ X, ρ (N(x)) = Hx for all Ce x closed points x ∈ |X|.

70 ρ ρ C Ce / ab x / ab Ce π1 (Ce) Cx π1 (x)

iC∗ eiC ix∗ eix ρ ρ X / ab X / ab CX π1 (X) CX π1 (x)

In particular, HC is realised by N(C), Hx is realised by N(x) for all curves and

N N closed points x and we have D −1 = D . Since N realises D by deﬁnition, this ρX (N) shows that N realises D −1 . ρX (N)

We can now state the Main Theorem of this thesis. Let X be a regular arithmetical variety, and recall from that an open subgroup H ≤ CX induces a covering datum DH .

n Main Theorem 4.3.4. Let k be a finite field, and let X ⊂ Pk be an open k- subariety. Then an open subgroup H ≤ CX is finitely realisable if and only if the induced covering datum DH geometrically bounded.

Remark 4.3.5. Lemma 4.2.13 showed that ﬁnitely realisable subgroups are geomet-

n rically bounded, and the converse will be shown for open aﬃne subsets of Pk in the next chapter.

Remark 4.3.6. In the ﬂat case of an arithmetical scheme X, the analogue of this theorem is the content of Wiesend’s paper [21], supplemented by the work of Kerz-

Schmidt ([5], [6]).

71 Let X be an arithmetical scheme with idèleclass group CX . Recall from 3.1.9 that a subgroup is a norm subgroup iff H = f∗(NCY ) for some étalecover Y −→ X.

n Corollary 4.3.7. Let X ⊂ Pk be an open subvariety, then the following hold:

1) There exists a one-to-one correspondence between open and geometrically bounded subgroups H < CX of ﬁnite index in the Wiesend id`eleclass group and open sub-

ab −1 groups N of π1 (X); it is given by ρX (N) 7→ N. Then ρX is a continuous injection

ab with dense image in π1 (X).

2) A subgroup of ﬁnite index CX is a norm subgroup if and only if it is realisable with a ﬁnite realisation, which is the case if and only if it is open and geometrically bounded.

3) If f : X00 −→ X is an ´etaleconnected cover, X0 −→ X the maximal abelian subcover, then NCX00 = NCX0 and the reciprocity map gives rise to an isomorphism

' 0 CX /NCX00 −→ Gal(X /X).

Proof. We begin by proving Part 2 and note that the ﬁrst equivalence is trivial. Let

H < CX be a norm subgroup of ﬁnite index, then H = N (f∗) for some ´etalecover

Y −→ X. Note hat norm subgroups are open since the local norm is an open map, and the induced map f∗ : CY −→ CX was deﬁned as the sum of local norms.

Since a norm subgroup N (f∗) is realisable the cover f it is induced from, by

Lemma 4.2.9, the norm subgroup N (f∗) < CX is geometrically bounded. In particular, a norm group of ﬁnite index is open, of ﬁnite index and geometrically bounded, as claimed. The Main Theorem 4.3.4 gives the converse statement.

72 Part 3 follows from 2 : Let N (f∗) = f∗(CX00 ) denote the f-norm subgroup, and

let DN (f∗) be the induced covering datum. Then by Part 2 of this corollary, DN (f∗) is realisable, and the realisation fN : XN −→ X is ﬁnite since it must be a subcover of f. By Remark 4.1.5, we have

ab 0 CX /N (f∗) ' π1 (X)/N ' Gal(X /X)

In particular, fN is abelian.

Lastly, we show Part 1: The one-to-one correspondence follows from

Proposition 4.3.3 and Part 2 of the corollary. The kernel of ρX is the connected component of the identity (cf. [21, Section 8]), which is equal to {1} for arith-

ab metic varieties by Remark 3.1.7. As π1 (X) is a proﬁnite group, the one-to-one correspondence implies that ρX has dense image.

4.4 The Key Lemma

Let X be a regular arithmetical variety, and D a covering datum on X. We recall that a cover of curves is tamely ramified if it is tamely ramified at all closed points x (cf. [9, Definition 4.15]).

Deﬁnition 4.4.1. Following Wiesend, we say that a covering datum D is curve-

D tame or curve-tamely ramified if the covers YC −→ Ce are tamely ramified for all curves C ⊂ X. If D is not tamely ramified, it is called wildly ramified.

73 Remark 4.4.2. If D is tame, the covers YC −→ Ce must have degrees prime to p, or be ´etalecovers.

Definition 4.4.3. A cover YC −→ Ce which has no non-trivial tamely ramified subcovers is called purely wildly ramified.

In this section, we shall reduce the proof of the Main Theorem 4.3.4 to a Key

Lemma, which is split into two statements - one about at most tamely ramiﬁed covering data, the other dealing with index-pm covering data which may be purely wildly ramiﬁed:

n Key Lemma 4.4.4. Let k be a ﬁnite ﬁeld of characteristic p, and let X ⊂ Pk be an open subvariety. Then the following hold:

1. If D is a geometrically bounded covering datum of cyclic prime-power index

pm, then D is eﬀective with a ﬁnite realisation.

2. If D is a covering datum with cyclic prime-power index lm, where l 6= p, then

D is eﬀective with ﬁnite realisation.

Part 1) of the Key Lemma shall be shown in the next chapter using Artin-

Schreier-Witt Theory, and Chapter 6 is devoted to proving 4.4.4.2 using Kummer

Theory.

In the remaining part of this chapter, we show that Key Lemma 4.4.4 implies

Main Theorem 4.3.4. Actually, once can prove a stronger assertion as follows:

74 Proposition 4.4.5. Assume that the Key Lemma 4.4.4 holds for a regular arithmetical k-variety X. Then the Main Theorem 4.3.4 holds for X.

Proof. By Fact 2.1.10 and Lemma 2.3.13, we may assume without loss of generality that X is a regular quasi-projective arithmetical variety. We start by proving the following lemma:

Lemma 4.4.6. Let D be a tame covering datum on any regular quasi-projective arithmetical variety X. Then D is geometrically bounded.

D Proof. Let C ⊂ X be a regular curve as in Notation 4.2.1, and let fC : YC −→ C be the cover of C defined by D. Recall that we let m(C,y) denote the ramification number of a regular point y above a point x ∈ C. Since fC is tame, we have m(C, y) ≤ 1, the ramification numbers of D are uniformly bounded

(cf. Deﬁnition 4.2.7).

The lemma then follows directly from Proposition 4.2.8.

Lemma 4.4.7. Let X be an arithmetical scheme, and let H < CX be an open subgroup of ﬁnite index of the Wiesend id`eleclass group. Let H be an open subgroup

0 0 of ﬁnite index such that all H containing H with cyclic factor group CX /H are

ﬁnitely realisable. Then H is ﬁnitely realisable.

Proof. If H < CX is open and of ﬁnite index, then CX /H is a ﬁnite abelian group.

By the structure theorem for ﬁnite abelian groups, we have CX /H ' ΠiCi where

C1,...,Cr are cyclic groups of prime power order. Let Hi < CX be the kernel of

75 CX −→ Ci. Then Hi is of cyclic prime-power index, and we show that H is realisable if and only if Hi is realisable for all i.

If H is ﬁnitely realisable, then the realisation f : X0 −→ X is a ﬁnite cover which, in particular, trivialises DH . Recall that iC : Ce −→ X denotes the composition of the normalisation with the inclusion morphism, and that iC∗ : denotes the induced

−1 −1 morphism on the Wisend id`eleclass groups. Let HC := iC∗(H), HiC := iC∗(Hi) denote the preimages in C , and set N(C) := ρ (H ), N (C) := ρ (H ). Ce Ce C i Ce iC

For x ∈ |X| a closed point, deﬁne similarly Hx, Hix, Nx := ρx(Hx), Nix :=

ρx(Hix). Note that in this notation, the covering datum induced by H consists of

the groups N(C), N(x), and DHi = (Ni(C),Ni(x))(C,x).

Since HC ⊂ HiC for all curves C ⊂ X, we also have N(C) ⊂ Ni(C) for all

i. In particular, by Corollary 2.3.9, if f trivialises DH , f also trivialises DHi . By

Theorem 2.3.15, this implies that Hi is realisable.

Conversely, assume that Hi is realisable for all i. As CX /Hi ' Z/miZ is cyclic, we may assume that it is realised by an open normal subgroup Ni < π1(X): We

−1 Tn have ρX (Ni) = Hi. Then N := i=1 Ni is open and normal as a ﬁnite intersection of open normal subgroups of π1(X). Let YN be the corresponding ´etalecover of X.

Then we have

−1 −1 −1 ρX (N) = ρX (∩iNi) = ∩iρX (Ni) = ∩iHi = H, so N realises H.

Coming back to the proof of Proposition 4.4.5, let X be an arithmetical scheme

76 such that the Key Lemma holds, and let D be a geometrically bounded covering datum on X. Now let Hi < CX be the subgroups deﬁned in the proof of

Lemma 4.4.7, i.e. Hi = ker(CX −→ Ci), where CX /H ' ΠiCi is written as the product of ﬁnite cyclic groups.

DHi Then DH is a subdatum of DH (cf. Deﬁnition 2.3.4, i.e. the cover Y −→ Ce i Ce

D is always a subcover of Y H −→ Ce. Ce Recall from Deﬁnition 4.2.5 that the ramiﬁcation numbers of a covering datum

D on X were defined as the last jumps of the lower ramification filtrations associated to the cover of curves defined by D. Also recall that the lower ramification filtration is well-behaved for subcovers (cf. [17, Prop IV.2]); thus the ramification numbers of

DHi are bounded by those of DH . Since DH is geometrically bounded by assumption,

Lemma 4.2.8 implies that DHi is also geometrically bounded. Thus we can apply the Key Lemma 4.4.4, and assume that Hi is realisable.

Then Lemma 4.4.7 gives the desired result.

77 Chapter 5

Wildly ramiﬁed Covering Data

In this chapter, we begin with a review of the Artin-Schreier-Witt Theory of ﬁnite k-algebras and its connection to cyclic Z/pmZ-´etalecovers of the corresponding

n aﬃne schemes. We then prove the Key Lemma in the case of aﬃne space X = Ak ,

n and proceed to the case where X ⊂ Ak is any open subset.

Speciﬁcally, in both cases we construct several pro-´etalecovers which weakly trivialise the given covering datum D of cyclic index pm on X. Then, we show that the associated ”intersection cover”, as described in the proposition below, is in fact,

n finite over X. This way, we obtain an étalecover of X ⊂ Ak which trivialises the covering datum, and by Proposition 2.3.13 this means that D is also effective with a finite realisation. Thus, D is realised by an étalecover, which finishes the proof of Part 1) of the Key Lemma.

We then construct several pro-´etalecovers which weakly trivialise the given

78 covering datum D of cyclic index pm on X. Then, we show that the associated

”intersection cover”, as described in the proposition below, is in fact, ﬁnite over

n X. This way, we obtain an étalecover of X ⊂ Ak which trivialises the covering datum, and by Proposition 2.3.13 this means that D is also effective with a finite realisation. Thus, D is realised by an étalecover, which finishes the proof of

Part 1) of the Key Lemma.

A key observation for all of these proofs is the following:

Proposition 5.0.8. Let {Ni}i∈I be a family of closed subgroups of π1(X) such

that each of the associated pro-´etalecovers Xi = XNi −→ X weakly trivialises the covering datum D. Then the intersection cover XN −→ X corresponding to the closed subgroup

N = < Ni >i∈I

generated by the Ni also weakly trivialises D.

Proof. Using characterisation 2) of Corollary 2.3.9, the proof is straightforward:

−1 XM −→ X weakly trivialises D if and only if we have an inclusion Nx ⊇ eix (M) in

−1 π1(x) for all closed points x ∈ X. By assumption, we have Nx ⊇ eix (Ni) for all i.

Since Ni is closed, it follows that

−1 Nx ⊇ < eix (Ni) >i , proving the Proposition.

79 5.1 A Review of Artin-Schreier-Witt Theory

5.1.1 Artin-Schreier Theory

Let k be a ﬁeld of characteristic p > 0, and let A be a ﬁnitely generated k-algebra.

Let Ae be a universal cover of A, i.e. the integral closure of A inside the maximal separable algebraic extension K/Ke of the function ﬁeld in which A is unramiﬁed.

Let ℘ : Ae −→ Ae be the map sending x 7→ xp − x then ker ℘ = Z/p, then we obtain a short exact sequence of π1(Spec A)-modules

℘ 1 −→ Z/pZ −→ Ae −→ Ae −→ 1 .

The long exact sequence of cohomology (cf. [11, Prop. 4.12]) then gives rise to a short exact sequence

0 0 −→ A/℘(A) −→ π1(Spec A ) −→ Z/pZ −→ 0

Thus we get a canonical isomorphism

m ' Ψ: A/℘(A) −→ Hom(π1(Spec A), Z/pZ) .

Note that a group homomorphism f : π1(Spec A) −→ Z/p has kernel of index p if and only if its image is non-trivial. Conversely, any normal index-p subgroup of π1(Spec A) gives rise to such a homormisphm (i.e. the induced canonical projection). Thus, we see that Hom(π1(Spec A), Z/p) and thus A/℘(A) classiﬁes all

Z/p-covers of Spec A.

80 The natural Fp-vector space structure on A/℘(A) is given by a.[f] := [af]

0 for a ∈ Fp,[f] ∈ A/℘(A). Indeed, if f ∈ [f] is another representative, then

0 p p f = f + h − h for some h ∈ A. Since a = a for all a ∈ Fp, we then have that af 0 = af + ahp − ah = af + (ah)p − (ah) ∼ af, as required. Recalling the natural

Fp-vector space structure of Hom(π1(Spec A), Fp), it follows immediately that Ψ is an Fp-linear map.

p p Remark 5.1.1. Let a ∈ Fp, f ∈ A, Hf (z) = z − z − f. Since a = a for all a ∈ Fp,

−1 we have Haf (z) = aHf (a z), so (Hf ) = (Haf ) as ideals in A[z]. Thus f and af deﬁne the same cover of Spec A, which we denote by Y[f].

Conversely, if Y[f] = Y[f 0] is a non-trivial Z/p-cover of Spec A, then Nf = ker(πf ) and Nf 0 = ker(πf 0 ) are equal. Thus the surjection πf : π1(Spec A) Fp gives rise to an isomorphism πf : π1(Spec A)/Nf 0 ' Fp.

πf π1(Spec A)/Nf 0 / Fp MMM MM ·a MMM πf0 MM MM& Fp

−1 Then, since πf 0 also gives such an isomorphism, πf 0 ◦ πf is an automorphism of

∗ the additive group Fp, and as such representable by an element a ∈ Fp. Therefore,

0 πf 0 = a·πf . As Ψ is an Fp-linear map, aπf corresponds to [af], so we have [f ] = [af].

We thus obtain a one-to-one correspondence between Fp-vector subspaces of the form Fp.[f] and Z/p covers of Spec A:

Fp.[f] 7→ Y[f] ↔ Nf = ker(χf )

81 More generally, we have:

Proposition 5.1.2. Let A be a finitely generated algebra over a finite field k.

Then there is a one-to-one inclusion-reversing correspondence of Fp-vector subspaces L M = i∈I Fp.[fi] of A/℘(A) and exponent-p ´etalecovers of Spec A:

M 7→ YM ↔ NM := ∩f∈M Nf .

n n M is ﬁnitely generated of rank n if and only if YM is a (Z/p) -extension of Ak .

Proof. For finitely generated modules M, the assertion is clear from the remarks preceding the proposition and the fact that the isomorphism ψ is compatible with addition. To get the general case, note that an infinitely generated submodule of is the direct limit of its finitely generated submodules, and that a p-exponent cover of An is the inverse limit of its finite subcovers.

Some relevant properties of the correspondence are as follows:

T 1. If {fi}i∈I is any generating set for M, then NM = i∈I Nfi : Indeed, if

Pm f = i=1 ai.fi with ai ∈ Fp, then by the additive property of Ψ, Nf ⊂ ∩iNfi , T so NM ⊂ i∈I Nfi . The other inclusion is trivial.

In particular, the submodule Mf,g generated by two elements f, g corresponds

to the cover Yf,g deﬁned by Nf ∩ Ng. Furthermore, since f + g ∈ Mf,g, Yf+g

is a subcover of Yf,g.

2. The correspondence is inclusion-reversing: If M ⊂ M 0, let I be a basis of

0 0 M over Fp, and I a basis of M containing I (such a basis can be found

82 by completing I to a basis of M 0 - see [8, Theorem III.5.1]). Then clearly,

T T NM = f∈I Nf ⊃ f∈I0 Nf = NM 0 as claimed.

3. For any two submodules M, M 0, we have

a) NM+M 0 = NM ∩ NM 0 , and

0 b) NM∩M 0 =< NM ,NM >

We ﬁnish by summarising some properties of Z/pZ-covers of regular curves:

Let C be a regular curve over a finite field k, and let A = OC (C). Assume we are given a Z/pZ- étalecover φ : Y[f] −→ C with Artin-Schreier representative

[f] ∈ A/℘(A). Let φ : Y [f] −→ C denote the induced cover on their regular compactiﬁcations, and recall that Y[f], Y [f] are the normalisations of C, C in the

p function ﬁeld K(Y[f]) := K(C)[z]/(Hf (z)), where Hf (z) = z − z − f.

Lemma 5.1.3. In the situation above, let x ∈ C be a closed point, and let νx denote the associated valuation. Recall that Y[f] can be speciﬁed by any of the elements in the equivalence class [f] ∈ A/℘(A).

0 0 1. φ is ´etaleat x if and only if there exists f ∈ [f] such that νx(f ) ≥ 0. If this

0 0 is the case, then νx(f ) ≥ 0 for all f ∈ [f].

2. φ is totally ramiﬁed above x if and only if there exists f 0 ∈ [f] is such that

0 νx(f ) < 0.

83 3. If φ is totally ramiﬁed above x, then let f 00 ∈ [f] be such that

00 0 0 νx(f ) = minf 0∈[f]{νx(f )}. Then gcd(νx(f ), p) = 1, and the ramiﬁcation

0 number of the unique point y above x is given by m(C,y) = −νx(f ).

Proof. If C is a curve over the ﬁnite ﬁeld k, then K(C) is an algebraic function

ﬁeld over k. Then 1 is shown in [18, 3.7.8.a)], and 2 follows from [18, 3.7.7] and

[18, 3.7.8.b,c)].

1 Corollary 5.1.4. Let k be any ﬁnite ﬁeld, and let φ : Y [f/ga] −→ Pk be a Z/pZ-cover

1 of curves that is ´etale over the open subscheme D+(g) ⊂ Pk. If

1 Pk\D+(g) = {x1, . . . , xl} denotes the set of closed points at which φ is ramiﬁed.

Let K(Y[f/ga]) denote the function ﬁeld of Y[f/ga]. Let νj ∈ V (K(Y[f/ga])) denote the valuation associated to xj, and let mj denote the ramiﬁcation number associated to a point above xj. Then

a a 1. φ is étaleat xj if and only if νj(f/g ) ≥ 0, and totally ramified iff νj(f/g ) <

a a 2. If νj(f/g ) < 0 is relatively prime to p, then we have mj = −νj(f/g ).

3. One can always ﬁnd a representative of f 0/ga0 of [f/ga] such that for all j,

0 a0 νj(f /g ) is either zero or relatively prime to p.

84 5.1.2 Artin-Schreier-Witt Theory

We begin with a quick primer on Witt vectors, further details of which can be found in [12, II.4] and [13, VI.I]. As in the previous section, k is a finite field of characteristic p, and A a finitely generated k-algebra with separable closure Ae.

In the following sections, we shall either set A = k[x1, . . . , xn], or

a A = AG := k[x1, . . . , xn]G, the localisation of k[x1, . . . , xn] at the set {G : a ∈ Z}

Recall that the m-th Witt polynomial Wm ∈ A[X0,...,Xm] is deﬁned as

pm pm−1 m Wm(X0,...,Xm) = X0 + pX1 , . . . , p X0 .

By induction on m, one can show that there exist universal polynomials

Sm ∈ A[X0,...,Xm,Y0,...,Ym,S1,...Sm−1] such that

Wm(X0,...,Xm−1) + Wm(Y0,...,Ym−1) = Wm(S0,...,Sm−1) .

Indeed, we may deﬁne

S0 = X0 + Y0 , 1 1 S = (Xp + Y p − Sp) + X + Y = (Xp + Y p − (X + Y )p) + X + Y , 1 p 0 0 0 1 1 p 0 0 0 0 1 1 and in general

1 S = (W (X ,...,X ) + W (Y ,...,Y ) − W (S ,...,S )) + X + Y . m pm m−1 0 m−1 m−1 0 m−1 m 0 m−1 m m

By induction, it is now clear that Sm ∈ A[X0,...,Xm,Y0,...,Ym]. Similarly, one can show that there exist polynomials Pm ∈ A[X0,...,Xm,Y0,...,Ym] such that

Wm(X0,...,Xm) · Wm(Y0,...,Ym) = Wm(P0,...,Pm) .

85 Definition 5.1.5. Let A be a finitely generated k-algebra, where k is a finite field.

The ring Wm(A)of Witt vectors of length m over A is deﬁned to have underlying set consisting of systems of length m over A, with ring operations as follows:

m Letting X = (X0,...,Xm−1) and Y = (Y0,...,Ym−1) be two elements of A , we set

X ⊕ Y := (S0(X, Y),...,Sm−1(X, Y)) , and

X · Y := (P0(X, Y),...,Pm−1(X, Y)) .

An element of Wm(A) is denoted by f = [f0, . . . , fm−1].

p p p Remark 5.1.6. Since char(A) = p, we have [f0, . . . , fm−1] = [f0 , . . . , fm−1] for all

Witt vectors [f0, . . . , fm−1] ∈ Wm(A).

Since the Witt polynomials Sm and Pm are universal, we clearly have an inclusion

Wk(A) ⊂ Wm(A) given by [f0, . . . , fk−1] 7→ [f0, . . . , fk−1, 0,..., 0]. Conversely, we also have a projection Wm(A) −→ Wk(A):

(k) Notation 5.1.7. For f = [f0, . . . , fm−1] ∈ Wm(A), k ≤ m, let f denote the truncated Witt vector [f0, . . . , fk−1] ∈ Wk(A).

Now let σm(X0,...,Xm−1) = Sm − Xm − Ym, then σm ∈ A[X0,Y0,...,Ym−1].

The following two Lemmas are immediate (using induction):

Lemma 5.1.8. The polynomial σj(X0,...,Xj−1,Y0,...,Yj−1) is homogeneous of degree pj.

k−j Lemma 5.1.9. degXj σk = degYj σk ≤ p .

86 Deﬁne a homomorphism ℘ : Wm(A) −→ Wm(A) by setting

p ℘([a0, . . . , am]) = [a0, . . . , am] [a0, . . . , am], where denotes the subtraction op- eration of Witt vectors. By Remark 5.1.6, we have

p ker ℘ = {[a0, . . . , am]: ai = ai for all i}

= {[a0, . . . , am]: ai ∈ Fp for all i}

= Wm(Fp)

m ' Z/p Z .

Then we have an exact sequence of π1(Spec Ae)-modules

m ℘ 0 −→ Z/p −→ Wm(Ae) −→ Wm(Ae) −→ 0 .

Applying the long exact sequence of ´etalecohomology, since Wm(A) is aﬃne, the connecting homomorphism induces an isomorphism

' m Ψ: Wm(A)/℘(Wm(A)) −→ Hom(π1(Spec A), Z/p Z) . (5.1.1)

The isomorphism Ψ makes it possible to associate to each Witt vector f an exponent-pm cyclic cover of Spec A:

Let f = [f0, . . . , fm] ∈ Wm(A). The homomorphism

m Ψ(f): π1(Spec A) −→ Z/p Z has image pm−sZ/pmZ ' Z/psZ, where s ≤ m, and corresponds to the normal

s s subgroup ker ψf of index p , which in turn deﬁnes a Galois Z/p Z-cover of Spec A, and denoted by Y[f].

87 Deﬁnition 5.1.10. Let f = [f0, . . . , fm] ∈ Wm(A). The cover Y[f] −→ Spec A deﬁned above is called the cover associated to [f].

Remark 5.1.11. For f = [f0, . . . , fm−1] ∈ Wm(A)/℘(Wm(A)), the associated cover

m Y[f] −→ Spec A is a proper Z/p Z-cover if and only if f0 6∈ ℘(A).

m m Noting that Hom(π1(Spec A), Z/p Z) has a natural structure as a Z/p Z-

m module, we can deﬁne a natural Z/p Z-module structure on Wm(A)/℘(Wm(A)) so that Ψ becomes a Z/pmZ-module homomorphism:

p For a ∈ Wm(Fp), f ∈ Wm(A), we have a = a, and deﬁne a · [f] := [af].

Now if f 0 is another representative of [f], then f 0 = f ⊕ hp h for some h ∈ Wm(A). Then

af 0 = a · (f ⊕ hp h) = af ⊕ ahp h = af ⊕ (ah)p (ah) ∼ af , i.e. the action is well-deﬁned.

As in the previous section, we can then generalise the correspondence induced by Ψ to the following proposition:

Proposition 5.1.12. Let A be any finitely generated algebra over a finite field k.

Then there is a one-to-one inclusion-reversing correspondence of Z/pmZ-submodules

L s M = i∈I Wm(Fp).[fi] of Wm(A)/℘(Wm(A)) and cyclic p ´etalecovers of Spec A, s ≤ m:

M 7→ YM ↔ NM := ∩f∈M Nf .

88 M is ﬁnitely generated of rank n if and only if YM is a ﬁnite cover of Spec A.

Proof. For f ∈ Wm(A)/℘(Wm(A), let Y[f] −→ Spec A denote the cover associated to f. We begin by analysing the structure of Y[f].

As Y[f] is integral, it is given as the normalisation of X inside the ﬁeld extension deﬁned by adjoining to K(A) elements {y0, . . . , ym−1} such that ℘(y0, . . . , ym−1) = f, i.e. such that

p y0 − y0 = f0,

p p y1 − y1 = −σ1(y0, −y0) + f1,

...,

p p p ym−1 − ym−1 = −σm−1(y0, . . . , ym−2, −y0,..., −ym−2) + fm−1 .

The ﬁrst equation determines a Z/pZ-cover Y[f0] of Spec A, the second equation a

Z/pZ cover of Y[f0], etc. Thus we get Y[f] as a tower of successive Z/pZ extensions over the original scheme Spec A (see the graphic below).

(k) Notation 5.1.13. Let f ∈ Wm(A) be a Witt vector of length m, and recall that f denotes the truncated Witt vector [(f0, . . . , fk−1] of length k associated to f. Then the intermediate scheme deﬁned by f (k) is a Z/pk-cover of Spec A, and denoted by

Y[f (k)].

Then the tower of intermediate covers can be written as follows:

89 Y[f] = Y[(f0,...,fm−1)]

Z/pZ Y[f (m−2)]

Z/pZ . m . Z/p Z m−1 Z/p Z Z/pZ Y[f0]

Z/pZ Spec A

Here, at each intermediate step, Y[f (k+1)] −→ Y[f (k)] is equal to the normalisation of Y[f (k)] in K(Y[f (k)])[yk+1]/(H[f (k+1)]), where

p p p Hf (k+1) (y0, . . . , yk−1) = yk − yk + σk(y0, . . . , yk−1, −y0, . . . , yk−1) − fk−1 and

K(Y[f (k)]) = A[y0, . . . , yk−1]/ H[f0](y0),...,H[f (k−1)](yk−1) .

× Now recall from Remark 5.1.1 that in the prime case m = 1, for a0 ∈ Fp we have −1 −1 Hf0 (a0 y0) = a0 Ha0f0 (y0). Similarly, using Lemma 5.1.8, if ak−1 6= 0, we have

−1 −1 −1 Hf (k) (ak−1y0, ak−1y1, . . . , ak−1fk−1)

−1 p p −1 −1 p −1 p −1 −1 = (ak−1) yk−1 − ak−1yk−1 + σk((ak−1) y0,..., (ak−1) y0, −ak−1y0,..., −ak−1yk−1) − fk−1

−1 p p p = ak−1 yk−1 − yk−1 + σk(y0 , . . . , yk−1, −y0,..., −yk−1) − ak−1fk−1

−1 = ak−1H(af)(k) (y0, . . . , yk−1)

m × Therefore, whenever a ∈ (Z/p Z) , we have (H(af)(k) ) = (Hf (k) ) as ideals for all k = 1, . . . , m, and thus Y[f (k)] = Y[(af)(k)] for all k. In particular, we have Y[f] = Y[af]

m for all a ∈ Wm(Fp). Conversely, if the Z/p Z-covers Y[f], Y[f 0] are equal for two

90 0 elements f, f ∈ Wm(A)/℘(Wm(A)), we must have

0 m Ψ(f) = Ψ(f ): π1(Spec A) Z/p Z .

0 Then Nf = ker Ψ(f) and Nf 0 = ker Ψ(f ) are equal, and Ψ(f) gives rise to an

m isomorphism Ψf : π1(Spec A)/Nf 0 ' Z/p Z.

Ψf m π1(Spec A)/Nf 0 / Z/p Z PPP PPP ·a PPP Ψf0 PP' Z/pmZ Then, since Ψ(f 0) also gives such an isomorphism, Ψ(f 0) ◦ Ψ(f)−1 is an automorphism of the additive group Z/pmZ, and as such representable by an element a ∈ (Z/pmZ)×. Therefore, Ψ(f 0) = a·Ψ(f). As we showed Ψ to be an Z/pmZ-linear map, aΨ(f) corresponds to [af], so we have [f 0] = [af].

m Thus we get a one-to-one correspondence between Z/p Z-submodules Wm(Fp).[f]

s of Artin-Schreier-Witt space Wm(A)/℘(Wm(A)), and Z/p Z-covers of Spec A, where s = |{i : fi 6∈ ℘(A)}| ≤ m.

Wm(Fp).[f] 7→ Y[f] ↔ Nf = ker(Ψ([f]))

The statement about ﬁnitely generated submodules and their ranks follows immediately, as do the statements about general submodules when taking direct limits

(see the proof of Proposition 5.1.2 for details).

m Remark 5.1.14. Y[f] has degree p over Spec A if and only if f0 6∈ ℘(A). If this is the case, we call [f] a primitive element of Wm(A)/℘(Wm(A)), and Wm(Fp).[f] a primitive simple submodule of Wm(A)/℘(Wm(A)).

91 Deﬁnition 5.1.15. A submodule M ⊂ Wm(A)/℘(Wm(A)) is called primitive if every simple submodule is contained in a primitive simple submodule.

We note that under the correspondence, primitive submodules correspond to covers with Galois group a direct sum of copies of Z/pmZ.

We ﬁnish by deﬁning some submodules of Artin-Schreier-Witt space:

Deﬁnition 5.1.16. Let ∆, m be positive integers, and let AG denote the localisation

a of k[x1, . . . , xn] at the set {G : a ∈ N}. Deﬁne subsets of Artin-Schreier-Witt space

Wm(AG)/℘(Wm(AG)) by

F0 Fm−1 m−j−1 M∆,m(AG) = ,..., : p aj ≤ ∆ for all j, Ga0 Gam−1

m−j−1 p degxi Fj − aj degxi Gj ≤ ∆ for all i .

i F0 Fm−1 m−j−1 M∆,m(AG) = ,..., : p aj ≤ ∆ for all j, Ga0 Gam−1

m−j−1 p degxi Fj − aj degxi Gj ≤ ∆ for all j 6= i .

i Proposition 5.1.17. Let M∆,m(AG), M∆,m(A)G) be the subsets of Artin-Schreier-

i m Witt space deﬁned above. Then M∆,m(AG) and M∆,m(AG) are Z/p -submodules of

Wn(AG)/℘(Wn(AG)).

Proof. The addition in Wm(AG)/℘(Wm(AG)) is inherited from the addition of Witt Vectors, so

" !# »„ F F «– F 0 F 0 0 ,..., m−1 + 0 ,..., m−1 a am−1 a0 a0 G 0 G G 0 G m−1 »„ 0 0 0 0 0 !# F0 F0 F1 F1 F0 F0 Fm−1 Fm−1 Fm−2 Fm−2 = + , + + σ1( , ),... + + σm−1(..., ,..., ) a a0 a a0 a a0 am−1 a0 am−2 a0 G 0 G 0 G 1 G 1 G 0 G 0 G G m−1 G G m−2

92 Now Aj be the power of G in the denominator of

0 Fj−1 Fj−1 j−l j−l 0 σj(..., aj−1 ,..., a0 ). By Lemma 5.1.9, Aj ≤ maxl

m−l−1 m−l−1 0 m−j−1 assumed p al, p al < ∆ for all l, this implies that p Aj ≤ ∆.

Now let eaj be the power of G in the denominator of the j-th coordinate in the sum, then a is bounded by a ≤ max{a , a0 ,A } ≤ ∆ , as required. ej ej j j j pm−j−1

al Similarly, since by assumption degxj (Fl/G ),

0 0 al m−l−1 degxj (Fl /G ) ≤ ∆/p for all j and l, Lemma 5.1.9 implies that

F F 0 deg (σ (..., j−1 ,..., j−1 )) ≤ max pl∆/pm−l−1 = ∆/pm−j−1 xj j a a0 G j−1 G j−1 l

i Thus, M∆,m(AG) and M∆,m(AG) are additive subgroups. It’s clear that the

m action of Z/p on Wm(A)/℘(Wm(A)) leaves the degree of numerators and denom-

F i inators of any element Ga unchanged, so M∆,m(AG), M∆,m(AG) are submodules.

Deﬁnition 5.1.18. For m = 1, we obtain the vector subspaces

a M∆(AG) := M∆,1(AG) = [F/G ]: a ≤ ∆, degxi F ≤ a degxi G + ∆ for all i ⊂ AG/℘(AG)

i i D a E M∆(AG) = M∆,1 = [F/G ]: a ≤ ∆, degxj F ≤ a degxj G + ∆ for all j 6= i ⊂ AG/℘(AG) .

93 Similarly, if we set G = 1, then AG = A := k[x1, . . . , xn] and we obtain analogous submodules for A:

m−s−1 M∆,m(A) := [F0,...,Fm−1]: p degxi Fs ≤ ∆ for all i, s

i D m−s−1 E M∆,m(A) := [F0,...,Fm−1]: p degxj Fs ≤ ∆ for all s, for all j 6= i ,

m both of which are Z/p -submodules of Wm(A)/℘(Wm(A)).

5.2 Covering data of Prime Index

In this section, we consider the case m = 1, where D is a bounded covering datum

n n of prime index p. We show that for X = Ak and, more generally, for any X ⊂ Pk an open subset, a geometrically bounded covering datum is realisable.

n 5.2.1 Aﬃne n-space Ak

n We begin by considering the aﬃne n-space X = Ak . Let A = k[x1, . . . , xn] and let D be a geometrically bounded covering datum of cyclic index p on X = Spec A. This special case formed the starting point of the investigation, and already contains most of the strategies used for the more general cases. The goal will be to show the following Proposition:

Proposition 5.2.1. Let k be a ﬁnite ﬁeld of characteristic p and let D be a

n n bounded covering datum of index p on Ak . Then there exists an ´etalecover of Ak

94 realising D.

Throughout the section, we shall make much use of the following, particularly simple ﬁbrations of aﬃne space:

Deﬁnition 5.2.2. Let Ai = k[x1,..., xˆi, . . . , xn] and let φi : Ai −→ A, φi(xj) = xj be the natural inclusions, then the induced morphism

n n−1 Φi : Ak −→ Ak are called the projection ﬁbrations onto the coordinate hyperplanes.

n−1 Fix a projection fibration Φi. A closed point ω ∈ Ak corresponds to a maximal ideal mω = (g1,..., gî, . . . , gn), where gj = gj(x1,..., xî, . . . , xj) is a polynomial in j variables if i > j, and in j − 1 variables if i ≤ j. Letting k(ω) ' A/mω denote the

1 residue ﬁeld of ω, the ﬁber Cω,i of Φi above ω is isomorphic to Spec k(ω)[xi] ' Ak(ω).

As before, we let Y D −→ C denote the cover induced by the subgroup Cω,i ω,i

NCω,i /π1(Cω,i) of the covering datum D. Recall the notations established in Section

5.1, then by Proposition 5.1.2, there exists a polynomial fω,i ∈ k(ω)[xi] such that

YCω,i = Y[fω,i] is the Artin-Schreier cover associated to [fω,i] ∈ A/℘(A). Y[fω,i] will

be referred to as an Artin-Schreier representative of the cover YCω,i .

Now consider the natural surjection

pi : A k(ω)[xi]/℘(k(ω)[xi]) which gives a map whose kernel clearly contains ℘(A), thus inducing a surjection

πi : A/℘(A) k(ω)[xi]/℘(k(ω)[xi]) .

95 For F ∈ A mapping to fω,i ∈ k(ω)[xi], we thus have πi([F ]) = [fω,i]. For any such

n F , the base change of the cover YF over Cω ,→ A is just Y[fω,i] by construction. In

particular, if Y[fω,i] is the cover induced by an index-p normal subgroup NCω,i of the covering datum D, then Y[F ] trivialises the covering datum D over Cω.

Let F ∈ A, then we denote by degxi (F ) the degree of F considered as a polynomial of k(x1,..., xˆi, . . . , xn)[xi].

Lemma 5.2.3. In the notation established above, let f ∈ [fω,i] be a representative of degree d. Then there exists a class [F ] ∈ π−1(f) containing an element F such

that degxi (F ) ≤ d. ˆ Proof. Write k(ω) = k(θ1,..., θi, θn), where θj+1 is such that

k(x1,..., xî, . . . , xj+1)/(g1,..., gî, . . . , gj+1) = k[x1,..., xî, . . . , xj]/(g1,..., gî, . . . , gj) θj+1 .

P I Then any a ∈ k(ω) can be written as a = aI θ , where I = (α1,..., αˆi, . . . , αn−1) P is a multi-index such that αj ≤ [k(ω): k]. If we now deﬁne an element of

P I A by Fa(x1,..., xˆi, . . . , xn) := aI x , then p(Fa) = a in k(ω). Moreover, given

d d f(xi) = adxi + ... + a0 ∈ k(ω)[xi], we can deﬁne F (x1, . . . , xn) = Fad xi + ...Fa0

such that pi(F (x1, . . . , xn)) = f(xi). Since deg(f) = d, we have degxi (F ) = d, and

πi([F ]) = f by the remark preceding the claim.

Recall that we set A = k[x1, . . . , xn], and recall from 5.1.18 the deﬁnitions of

i M∆(A) and M∆(A).

i Lemma 5.2.4. Let A = k[x1, . . . , xn] and ﬁx a positive integer ∆. Let Mi ⊂ M∆(A) be vector subspaces in Artin-Schreier space A/℘(A). If we set M = ∩Mi, then

96 M ⊂ M∆(A), i.e. every equivalence class in M contains an element G such that

degxi G ≤ ∆.

i Proof. As M ⊂ Mi ⊂ M∆(A) for all i, every equivalence class of M contains an

p element Fi such that degxi (Fi) ≤ D. We have Fi = Fj + h − h for some h ∈ A.

Write

X α Fi = aαxi , α

X α0 Fj = bα0 xi , α0 where the coeﬃcents aα and bα0 are polynomials in k[x1,..., xˆi, . . . , xn], and consider the terms of Fi that are of highest xi-degree D. Note that for any non-constant

p h ∈ k[x1, . . . , xn], the monomials of highest xi-degree in h − h contain all xk to a power divisible by p. Thus, if D is not divisible by p, or if aD contains a monomial

p p whose exponents are not all multiples of p, and Fj +h −h is equal to Fi, then h −h

D must be of xi-degree strictly smaller than D. In particular, aDxi is left unchanged

p by subtracting h − h, and must thus be equal to the corresponding terms in Fj. In

particular, we must have equality of degrees in xi: degxi (Fi) = degxi (Fj).

Now consider the case where p divides D, and all exponents in the terms of aD are divisible by p: Let c be the highest power of p dividing all exponents of terms

c in aD, and such that p divides D. Then we can write

c c c c 0 p p X p α1 p αn aD = aD(x1 , . . . , xn ) = aI x1 . . . xn , I

c c where the sum goes over all multi-indices I = (p α1, . . . , p αn) of size n − 1. Then

97 0 0pc 0 P 0 D0 c 0 let aI be such that a I = aI , and deﬁne h = I aI x , where D = p D . Clearly,

pc pc pc−1 pc−1 pc−2 p h − h = (h − h ) + (h − h ) + ... + (h − h) ∈ ℘(k[x1, . . . , xn])

0 pc 0 and Fi = Fi + (h − h) ∼ Fi has degxi (Fi ) < d. Repeating the process if necessary,

0 0 0 we thus obtain Fi of xi-degree D such that either p does not divide D , or such that aD0 contains a term in which a variable occurs with an exponent not divisible by p. We are thus reduced to the ﬁrst case, and get

0 degxi (Fj) = degxi (Fi ) ≤ degxi (Fi) ≤ D.

Now use this procedure to compare F1 to all Fi, then F1 must satisfy

degxi (F1) ≤ degxi (Fi) ≤ D for all i, as required.

n n−1 Proposition 5.2.5. Let k be a finite field, and let Φi : Ak −→ Ak be the ith coordinate fibration. Let D be a geometrically bounded covering datum of index p on n, and let Y D −→ C be the cover of C defined by D. Then for all ω and i, Ak Cω,i ω,i ω,i there exists an Artin-Schreier representative f ∈ k(ω)[x ] such that Y = Y D ω,i i [fω,i] Cω,i

and a positive integer ∆ such that degxi (fω) ≤ ∆ .

Proof. Since D is a covering datum of index p, for all ω and i, we have Y D = Y Cω,i [fω,i]

for some fω,i ∈ k(ω)[xi]. Let s = degxi (fω,i). Whenever s > 0 and p|s, the

ps0 highest term is of f is of the form axi . We may then make a change of variables

0 s0 0p z 7→ z − a xi , where a = a, and thus replace the term of highest degree of f by its pth root. Repeating if necessary, we may without loss of generality assume that

(s, p) = 1.

98 1 Recall that Y[fω,i] denotes the normalisation of Ak(ω) in

p K(fω,i) := k(ω)(xi)[z]/(z − z − fω,i(xi)) .

1 1 The regular compactiﬁcation of Ak(ω) is Pk(ω); let Y [fω,i] denote the normalisation

1 of Pk(ω) in K(fω,i). We have a commutative diagram

Y / [fω,i] Y [fω,i]

fC f ω,i Cω,i 1 1 Ak(ω) / Pk(ω) ,

where all covers are defined over k(ω). The ramification locus RCω,i either consists of the point x∞ at infinity corresponding to the fractional ideal (1/xi) ⊂ k(ω)(xi), or is empty.

Since D is assumed to be geometrically bounded, by Proposition 4.2.8, there

exists a positive integer such that the ramiﬁcation numbers my∞ (ω, i) ≤ ∆. By

Corollary 5.1.4, we get that degxi fω,i ≤ ∆ for all ω, i.

Remark 5.2.6. Using Lemma 5.1.4 and the fact that the genera of degree-p covers of aﬃne lines can easily be computed, we can also provide a direct proof of Proposition

5.2.5 which does not make use of Proposition 4.2.8:

n Indeed, the degree of Cω,i as a subvariety of Pk is given by degk Cω,i = [k(ω): k],

and fCω,i : YCω,i −→ Cω,i is defined over kCω,i = k(ω), so we have deg Cω,i = 1 kCω,i for all fibers Cω,i of one of the coordinate fibrations Φi.

As above, we let y∞ ∈ Y[fω,i] denote the unique point above the point x∞ of

99 1 1 Pk(ω)\Ak(ω), then [k(y∞): k(x∞)] = 1. Note also that k(x∞) = k(ω), and that we

1 1 have gC = g 1 = 0 for all ﬁbers Cω,i, Cω,i\Cω,i = \ . Then Hurwitz’s ω,i Ak(ω) Pk(ω) Ak(ω) formula (cf. 4.2.1) gives

1 gY = p(gCω,i − 1) + 1 + degk(ω) RC [fω,i] 2 ∞ 1 X = p(g − 1) + 1 + (|G | − 1) [k(y ): k(ω)] Cω,i 2 y∞ ∞ i=0 my 1 X∞ = 1 − p + (|G | − 1) 2 y∞ i=0 (m − 1)(p − 1) = y∞ 2

By Lemma 5.1.4, we have

(s − 1)(p − 1) gY = (5.2.1) [fω,i] 2

Since D is geometrically bounded, Deﬁnition 4.2.2 gives a constant δ = δ(1) such that g ≤ δ for all ω ∈ n−1, for all coordinate ﬁbrations i. Thus, s is YCω,i Ak bounded by 2δ s ≤ + 1 , p − 1 which proves the Proposition as claimed.

Proof of Proposition 5.2.1. Following the notation established above, let

Y D −→ C denote the induced by the covering datum D, and for each ω and i, Cω,i ω,i let f ∈ Aω,i be an Artin-Schreier representative of Y D . ω,i Cω,i

1 Recall that Cω,i ' Ak(ω) denotes the ﬁber of the ith projection ﬁbration onto a

coordinate hyperplane, and that Y[fω,i] −→ Cω,i is thus deﬁned over k(ω). Thus, all

100 the fibers have degree deg C = deg 1 = 1 over their field of definition kC ω,i k(ω) Ak(ω) kC = k(ω). Thus, by Proposition 5.2.5, there exists a positive integer ∆ such that deg(fω,i) ≤ ∆ for all ω,i.

−1 From Lemma 5.2.3, we may now ﬁnd lifts Fω,i ∈ π ([fω,i]) such that

degxi Fω,i = degxi fω,i ≤ ∆ for all ω, for all i = 1, . . . , n. Let Mi = ⊕ωFp.[Fω,i] be the subspace of A/℘(A) generated by the lifts {Fω,i} and recall the deﬁnition

i i of M∆(A) and M∆(A) from Deﬁnition 5.1.18. Then Mi ⊂ M∆(A) for all i and

n YMi −→ A trivialises all NCω,i by construction.

The following proposition shows that YMi is a weak trivialisation of D, i.e. YMi trivialises Nx for all closed points x contained in a ﬁber Cω,i of Φi:

Proposition 5.2.7. Let f : X −→ X0 be a fibration with regular fibers, D a covering datum on X. If Y −→ X is a pro-étalecover trivialising D on each fiber of f, then

Y weakly trivialises D.

Proof. Every closed point x ∈ |X| is contained in some ﬁber, say Cω for some

0 ω ∈ X . As all the ﬁbers are regular, x is always a regular point of Cω. Then the proposition follows directly from Lemma 2.3.18.

Now let M = hMiii, then the associated p-elementary cover YM also weakly trivialises D by Theorem 5.0.8.

By Lemma 5.2.4, M is contained in M∆(A) = h[F ] : degxi F ≤ ∆ for all ii.

Any class in M∆(A) has a representative of total degree ≤ n∆, so M∆ is ﬁnitely

101 n generated. Thus M is of ﬁnite rank, so YM −→ Ak trivialises D by Theorem 2.3.10.

Then D is realisable with a ﬁnite realisation by Theorem 2.3.15.

n 5.2.2 The Case of Open Subsets of Pk

n Let k be a finite field. In this section, we consider open subsets X ⊂ Pk , and show that bounded covering data of prime index p on such X are finitely realisable

(Theorem 5.2.11).

n Since Pk is Noetherian, we can write X as a ﬁnite union of principal open

m subsets: X = ∪j=1D+(Gj), where for any j, Gj ∈ k[X0,...,XN ] is a homogeneous

n polynomial and deﬁnes a hypersurface of Pk . We let A = k[x1, . . . , xn], and consider

n n the open aﬃne subset Ak = D+(X0) ' Spec A of Pk . The intersection of a simple

n open D+(G1) with A is given by D(G), where G ∈ A is the dehomogenisation of

G1.

a Let AG denote the localisation of A at the set {G : a ∈ N}. Then

n Spec AG ' D(G) ⊂ Ak .

Deﬁnition 5.2.8. Let φi : k[xi] ,→ A be the inclusion, then we shall call the

n 1 induced morphism Φi : Ak −→ Ak the projection ﬁbration onto the i-th coordinate axis.

1 ω,i If ω ∈ Ak is a closed point with residue ﬁeld k(ω) = k[xi]/(hi(xi)), let A denote

1 the ring k(ω)[x1,..., xˆi, . . . , xn]. Then the ﬁber above the closed point ω ∈ Ak is

102 ω,i n given by Cω,i ' Spec A ⊂ Ak .

ω,i Recall from Section 5.1.1 that the surjection pω,i : A A induces a surjection of Artin-Schreier spaces

ω,i ω,i πω,i : A/℘(A) A /℘(A )

−1 such that πω,i([F ]) = f for any F ∈ pω,i(f).

ω,i Now let g = g be the image of G under pω,i, and let f = pω,i(F ) be the image of a polynomial F ∈ A. Then pω,i induces a natural surjection

0 ω,i pω,i : AG Ag given by sending F/Ga 7→ f/ga. Composing with the natural projection, we get a

0 ω,i ω,i surjection pω,i : AG Ag /℘(Ag ) whose kernel contains ℘(AG). Thus we get a surjection of Artin-Schreier spaces

0 ω,i ω,i πω,i : AG/℘(AG) A /℘(A ) (5.2.2)

0 a a 0−1 such that πω,i([F/G ]) = f/g for any F ∈ pω,i (f).

a ω,i ω,i Lemma 5.2.9. Given [f/g ] ∈ Ag /℘(Ag ), there exists a preimage

a −1 a [F/G ] ∈ πi ([f/g ]) which has a representative such that degxj F = degxj f for all j 6= i.

−1 Proof. We prove the claim by constructing F ∈ pi (f) such that degxj F = degxj f

0 a a a for all j 6= i. Then πω,i([F/G ]) = f/g so F/G is the required representative.

103 Let θ be a primitive element of k(ω) over k, i.e. such that k(ω) = k(θ), then we

Pm−1 k can write any a ∈ k(ω) as a = k=0 αkθ , where m = [k(ω): k] and αk ∈ k. Now

Pm−1 k let Fa(xi) = i=0 αkxi , then Fa maps to a under the surjection k[xi] k(ω).

P I k Writing f as f(x1,..., xˆi, . . . , xn) = I aI x xi , where the sum goes over multi-

I α1 αn−1 indices I of size n − 1, and x = x1 ... xˆi . . . xn for the multiindex

P I k I = (α1, . . . , αn−1), we may deﬁne F by F (x1, . . . , xn) = I FaI x xi . Then pi(F ) = f and therefore πi([F ]) = [f] as claimed.

Noting that we have constructed F such that degxj F = degxj f for all j 6= i,

and such that degxi F ≤ [k(ω): k] concludes the proof of the claim.

Recall from Section 5.1.1 that Artin-Schreier space AG/℘(AG) classiﬁes p-exponent

i covers of Spec AG. Also recall the special vector subspaces M∆(AG), M∆(AG) of

AG/℘(AG) (Deﬁnition 5.1.18).

Lemma 5.2.10. Let AG be the localisation of A = k[x1, . . . , xn] at the set

a i {G : a ∈ Z}, and ﬁx a positive integer ∆. Let Mi ⊂ M∆(AG) be vector subspaces of the associated Artin-Schreier space AG/℘(AG). If we set M = ∩Mi, then

a M ⊂ M∆(AG), i.e. every equivalence class in M contains an element F/G such

that a ≤ ∆, and degxj F ≤ a degxj (G) for all j.

ai Proof. As M ⊂ Mi for all i, any equivalence class contains an element Fi/G such

that degxj Fi ≤ ai degxj G for all j 6= i. For all pairs (i, j) such that j 6= i, there thus exists some element F 0/Gb such that

104 F F 0p F 0 F i + − = j Gai Gpb Gb Gaj

Let eai = max{ai, bp}, then aj ≤ eai and the above is equivalent to

ai−ai 0p ai−bp 0 ai−p ai−aj FiGe + F Ge − F Ge = FjGe (5.2.3)

0 Bringing the terms containing F to the other side, we have equality of xi-degrees as follows:

0 0 degxi Fi = max{p degxi F +(ai−bp) degxi G, degxi F +(ai−p) degxi G, degxi Fj+(ai−aj) degxj G} .

0 So if degxi F ≤ b degxi G, then this implies that

degxi Fi ≤ max{ai degxi G, degxi Fj + (ai − aj) degxi G} . and therefore

degxi Fi ≤ ai degxi G.

a ai If this is the case, we can take F/G = Fi/G and are done. So now assume that

0 degxi F > b degxi G. Then the middle term of the left-hand side in (5.2.3) has bigger xi-degree than the term to its right.

If degxi Fi ≤ degxi Fj + (ai − aj) degxi G, then degxi Fi ≤ ai degxi G, so there

is nothing to be shown. Thus assume that degxi Fi > degxi Fj + (ai − aj) degxi G.

Then the middle term has to cancel the terms of highest xi-degree of the term to the left; in particular, the xi-degree of the ﬁrst and middle term must be equal.

This implies that

0 degxi Fi = (ai − bp) degxi G + p degxi F .

105 In particular, p must divide degxi Fi − ai degxi G. Now, we define the i-order, an order on the terms of a polynomial ∈ k[x1, . . . , xn], as follows. The lexicographical order on polynomials in k[x1, . . . , xn] has considers the x1-degree first, then the x2-degree etc. In the i-order of terms in a polynomial, we firstly then consider the degree in xi first, and the remaining degrees in lexicographical order. Then we can write

α1 αn Fi = α0x1 . . . xn + (terms of lower i-order, same xi-degree) + (terms of lower xi-degree) ,

β1 βn G = β0x1 . . . xn (terms of lower i-order, same xi-degree) + (terms of lower xi-degree) ,

0 0 0 α1 αn F = α0x1 . . . xn + (terms of lower i-order, same xi-degree) + (terms of lower xi-degree) ,

0 and we must have that αl −aiβl = p (αl − bβl) for all l. In particular, p must divide

αl − aiβl for all l.

0 0 ai Thus we can write αl −aiβl = pNl, and let Nl = Nl −βl. Then setting b = b p c,

0 0 0 0 0 N1 Nn 0 0p (bp−ai) we have b p ≤ ai. Let F = γ0x1 . . . xn , where γ0 is such that α0 = γ0 β0 .

Fi Then we can replace Gai by

0p ai−bp 0 ai−b Fi + F G − F G ai , Gi

0 thereby replacing the term of highest i-order in the numerator Fi by a lower-order

0 term. For all j 6= i, one now checks that the xj-degree of Fi is still bounded by

0 0 0 the xj-degree of the denominator: degxj Fi ≤ ai degxj G. Also, we have chosen b

0 so that b p ≤ ai,k, so that the power of G in the denominator remains unchanged:

0 ai = ai ≤ D. Thus we may repeat this process until we have replaced all terms of highest xi-degree by terms of lower xi-degree, and we repeat further until we get

0 Fi 0 a0 such that p 6 |αl − aiβl for some l, or such that degx Fi is less than that of the G i i

106 0 denominator. In the latter case, we get that degxi Fi ≤ ai degxi G, and can take

0 0 0 ai 0 ai Fi /G as the required representative. In the former case, we again compare Fi /G

aj to the representative Fj/G :

0 0 eai−ai 0p eai−bp 0 eai−b eai−aj Fi G + F G − F G = FjG .

0 0p a0i−b The condition p 6 |αl −aiβl ensures that no term of F G can match the term

eai−ai 0 of highest xi-degree of FiG ; thus degxi Fi = degxi Fj ≤ aj degxi G and we may

0 0 ai take Fi /G as the required representative.

We can now state and prove the following Theorem:

Theorem 5.2.11. Let k be a ﬁnite ﬁeld of characteristic char(k) = p, and let X

n be an open subset of Pk , and D a geometrically bounded covering datum of prime index p on X. Then there exists an ´etalecover Y −→ X realising D.

n m Proof. We have Pk ' P roj k[X0,...,Xn]. As above, we write X = ∪j=1D+(Gj), where for any j, Gj is a homogeneous polynomial in X0,...,XN and deﬁnes a

n hypersurface of Pk . By Propositions 2.3.13 and 2.3.15, it will suﬃce to ﬁnd a trivialisation of D over D+(G1), an open subscheme.

n 1 Now let Φi : Ak −→ Ak be the projection onto a coordinate axis, as deﬁned in

1 n−1 Deﬁnition 5.2.8. For a closed point ω ∈ Ak, let Hω,i ' Ak(ω) denote the ﬁber above

0 ω, and set Hω,i = Hω,i ∩ X.

0 1 Then Hω,i is the ﬁber of Φi|X : X −→ Ak, the restriction of Φi to X.

107 0 Construction 5.2.12 (Induction Construction). The ﬁber Hω,i of Φi|X above a closed

1 n−1 ω,i point ω ∈ Ai is of dimension n − 1 and isomorphic to D(g) ⊂ Ak(ω), where g = g

ω,i is the image of G under the canonical surjection pω,i : A A , which is given by

0 ”modding out by (hi(xi))”. Let Dω,i denote the restriction of D to Hω,i. Then Dω,i

0 is a geometrically bounded covering datum of index bounded by p on Hω,i.

Thus we may use induction on the dimension n of X to prove Theorem 5.2.11.

0 The induction hypothesis is that Dω,i has a ﬁnite realisation Yω,i −→ Hω,i, and the base case is given by n = 2, where the ﬁbers of Φi are curves on which Dω,i will be

1 0 1 trivially realisable. Indeed, we have Hω,i ' Ak(ω), Hω,i ' D(g) ⊂ Ak(ω) and Dω,i is

D 0 just the p-exponent cover Y 0 −→ H cover of regular curves induced by D. Hω,i ω,i

We note that since Dω,i is of index bounded by p, the realisation Yω,i of Dω,i is either trivial or a Z/p-cover of D(g). By the Artin-Schreier Theory (cf. Section

aω,i a ω,i 5.1.1), we have Yω,i = Y ω,i for some element fω,i/gω,i ∈ Ag . [fω,i/gω,i ]

Key Properties of this construction are summarised in the following lemma:

Lemma 5.2.13. Let k be a ﬁnite ﬁeld, let D be a geometrically bounded covering

n n datum of index p = char(k) on X = D+(G) ⊂ Pk , and identify Ak with V+(X0).

Let Φi be the projection ﬁbration onto the ith coordinate axis, let Hω,i be the ﬁber

1 0 n above a closed point ω ∈ Ak. Set Hω,i = Hω,i ∩ X ⊂ Ak , and let Dω,i denote the

0 0 restriction of D to Hω,i. If Dω,i is eﬀective with a ﬁnite realisation Yω,i −→ Cω,i, then there exists a positive integer ∆ such that

aω,i 1. Yω,i = Y[fω,i/g ] with aω,i ≤ ∆ for all ω,i.

108 2. degxj fω,i ≤ aω,i degxj gω,i + ∆ for all j 6= i.

Proof of 5.2.13.1. If D is geometrically bounded, let ∆ be the constant from Propo- sition 4.2.2, i.e. such that the ramiﬁcation number m(C,y) associated to any closed point y above C\C on the compactiﬁcation of a curve C ⊂ X is bounded by ∆: m(C,y) ≤ ∆ for all pairs (C, y).

Note that if D is of index p, then Dω,i is of index bounded by p. If Dω,i is trivial,

the lemma is trivially true: Let g = 1, a = 0, fω,i ∈ ℘(A) then Y[fω,i] −→ C realises

Dω,i, and any positive integer ∆ satisﬁes the condition of the lemma.

So if there exist ω and i such that aω,i > ∆, then Dω,i is of index p, and its

0 realisation is a proper Z/pZ-cover Y aω,i −→ Hω,i. We want to ﬁnd a curve [fω,i/gω,i ]

0 C ⊂ Hω,i on which we have m(C,x) > ∆ for some closed point x ∈ C\C.

We let f = fω,i, g = gω,i and a = aω,i, and let j ∈ {1, . . . , n}, j 6= i. Consider

n−1 n−2 the projection ﬁbration onto the jth coordinate hyperplane Φj : Ak(ω) −→ Ak(ω).

n−1 n−2 Let Cη,j ⊂ Ak(ω) be the ﬁber above a closed point η ∈ Ak(ω).

η,j Deﬁne A := k(η)[x1,..., xˆi, xˆj, . . . , xn], then we have a surjection

ω,i η,j A A .

ω,i η,j Denote the images of f, g by fη, gη. Now let Ag , Agη denote the localisations at

b b the sets {g : b ∈ Z} and {gη : b ∈ Z}, respectively, then we can extend the above surjection to a surjection

p : Aω,i Aη,j , which maps η,j g gη

109 f fη a 7→ a . g gη

0 0 Then since Y[f/ga] −→ Hω,i realises Dω,i on Hω,i, the pullback Y[f/ga] −→ Cη,j realises the covering datum over Cη,j.

Y[f/ga] = Y ×Cη,j Y[f/ga] / Y[f/ga]

0 Cη,j / Hω,i .

a 0 a1 al 0 Now write f/g = f /g1 ··· gl , where f and gj are pairwise relatively prime for all j. Then there is a one-to-one correspondence between the gj and the points

n−1 n−1 of codimension one xj in Pk(ω)\D+(g) which correspond to ﬁnite primes of Ak(ω).

a Let νxj be the valuation associated to gj and xj, then νxj (f/g ) = −aj ≤ −∆.

0 0 Claim 5.2.14. Let f := pη,j(f ), gr = pη,j(gr) for r = 1, . . . , l denote the images of

0 η,j f and gr in A . Then f is relatively prime to gr for all r.

0 ω,i Proof. Since f and gr are relatively prime in A , there exist polynomials

0 ω,i 0 0 0 hr,hr ∈ A such that hr f + hr gr = 1. Letting hr, hr denote the images of

0 η,j 0 0 hr, hr in A , we have hr f j + hr gr = 1, as claimed.

f f 0 f Now we have pη,j( ga ) = a1 al = ga . Then if y is a closed point of Cη,j\Cη,j g1 ···gl 0 a a1 al lying in the closure V (gr) of xr, we thus have νy(f/g ) = νy(f /g1 ··· gl ) ≤ −ar

By Lemma 5.1.4, if y0 denotes a point of lying above y we then have

a 0 mCη,j ,y = −νy(f/g ) = aj > ∆, which contradicts the uniform bound on rami-

ﬁcation numbers of Dω,i.

110 n−1 0 n 0 Proof of 5.2.13.2. Let Hω,i ' Pk(ω) denote the closure of Hω,i in Pk , and let Hω,i

0 denote the closure of Hω,i in X = D+(G).

Recall that D is a covering datum defined on all of X = D+(G), and not just on the affine part X ∩ V+(X0) ' D(G). Similarly, Dω,i is defined on all of Hω,i.

0 0 So if Yω,i −→ Hω,i realises Dω,i then the normalisation Y ω,i of Hω,i is ´etaleat

0 0 0 Hω,i\Hω,i ' V+(X0) ∩ Hω,i.

Since D is geometrically bounded by assumption, we let ∆ be the positive integer deﬁned in Proposition 4.2.2.

We argue by contradiction, and assume that there exist ω, i, j 6= i be such that degx fω,i > aω,i degx gω,i + ∆ in Y aω,i . j j [fω,i/gω,i ]

0 n−1 n−2 Now let Φj : Ak(ω) −→ Ak(ω) be the projection ﬁbration onto the jth hyperplane.

n−2 1 For η ∈ Ak(ω) a closed point, let Cη,j = Spec k(η)[xi] ' Ak(η) be the ﬁber above η,

1 0 η,j and note that Cη,j ' Pk(η). Denote Cη,j ∩ X by Cη,j. Let A = k(η)[xj], then we have the canonical surjection

ω,i η,j pη,j : A A .

n−1 Since ∪ηCη,j = Ak(ω),we have ∪ηCη,j = Hω,i and ∪η Cη,j\Cη,j = Hω,i\Hω,i.

Therefore

∪η Cη,j\Cη,j ∩ X = Hω,i\Hω,i ∩ X.

n−2 We have two cases: If the left-hand side is non-empty, there exists an η ∈ Ak(ω)

1 1 such that Cη,j\Cη,j ∩ X = Pk(η)\Ak(ω) ∩ X is non-empty; let x∞ denote its unique

111 point. Note that since x∞ ∈ X, Y [f/ga] −→ Hω,j must be ´etaleat x∞. In particular, if f := pη,j(f), g := pη,j(g), then the pullback to Cη,j, given by Y [f/ga] −→ Cη,j, must be ´etaleat X∞ ∈ Cη,j.

We have Cη,j ' Spec k(η)[xj], so we let νx∞ = ν1/xj denote the valuation associated to point x∞ at inﬁnity, then

a a a ν1/xj (f/g ) = − degxj (f/g ) = − degxj (f/g )

= a degxj g − degxj f < −∆ < 0

Then by Lemma 5.1.3, Y [f/ga] −→ Cη,j is not ´etaleat x∞, a contradiction.

Now consider the case where Y [f/ga] −→ Hω,j is not necessarily étaleat the point at infinity of any fiber Cη,j. By definition of ∆ in Proposition 4.2.2, we must still have that m(C,x) ≤ ∆ for all pairs of closed points x on the closure of a curve

C ⊂ X.

a But by Lemma 5.1.4, mC,x∞ = − degxj (f/g ) > ∆, a contradiction.

Returning to the Proof of Theorem 5.2.11, by Lemmas 5.2.13 and 5.2.13, there

0 exists a positive integer ∆ such that each cover Yω,i −→ Hω,i has an Artin-Schreier

aω,i representative fω,i/gω,i such that for all ω and i, we have aω,i ≤ ∆ and

degxj fω,i ≤ aω,i degxj gω,i + ∆ for all j 6= i.

aω,i aω,i By Lemma 5.2.9, we may always choose a lift [Fω,i/G ] of [fω,i/gω,i ] such that

aω,i 1 degxj Fω,i = degxj fω,i for all j 6= i. So ﬁx i and let Mi =< [Fω,i/G ]: ω ∈ Ak > be

112 the vector subspace of AG/℘(AG) generated by these lifts. Then the associated cover

0 YMi −→ X trivialises D on each of the ﬁbers Hω,i by construction. By Lemma 5.2.7,

it follows that YMi weakly trivialises the covering datum D for all i. Theorem 5.0.8 shows that the cover YM associated to M = < Mi >i also weakly trivialises D.

i Recall Deﬁnition 5.1.18 of the vector subspaces M∆(AG), M∆(AG) of Artin

Schreier spaceAG/℘(AG) and note that since k is a finite field, M∆(AG) is finite.

By Lemma 5.2.10, M ⊂ M∆(AG), so M is also ﬁnite.

Therefore, fM : YM −→ X is a ﬁnite ´etalecover of X weakly trivialising D. By

Lemma 2.3.10, fM trivialises D, so the covering datum D is effective with a finite realisation by Theorem 2.3.15. The realisation is étaleabove X by Lemma 2.1.15, so all induction assumptions are satisfied.

5.3 Covering data of Prime Power Index

n In this section, we let X ⊂ Pk be an open subset of, and consider geometrically bounded covering data D of cyclic index pm on X. The goal is to show the following:

Theorem 5.3.1. Let k be a ﬁnite ﬁeld with characteristic char(k) = p, and let X

n be an open subset of Pk . Let D a geometrically bounded covering datum of cyclic index pm on X. Then there exists an ´etalecover of Y −→ X realising D.

n n Remark 5.3.2. Setting X = Ak ⊂ Pk , the theorem equals includes the case where

113 m n D is a covering datum of cyclic index p on Ak .

n 1 As before, we let Φi : Ak −→ Ak denote the projection ﬁbration onto the ith

n−1 1 coordinate axis. Denote by Hω,i ' Ak(ω) the ﬁber above a closed point ω ∈ Ak, and

0 by Hω,i = Hω,i ∩D(G) ' D(g) the intersection of the ﬁber with X. If G denotes the

ω,i ω,i dehomogenisation of G at X0, then g = g is the image of G under pω,i : A A .

Then as in the Induction Construction 5.2.12, we assume by induction on n

0 that the restriction Dω,i of D to Hω,i is realisable. As such, it is either trivial

s 0 or a Z/p -cover of Cω,i for some s ≤ m; in either case it is represented by some

f0 fm−1 [f] = [ ga0 ,..., gam−1 ] ∈ Wm(Ag). As in 5.2.12, the base case is given by the covering datum itself.

n Lemma 5.3.3. Let X = D+(G) be a simple open subset of Pk , let D be a geometri-

m cally bounded covering datum of cyclic prime power index p on X, and let Dω,i be

0 the restriction of D to Hω,i. Assume that Dω,i is realised by an Artin-Schreier-Witt

f0 fm−1 cover Yω,i = Y[f], where [f] = [ ga0 ,..., gam−1 ] ∈ Wm(AG)/℘(AG).

Then there exists a postive integer ∆ such that

m−s−1 1. p as ≤ ∆ for all s and

m−s−1 2. p degxj fs − as degxj g ≤ ∆ for all s, for all j 6= i.

Proof. We proceed by induction on the exponent m: For m = 1, it was shown in Lemmas 5.2.13 and 5.2.3 that we can take ∆ to be the bound on ramiﬁcation numbers given by Lemma 4.2.2.

114 So now assume that the lemma holds for m − 1: If D is a geometrically bounded

0 covering datum on X, then there exists a positive integer ∆ such that whenever Dω,i

0 0 m 0 0 f0 fm−2 0 is realisable by a Z/p Z-cover Y[f ] −→ Hω,i for some element [f ] = [ a0 ,..., a0 ] g 0 g m−2 of Wm−1(AG)/℘(Wm−1(AG)), we have that

m−s−2 0 0 p as ≤ ∆ for all s, and

m−s−2 0 0 0 p degxj fs − as degxj g ≤ ∆ for all s, for all j 6= i . (5.3.1)

(k) Recall that for f ∈ Wm(AG), k = 1, . . . m − 1, we let f denote the truncated

Witt vector of length k, and that we have a tower of succcessive Z/pZ-covers

0 Y[f] −→ Y[f (m−1)] −→ ... −→ Y[f0] −→ Hω,i . (5.3.2)

Then applying (5.3.1) to the truncated Witt vector [f (m−1)], we get

m−s−2 0 p as ≤ ∆ for all s, and

m−s−2 0 p degxj fs − as degxj g ≤ ∆ for all s, for all j 6= i . (5.3.3)

0 n−1 n−2 Now let Φj : Ak(ω) −→ Ak(ω) be the projection ﬁbration onto jth coordinate

n−1 n−2 hyperplane of Hω,i ' Ak(ω). Let η ∈ Ak(ω) be a closed point, recall that we set

ω,i η,j η,j A = k(ω)[x1,..., xˆi, . . . , xn], and deﬁne A = k(η)[xj]. Let g = g is the image of G under the combined surjections:

ω,i η,j pη,j ◦ pω,i : A A A ,

1 let Cη,j ' Spec k(η)[xj] = Ak(η) denote the ﬁber above η and set

0 Cη,j := Cη,j ∩ X ' D(g).

115 ω,i For an element f ∈ A , we set f := pη,j(f), and note that degxj f = degxj f.

f 0 f m−1 For f ∈ Wm(AG), we set f := pη,j(f) = [ ga0 ,..., gam−1 ]. Then the pullback of

0 the tower (5.3.2) to Cη,j is given by

0 Y −→ Y (m−1) −→ ... −→ Y −→ C , (5.3.4) [f] [f ] [f 0] η,j where f (s) denotes the truncated Witt vector of length s associated to f.

0 Recall the deﬁnition 4.2.5 of the ramiﬁcation number m 0 of Y −→ C (Cη,j ,z) [f] η,j at a closed point z ∈ Y , then m 0 is equal to the last lower jump of the [f] (Cη,j ,z)

i ramiﬁcation ﬁltration {Gz} associated to z (cf. [14, 2.4]). Then by [14, Lemma

3.1], m is equal to the unique lower jump of the ramiﬁcation ﬁltration {Gi0 } (Y[f],z) z

0 of Y −→ Y (m−1)ω,i , i.e. the ramiﬁcation number m 0 of z in the uppermost [f ω,i] [f ] (Cη,j ,z)

Z/pZ-subcover of the tower (5.3.4).

Recall from the proof of 5.1.12 that Y −→ Y (m−1) is deﬁned by [f ω,i] [f ω,i ]

p p p fm−1 ym−1 − ym−1 = −σm−1(y0, . . . , ym−2, y0, . . . , ym−2) + , gam−1 where the ys are the elements generating K(Y (s) ) over K(Y (s−1) ). [f ω,i] [f ω,i ]

Now let z ∈ Y (l) be the image of z under Y −→ Y (l) for l = m, . . . , 1, l−1 [f ω,i] [fω,i] [f ω,i] and let x ∈ C0 be the image of y under Y −→ C0 . By Lemma 5.1.3, we have η,j [f ω,i] η,j

0 p p fm−1 m(C0 ,z) = −νzm−2 (σm−1(y0, . . . , ym−2, y0, . . . , ym−2) + ) . (5.3.5) η,j gam−1

Recall that D is geometrically bounded, and that we deﬁned ∆ to be the constant

0 from Lemma 4.2.2, then ∆ is such that m 0 = m(C0 ,z) ≤ ∆. Since σj is (Cη,j ,zm−2) η,j

j p p homogeneous of degree p , and νzm−2 (σm−1(y0, . . . , ym−2, y0, . . . , ym−2)) < 0, we have

116 p p m−1 p m−1 νzm−2 (σm−1(y0, . . . , ym−2, y0, . . . , ym−2)) = min {p νzm−2 (yl ), p νzm−2 (yl)} l

m = p min {νzm−2 (yl)} l

m 1 = p min { νzl (yl)} l

m ≥ p min {νz (yl)} l

m fl ≥ p min {pνz ( )} l

m+1 f l ≥ p min {νx ( )} l

m+1 1 f l = p min { νx( )} al l

m+1 f l ≥ p min {νx( )} l

0 fl ∆ By the induction assumption, we have νx( gal ) ≥ − pm−l−2 . This implies that

p p m+1 fl νzm−2 (σm−1(y0, . . . , ym−2, y0, . . . , ym−2)) = p min {νx( )} l

Thus by (5.3.5),

fm−1 p p 0 −νzm−2 ( ) = max{−νzm−2 (σm−1(y0, . . . , ym−2, y0, . . . , ym−2)), m(C0 ,z)} gam−1 η,j = max{pm+1∆0, m } (Y[f],z)

≤ max{pm+1∆0, ∆} .

117 and therefore,

fm−1 fm−1 −νx( ) = −e(zm−2/x)νz ( ) gam−1 m−2 gam−1

m−1 fm−1 ≤ p νz ( ) m−2 gam−1 ≤ max{p2m∆0, ∆} .

00 2m 0 Let us deﬁne ∆ := max{p ∆ , ∆}. For x the point at inﬁnity of Cη,j, we then

am−1 am−1 00 00 0 obtain degxj (fm−1/g ) = degxj (f m−1/g ) ≥ ∆ . Since ∆ > p∆ , we have

m−s−1 00 from 5.3.1 that p degxj fs − as degxj g ≤ ∆ as well, which proves 5.3.3.2.

fm−1 To show 5.3.3.1, assume that gam−1 is written in lowest terms: gm−1 6 | fm−1.

0 fm−1 f 0 Then write gam−1 = α1 αl such that gr is irreducible and relatively prime to f for g1 ···gl 0 f m−1 f η,j 0 all r. Then we have gam−1 = α1 αl ∈ A . Now let x ∈ Cη,j\Cη,j be a point in g1 ···gl

V (gr), then

! fm−1 f m−1 00 am−1 ≤ αr = νgr ≤ νx ≤ ∆ gam−1 gam−1

m−s−1 0 00 and we have p as ≤ p∆ < ∆ for all s < m − 1 from the induction assumption

5.3.1.

By Lemma 5.3.3, we may thus assume that Dω,i is realised by a cover with Artin-

f0 fm−1 Schreier-Witt representative [f] = [ ga0 ,..., gam−1 ] ∈ Wm(AG)/℘(AG) such that for

118 all ω, for all i, we have

m−s−1 0 0 p as ≤ ∆ for all s, and

m−s−1 0 0 0 p degxj fs − as degxj g ≤ ∆ for all s, for all j 6= i (5.3.6) for some ﬁxed positive integer ∆0.

Fω,i,s fs Now let Fω,i,s = Gaω,i,s be the representative of a preimage of [ gas ] deﬁned by Lemma 5.2.3, and set Fω,i = (Fω,i,0,..., Fω,i,m−1) ∈ Wm(AG)). Then since

Fω,i,s fω,i,s deg a = deg a , and the exponents a of G in the denominators remain xj G ω,i,s xj g ω,i,s s

1 unchanged, [Fω,i] ∈ M∆,m(AG) for all ω ∈ Ak.

Now let Mi := hWm(Fp).[Fω,i]iω be the Wm(Fp)-module generated by the Fω,i,

i m then Mi ⊂ M∆0,m(AG). By construction, the associated p - exponent cover YMi −→

0 X trivialises the covering datum D on the ﬁbers Hω,i of Φ|X . By Lemma 5.2.7, YMi

n thus weakly trivialises D over X ∩ Ak . The cover YM associated to M = ∩iMi then

n weakly trivialises D over X ∩ Ak by Theorem 5.0.8.

By Lemma 5.3.4 (see below), M ⊂ MD,m(AG) is ﬁnitely generated. Then

n Lemma 2.3.10 implies that YM trivialises D over X ∩ Ak , and the covering datum

D is effective with a finite realisation by Corollary 2.3.16 of Theorem 2.3.15. The realisation of a covering datum is étaleabove X by Lemma 2.1.15, so all induction assumptions are satisfied.

Lemma 5.3.4. Let AG be the localisation of A = k[x1, . . . , xn] at the set

{Ga : a ∈ Z}, and ﬁx a positive integer ∆. For i = 1, . . . , n, assume we are

119 i given vector subspaces Mi ⊂ M∆,m(AG)of the associated Artin-Schreier-Witt space

Wm(AG)/℘(Wm(AG)). If we set M = ∩Mi, then M ⊂ Mp∆,m(AG).

Proof. We must show that every equivalence class in M contains a representative

Fs (F0,..., Fm−1) such that if Fs = Gas ,

m−s−1 p degxj Fs − as degxj G ≤ p∆ for all j and

m−s−1 p as ≤ p∆ for all j and s , .

i Since M ⊂ Mi ⊂ M∆,m(AG) for all i = 1, . . . , n,[F] has representatives

Fi,s Fi = (Fi,0,..., Fi,m−1) such that if Fi,s = Gai,s , then

m−s−1 p ai,s ≤ ∆ and

m−s−1 p degxj Fi,s − ai,s degxj G ≤ ∆ for all s, for all j 6= i.

We proceed by induction on m: For m = 1, this was shown in Sections 5.2.1

F and 5.2.2: We found F = Ga such that [F] ∈ MD,1 realises D.

m−1 So assume that any Z/p Z-cover can be realised by an element of M∆,m−1(AG).

m−1 In particular, for [F] ∈ M, the Z/p Z-cover Y (m−1) has a representative [Fi ]

(H0,..., Hm−2) ∈ M∆,m−1(AG).

(m−1) Then we have (H0,..., Hm−2) Fi ∈ ℘(Wm(A)), so we can replace Fi by

0 (m−1) 0 Fi = Fi ⊕ (H0,..., Hm−1) Fi = (H0,..., Hm−2, Fm−1). Here,

0 0 Fm−1 Fm−1 = Fi,m−1 + σm−1(H0,..., Hm−2, Fi,0,..., Fi,m−2) = a0 . G m−1 120 Fi,m−1 By construction of Fi, for all j 6= i, we have Fi,m−1 = Gai,m−1 such that ai,m−1 ≤ ∆ and p degxj Fi,m=1 − ai,m−1 degxj G ≤ ∆.

Hs Writing Hs = Gbs , we can apply the induction assumption to get

∆ deg H − b deg G ≤ for all j, for all s = 0, . . . , m − 2 , xj s s xj pm−s−2

∆ b ≤ for all s = 0, . . . , m − 2 . s pm−s−2

By Lemma 5.1.9, this implies that the exponent of G in

σm−1(H0,..., Hm−2, Fi,0,..., Fi,m−2) is at most p∆, and that

degxj (σm−1(H0,..., Hm−2, Fi,0,..., Fi,m−2)) ≤ p∆ .

0 0 0 0 a0 0 am−1 It follows that Fi = [F0/G ,...,Fm−1/G ] satisﬁes

m−s−1 0 p as ≤ p∆ for all s and

m−s−1 0 0 p degxj Fs − as degxj ≤ p∆G for all j 6= i for all s ,

0 so that [Fi] ∈ Mp∆,1 ⊂ A/℘(A).

0 0 0 0 Now we compare Fi to Fj: We must have Fi Fj ∈ ℘(Wm(A)). But

0 0 0 0 0 0 Fi Fj = (0,..., 0, Fi,m−1 − Fj,m−1), so this implies that [Fi,m−1] = [Fj,m−1]

0 in A/℘(A). By Lemma 5.2.10, this implies that Fi,m−1 ∈ Mp∆,1(AG).

H Thus there exists an element Hm−1 ∈ [Fi,m−1] such Hm−1 = GA , where

0 degxj H − A degxj G ≤ p∆ and A ≤ p∆. Then Fi ∼ (F0,..., Fm−2, H), so

(F0,..., Fm−2, H) is the required representative of [F].

121 Chapter 6

Tamely ramiﬁed covering data revisited

In this chapter, we prove Part 2 of the Key Lemma 4.4.4:

n Theorem 6.0.5. Let k be a finite field of characteristic p, and let X ⊂ Ak be an open regular subscheme. If D is any covering datum with cyclic prime-power index lm, where l 6= p, then D is effective with finite realisation.

As noted previously, these results are already known from the works of Wiesend,

Kerz and Schmidt ([21], [5]). We give a new proof using Kummer Theory and families of ´etalecovers (cf. Theorem 5.0.8). The strategy of proof is modelled on that of Chapter 5, with Kummer Theory replacing Artin-Schreier-Witt Theory. By

Lemma 4.4.6, no considerations of geometric boundedness conditions are necessary.

122 6.1 A review of Kummer Theory

Let A be an integrally closed domain which is a finitely generated k-algebra. Given an integer m relatively prime to char(k) = p, enlarge k to a finite field k0 containing the m-th roots of unity, and consider A0 = A ⊗ k0. Let Ae0 be a universal cover of

A0, i.e. the integral closure of A in the maximal seperable algebraic extension of the function ﬁeld K = K(A) in which A0 does not ramify. Consider the short exact

0 sequence of π1(Spec A )-modules

× × 0 n 0 1 −→ µn −→ Ae −→ Ae −→ 1 .

The long exact sequence of cohomology then gives rise to a short exact sequence

0× 0×m 0 0 0 −→ A / A −→ Hom(π1(Spec A ), µm) −→ P ic(A ) −→ 0 , where P ic(A0) denotes the Picard group of the ring A0 (see [11, Prop. 4.11] for details). In particular, if P ic(A0) = 0, we get a canonical isomorphism

0× 0×m ' 0 Ψ: A / A −→ Hom(π1(Spec A ), µm) .

Remark 6.1.1. In our situation, the rings considered will always be aﬃne rings of arithmetical varieties. For such rings, the Picard group is isomorphic to the class group of Cartier divisors (cf. [9]), which is ﬁnitely generated. Therefore, after possibly inverting an element of the ring, the ring A0 can be assumed to have trivial

Picard group P ic(A0) = 0. From now on, we only consider ring A0 satisfying this hypothesis.

123 0 0× 0× m Deﬁnition 6.1.2. We call Km(A ) := A / (A ) the Kummer space of level m of

0 0 r r r A , and note the natural µm-module structure on K(A ) given by ζn ·[a] = [a] = [a ].

r 0 Now let ∆a = {a : r ∈ Z} be the subgroup of K(A ) generated by an element a

0× of A . Note that we have ∆a = ∆a0 for two cyclic subgroups of order n if and only if gma = a0rg0m for some elements g, g0 ∈ A0×, r such that (r, m) = 1. (Without loss of generality, r < m.)

In general, ∆a is a subgroup of order k dividing m. Ψ(a) then has kernel Na, a normal subgroup of index k, and image isomorphic to Z/kZ. Let Ya denote the

0 ´etalecover corresponding to Na, then Ya is equal to the normalisation of Spec A in √ m the ﬁeld K( ∆a).

0 If Y[a] = Y[a0] is a Z/nZ-cover of Spec A , then Na = ker(Ψ(a)) and Na0 = ker(Ψ(a0)) are equal, and of full index n. In particular, Ψ(a) also factors through

0 Na giving rise to an isomorphism Ψ(a): π1(Spec A )/Na0 ' µm. Thus we get

Ψ(a0) π1(Spec A)/Na0 / µm NNN NNN r NNN Ψ(a) NNN' µm

−1 Then Ψ(a) ◦ Ψ(a) is an isomorphism of µm, i.e. represented by exponentiating with an element r relatively prime to m. Thus we can write (Ψ(a))r = Ψ(a0). Since

r 0 Ψ is compatible with the µm-module structure, this implies that [a ] = [a ], which

0 is equivalent to ∆a = ∆a0 for the associated cyclic subgroups of order n in K(A ).

0× 0× n Conversely, if ∆a = ∆a0 ⊂ A / (A ) are cyclic subgroups of order m, then √ m pm clearly K( ∆a) = K( ∆[a0]), so Y[a] = Y[a0].

124 Thus Ψ induces a one-to-one correspondence of simple µn-submodules of order k dividing m with Z/kZ-covers of Spec A0:

0 ∆a 7→ Y[a] ↔ Na < π1(Spec A )

0 Here, Y[a] corresponds to the normal subgroup Na of index k in π1(Spec A ).

0 More generally, if M ⊂ Km(A ) is a µm-module, then we can associate to M the √ 0 n normalisation YM of Spec A in K( M), the ﬁeld obtained by adjoining to K all m-th roots of elements in M. Then YM is the ´etale cover associated to the normal subgroup NM = ∩a∈M Na.

If M is finite, then NM is a finite intersection of normal subgroups with finite

0 index, and thus also of ﬁnite index. If this is the case, then YM −→ Spec A is a

(ﬁnite) ´etalecover.

Conversely, if Y −→ Spec A is an ´etaleGalois ´etale cover of exponent m, then we use the Galois correspondence to write Y as the composition of r Z/kiZ-covers

Y[ai], where ki divides m for i = 1, . . . , r. (Here, r is the rank of the Galois group

as a Z/mZ-module.) Then Y corresponds to the subgroup N := ∩iNai and is equal to the normalisation of Spec A in the compositum of the function ﬁelds

pm 0 K( ∆ai ). Writing M = h[ai]: i = 0, . . . , ri ⊂ Km(A ), this means that the function ﬁeld of Y can be obtained by adjoining all m-th roots of elements in M to √ K, i.e. K(Y ) = K( m M).

We thus obtain the following:

125 Proposition 6.1.3. Let A0 be any finitely generated algebra over a finite field k0 containing the mth roots of unity. Then there is a one-to-one inclusion-reversing

correspondence of µm-submodules M = Σi∈I ∆ai of Km(A) and Galois pro-´etale covers of exponent m of Spec A0:

M 7→ YM ↔ NM := ∩a∈M Na .

M is ﬁnitely generated of rank r if and only if YM is an exponent m-´etalecover of

n Ak0 whose Galois group has r generators.

Proof. The inﬁnite case follows from the ﬁnitely generated case by taking inverse limits (as in Section 5.1).

The properties of the correspondence are summarised as follows:

0 1. For M ⊂ Km(A ), the associated pro-´etalecover YM is the normalisation of √ K in K( n M), the ﬁeld obtained by adjoining all m-roots of elements in M.

0 T 2. If {ai}i∈I is any generating set for M ⊂ Km(A ), then NM = i∈I Nai : Indeed,

Qm ri if f = i=1 fi with ri relatively prime to m, then since Ψ is multiplicative, T Na ⊂ ∩iNai , so NM ⊂ i∈I Nai . The other inclusion is trivial.

3. For any two submodules M, M 0, we have

0 a) MM ↔ NM ∩ NM 0 , and

0 0 b) M ∩ M ↔< NM ,NM >

126 0 We now deﬁne several important submodules of Kummer space Km(A ) for the

0 0 0 0 free k -aglebra A = k [x1, . . . , xn] and its localisation AG = k [x1, . . . , xn]G at some element G ∈ A.

0 For A = k[x1, . . . , xn], we let as above k be a ﬁnite extension of k containing

0 0 the mth roots of unity, and A = k [x1, . . . , xn]. We deﬁne a submodule of Kummer

0 m-space by setting MD(A ) =< [f] : degxj f ≤ D for all j >. Then MD is finitely generated and finite, as every element has a representative contained in the finite

r set {f ∈ A : degxj ≤ D for all j, r < m relatively prime to m}.

i 0 We also deﬁne MD(A ) =< [f] : degxj f ≤ D for all j 6= i >, an (inﬁnitely

0 generated) µm-submodule of K(A ), and show:

i 0 Proposition 6.1.4. In the above notations, the intersection of the spaces MD(A )

0 0 over all i equal to MD0 (A ) for some positive integer D , and thus ﬁnite.

0 Proof. By assumption, each equivalence class [F ] ∈ MD0 (A ) contains an element Fi

i m of MD(A), i.e. such that degxj Fi ≤ D for all j 6= i. If G |Fi for some polynomial

0 m 0 G, without loss of generality replace Fi by Fi = Fi/G . Then degxj Fi ≤ degxj Fi

0 r m for all j, and thus degxj Fi ≤ D for all j 6= i. Then we have Fi = Fj H for some

0 0 polynomial H ∈ A and some r < m that is relatively prime to m. But Fi is not divisible by any n-th power of a polynomial, so H must be constant, and we have

0 0 degxi Fi ≤ r degxi Fj ≤ mD. Thus we have Fi ∈ MmD(A).

The other inclusion is trivial by deﬁnition.

0 0 0 Let AG denote the localisation of A at some element of A , and consider the

127 Kummer space K(AG). Deﬁne µm-submodules

a MD(AG) =< [F/G ]: a ≤ D, degxj F ≤ a degxj G for all j > and

i a MD(AG) =< [F/G ]: a ≤ D, degxj F ≤ a degxj G for all j 6= i > , then MD(AG) is again ﬁnite as it is ﬁnitely generated as a Z-module. Similar to above, we may now show that:

i 0 Proposition 6.1.5. In notations as above, the intersection of the spaces MD(AG)

0 0 over all i contained in MD0 (AG) for some positive integer D , and thus ﬁnite.

a 0 Proof. By assumption, each equivalence class [F/G ] ∈ MD0 (AG) contains elements

ai i Fi/G ∈ MD(AG), i.e. such that Fi and G are relatively prime, and such that

m ai ≤ D and degxj Fi ≤ ai degxj G for all j 6= i. If H |Fi for some polynomial H,

ai 0 ai 0 m without loss of generality replace Fi/G by Fi /G , where Fi = Fi/H . Now we

0 may assume that Fi is not divisble by the m-th power of any polynomial, and since

0 degxj Fi ≤ degxj Fi for all j, it is still true that degxj Fi ≤ ai degxj G for all j 6= i.

Now we have F 0 F r H m i = j Gai Gaj Gb for some polynomial H ∈ A0 and some r < m that is relatively prime to m.

H Write Gb in lowest terms, then we have two cases: If b > 0, then H and G

ai are relatively prime and ai = raj + mb. Multiplying through by G , we have

0 r m 0 Fi = Fj H . But Fi is not divisible by any n-th power of a polynomial, so then H

0 F r Fi j must be constant, and we have Gai = Graj . As the Fl and G are relatively prime

128 by assumption, this implies that ai = raj ≤ mD. Taking degrees on both sides,

0 we also get that degxi Fi = r degxi Fj ≤ raj degxi G = ai degxi G. Thus we have

0 Fi ∈ MmD(AG), as required.

If b = 0, then write H = GcH0 with H0 relatively prime to G, then

0 r c 0 m c 0 0 Fi = Fj (G H ) As before, this implies that G H is contant, so c = 0 and H

0 is constant and we get that Fi ∈ MmD(AG) as above.

6.2 Tamely ramiﬁed covering data of cyclic factor

group

n n In this section, we shall show that for X = Ak or X ⊂ Pk an open subvariety, any

ﬁnite open subgroup H < CX whose index is prime to char(k) = p is realisable, and thus a preimage. Together with the results of the previous chapter, this proves the

Key Lemma 4.4.4.

We note that our construction will not use any theorems on geometric ﬁniteness, contrary to the proof given for the tame variety case in [5].

n Theorem 6.2.1. Let k be a ﬁnite ﬁeld of characteristic p, and let X ⊂ Pk be an open subvariety. If D is a covering datum of cyclic index m on X such that

(m, p) = 1, then D is realisable with a ﬁnite realisation.

Proof. We already showed in the previous chapter that it suﬃces to trivialise D

n over an aﬃne open subset of the form D(G) ⊂ Ak ' D+(X0). We again make use

129 n 1 1 of the fibrations Φi : Ak −→ Ak, let ω denote a closed point of Ak with residue field k(ω) ' k[xi]/(hi(xi)), and Cω,i the fiber of Φi over ω.

0 As before, we denote X ∩ Cω,i by Cω,i, and assume by induction that Dω,i is

aω,i a realisable by a Galois cover Y ω,i whose equivalence classe [fω,i/gω,i ] ∈ Km(AG) [fω,i/gω,i ]

has a representative such that aω,i ≤ D and degxj fω,i ≤ a degxj gω,i for all j 6= i. As before, the base case is given by our assumption that D is a covering datum that is defined on the projective closure D+(G), and thus étaleat the point at infinity of a

1 curve D(g) ⊂ Ak. This ensures that degx f ≤ a degx g in this case. Furthermore, as

D is tame and of index bounded by m, it is geometrically bounded by Lemma 4.4.6, so we may assume that a ≤ D universally for some D. Recalling the canonical surjections A Aω,i, where Aω,i = k(ω)[x ,..., xˆ , . . . , x ], and A Aω,i . They 1 i n G gω,i clearly induce surjections of Kummer spaces analogously to those of Artin-Schreier space:

ω,i pω,i : Km(A) Km(A ) and

ω,i πω,i : Km(AG) Km(Ag ) .

Then the lemma below is proven entirely analously to Lemma 5.2.9 of the Artin-

Schreier case:

a ω,i a −1 a Lemma 6.2.2. Given [f/g ] ∈ Km(Ag ), there exists a preimage [F/G ] ∈ πω,i([f/g ])

which has a representative such that degxj F = degxj f for all j 6= i.

So for each ω, i, we let Fω,i ∈ A denote this representative, and set

aω,i 1 Mi = h[Fω,i/G ]: ω ∈ Aki. Then as before, Mi corresponds to a pro-´etalecover

130 of X trivialising all the ﬁbers of Φi. As the ﬁbers are regular, every closed point

of X is a regular point of a ﬁber. So D is trivialised at all closed points, i.e. YMi weakly trivialises D on X. Combining Theorem 5.0.8 and Property 4) of the Kum- mer correspondence 6.1.3, this implies that the cover YM −→ X corresponding to

M = ∩iMi also weakly trivialises D. M ⊂ MD(AG) is ﬁnite, so YM is an ´etale cover. By Proposition 2.3.10, it gives a full trivialisation of D, and is thus re-

a alised by an element [F/G ] ∈ M ⊂ MD(AG). In particular, we have a ≤ D and

degxj F ≤ a degxj G for all j, so all the induction assumptions are satisﬁed.

Remark 6.2.3. As remarked in Section 5.5.2, this includes the case of aﬃne n-space

n X = Ak since we then have X = D+(G), G(X0,...,Xn) = X0.

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