Higher Dimensional Class Field Theory: The variety case

Linda M. Gruendken

A Dissertation

in

Mathematics

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

2011

Florian PopSupervisor of Dissertation

Jonathan BlockGraduate Group Chairperson

Dissertation Committee

Florian Pop , Samuel D. Schack Professor of Algebra, Mathematics

David Harbater, Christopher H. Browne Distinguished Professor inthe School of Arts and Sciences, Mathematics

Robin Pemantle, Merriam Term Professor of Mathematics

HIGHER DIMENSIONAL CLASS FIELD THEORY: THE VARIETY CASE

COPYRIGHT

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

Let k be a finite field, and suppose that the arithmetical variety X ⊂ Pnk is

an open subset in projective space. Suppose that CX is the Wiesend idele class

group of X, πab1 (X) the abelianised fundamental group, and ρX : CX −→ πab1 (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

subgroups of πab1 (X). Furthermore, we show that for an etale cover X ′′ −→ X

with maximal abelian subcover X ′ −→ X, the reciprocity morphism induces an

isomorphism CX/NCX′′ ' 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 Idele Class 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

5.2.1 Affine n-space Ank . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2.2 The Case of Open Subsets of Pnk . . . . . . . . . . . . . . . . 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

πab1 (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 alge-

braic 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 ”idele class group” CX in terms of the arithmetical curves and

closed points contained in X, and gets a canonical homomorphism

ρX : CX −→ πab1 (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 SpecZ (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-

procity law for the abelianised tame fundamental group πab,t1 (X), which classifies

all the tame etale abelian covers of X. A generalised Artin map can be defined

between

ρX,t : CtX −→ πab,t1 (X) ,

where CtX 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.

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

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

ρX : CX −→ πab1 (X)

be the Wiesend recprocity morphism. Then the following hold:

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

subgroups of CX and open subgroups N of πab1 (X); it is given by ρ−1X (N) 7→ N . The

reciprocity morphism ρX is a continuous injection with dense image in πab1 (X).

2) A subgroup of CX is a norm subgroup iff it is open, of finite index and geometri-

cally bounded.

3

3) If X ′′ −→ X is an etale connected cover, X ′ −→ X the maximal abelian subcover,

then NCX′′ = NCX′ and the reciprocity map gives rise to an isomorphism

CX/NCX′′'−→ 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 idele class group and

the reciprocity homomorphism, giving the right definitions as to guarantee commu-

tativity 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.

The Key Lemma is shown for open subvarieties X ⊂ Pnk 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 SpecZ:

Definition 2.1.1. X is said to be an arithmetical scheme over Z if it is integral,

separated and of finite type over SpecZ.

We assume that all schemes are arithmetical, unless otherwise stated, and dis-

tinguish two cases:

Definition 2.1.2. If the structural morphism X −→ SpecZ has open image, X

will be called flat. If this is not the case, then the image of X −→ SpecZ 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 degtrK denotes the

transcendence degree of a field K. For a scheme X, let dimX 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 −→ SpecZ, and let f(X) be the closure of the image of f in SpecZ. Denote

the function field of X by K(X). Then the Kronecker dimension d of X is defined

as

dimX = degtrK(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 etale cover of X is a finite etale cover

Y −→ X, and a pro-etale cover of X is the projective limit of etale covers of X.

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

curve, i.e. a one-dimensional integral closed subscheme of X, then C ⊂ Pnk 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-Poincare characteristic 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 .

Definition 2.1.8. If k is an arbitrary field, and X ⊂ Pnk 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 degkX.

Fact 2.1.9. Let k be an arbitrary field, X ⊂ Pnk 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 = C ⊂ X.

Note: The converse of this is not true: If C ⊂ P2k is the completion of

V (x− yn) ⊂ A2k ⊂ P2

k, 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 : Frobx ∈ 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 etale cover 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 ′ 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 ′, 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 Frobx = 1 for all Frobenius elements

above x, this follows directly from the Chebotarev Density Theorem 2.1.11. Let

S = x : Frobx = 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 etale cover 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,

X ′ ⊂ X a dense open subscheme, Y ′ −→ X ′ an etale cover and Y the normal-

ization of X in k(Y ′). Suppose that for every curve C on X with C ′ = C ∩X ′ 6= ∅,

the etale cover Y ′ × C ′ −→ X × C ′ extends to an etale cover of C. Then Y −→ X

is etale.

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 etale covers 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 fix s : SpecΩ −→ X, a geometric

point of X. If Y −→ X is an element of EtX , consider the geometric fiber Y ×X

SpecΩ, and let Fibs(Y ) denote its underlying set. Any morphism Y1 −→ Y2 in

EtX induces a morphism Y1 ×X SpecΩ −→ Y2 ×X SpecΩ. Applying the forgetful

functor, we get an induced set-theoretic map Fibs(Y1) −→ Fibs(Y2).

Thus, Fibs(·) 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 basepoint s is the

automorphism group of the fiber functor Fibs associated to s, and is denoted by

π1(X, s).

Lemma 2.2.2. The fundamental group is profinite and acts continuously on Fibs(X).

Proof. See [19, Theorem 5.4.2].

10

Proposition 2.2.3. The functor Fibs induces an equivalence of the category of

finite etale covers 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 let s, s′ : SpecΩ −→ X be two geometric points of X.

Definition 2.2.4. A path from s to s′ is an isomorphism of fiber functors

Fibs'→ Fibs′ .

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

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′ two geo-

metric points of X, then there always exists a path p from s and s′.

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

πab1 (X) does not depend on the choice of geometric basepoint [19, Remark 5.5.3].

Definition 2.2.6. The maximal abelian quotient πab1 (X) is called the abelianised

fundamental group of X.

Now let X, Y be schemes with geometric points s′ and s.

11

Definition 2.2.7. In the situation above, if f : Y −→ X is a morphism of schemes

such that f s = s′, then the morphism is said to be compatible with s and s′.

SpecΩ s //

s′##GG

GGGG

GGG Y

f

X

Proposition 2.2.8. In the situation above, a morphism that is compatible with s

and s′ induces a morphism on fundamental groups f : 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 point s′′ of Y so that s′ and

s′′ are compatible. Indeed, let x be the image point of s′ 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 point s′′ 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:

f : π1(X, s)p · p−1

' π1(X, s′′) −→ π1(Y, s′)

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

or not, the preimage f−1(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 etale covers of X, it is pro-representable and thus

representable in the larger category of profinite limits of finite etale covers 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-etale covers of X.

In particular, fixing a universal cover with geometric basepoint x amounts to

choosing a system of compatible basepoints for every pro-etale cover 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 X of X,

let x denote its geometric basepoint, and let f : Y −→ X be a finite connected etale

subcover. Then

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

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

strict to connected etale subcovers Z of X.

2. f is a trivial cover if and only if f 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 etale cover, and let w, y be the geometric basepoints

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

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

Z //

W

g

Yf

// 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 etale cover 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-etale cover f : Y −→ X is the inverse

limit of finite etale covers. Analogously to above, we then have the following propo-

sition:

Proposition 2.2.12. Let X be a connected scheme. Fix a universal cover X of X,

let x denote the associated geometric basepoint, and let f : Y −→ X be a pro-etale

subcover. Then

14

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

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

restrict to connected etale subcovers Z of X.

2. f is a trivial cover if and only if f 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-etale cover, let w, y denote the geometric

basepoints of W , respectively Y which are compatible with x. Let Z ⊂ W×XY

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

Then we have a commutative diagam

Z //

W

g

Yf

// 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 etale cover 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 chap-

ters:

15

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

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

is a morphism, then f : 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 C, and note that the

normalisation might lie outside the scheme X.

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

morphism with the inclusion of the curve in X.

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

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

iC : π1(C) −→ 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 ix.

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 = f × ix and f |C = f × iC 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:

π1(D)eiD //

ef |C

π1(Y )

ef

π1(C)eiC // π1(X)

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

then we get the commutative diagram:

π1(y)eiy

//

ef |x

π1(Y )

ef

π1(x)eix // 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 C 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(C) 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 C and x, respectively.

We notice that whenever C is any curve containing the closed point x, the fibered

product gives commutative diagrams

x× C //

C

iC

xix // X

in which x× C is finite. If x is a regular point of C, then the left vertical morphism

is an isomorphism.

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

groups

π1(x) //

π1(C)

iC∗

π1(x)ix∗ // π1(X) .

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

NC < π1(C), Nx < π1(x) is a covering datum if the NC and Nx satisfy the fol-

lowing compatibility condition:

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

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

subgroups of π1(x).

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

2. For a curve C ⊂ Y , we let NC = f−1C (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(C) :

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

Y DC −→ C and xD −→ 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,D′ be two covering

data on X. Let C ⊂ X denote a curve, and let x ∈ |X| denote a closed point of

X. Let fDC : Y DC −→ C, fD′

C : Y D′

C −→ C denote the covers of C defined by D

and D′. Also let fDx : xD −→, fD′

C : xD′ −→ x denote the cover of x defined by D,

respectively D′.

19

We say that D is a subdatum of D′ if fDC is a subcover of fD′

C for all curves

C ⊂ X, and if fDx is a subcover of fD′

x 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(C) 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 etale cover

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.

5. If the pullback i∗(D) to an open subset Ui→ X has one of the above proper-

ties, we say that D has that property over U .

Let f : XN −→ X be a Galois etale cover corresponding to the open nor-

mal subgroup N / π1(X), and recall Notations 2.2.13.1 and 2.2.13.4. For every

20

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

pullbacks N(C) := i−1C (N), N(x) = i−1

x (N). They correspond to the etale covers

fC : XN × C −→ C and fx : XN × x −→ x, respectively. Then for any x ∈ C × x,

the diagram

π1(x)

// π1(C)

π1(x) // π1(X)

commutes since it is induced from the commutative diagram

x

// C

x // X

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

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

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

Notation 2.3.7. Given a covering datum D on X, we let Y DC −→ C, y −→ x be the

etale covers of connected schemes corresponding to NC / π1(C) and Nx / π1(x).

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

Y × C × Y DC

// Y × C

Y × x× y

// Y × x

Y DC// C y // x

Notation 2.3.8. Now let Z, Z ′ be the connected components of Y×C and Y×C×Y DC ,

respectively, and let z, z′ 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:

Z ′

pC

qC // Z

fC

z′

px

qx// z

fx

Y DCgC // C y gx

// x

Following the remarks on the correspondence between connected pro-etale covers

and closed subgroups of the fundamental group of a scheme X, we note that the

covers Z ′ −→ C and z′ −→ x correspond to the subgroups NC∩N(C) and N(x)∩Nx

of π1(C) and π1(x), respectively. We also note that we can identify N(C) with

fC(π1(Z)), NC with gC(π1(Y DC )) 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(C) and π1(x), but also as

inclusions/equalities of subgroups in π1(YC), π1(Y DC ) 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-etale cover, 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(C) for all curves C ⊂ X.

22

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

f−1C (NC) = π1(Y × C) 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.

(c) Nx ⊃ N(x) for all closed points x ∈ |X|.

(d) We have f−1x (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-etale cover f : Y −→ X, TFAE:

(a) f realises D.

(b) We have f−1x (Nx) = π1(Y × x), g−1

x (N(x)) = π1(y) for all closed points

x ∈ |X| and f−1C (NC) = π1(Y × C), g−1

C (N(C)) = π1(Y DC ) for all curves

C ⊂ X.

(c) The covers pC and qC are trivial for any curve C ⊂ X and the covers

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

4. Analogously to 3), TFAE:

23

(a) f weakly realises D.

(b) We have f−1x (Nx) = π1(Y × x), g−1

x (N(x)) = π1(y) for all closed points

x ∈ |X|.

(c) The covers px, qx are trivial for any closed points x ∈ |X|.

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

X. An etale cover 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 pC . If x is a point of C×x, then the morphism pex induced by base

changing from x to x 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. LetX be an arithmetical scheme, and letD be a covering datum

on X. Then D is called effective if it is realised by a pro-etale 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 ex-

ists a curve C ⊂ X such that the canonical morphism iC : C −→ X

(cf. Definition 2.2.13) and the induced morphism iC : π1(C) −→ π1(X) induces

an isomorphism

π1(C)/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

N(C) := i−1C (N) for any curve C ⊂ X, and likewise N(x) := i−1

x (N) for closed

points x. Then we have natural inclusions

π1(C)/N(C) → π1(X)/N and π1(x)/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 × C −→ C has the same Galois group as f . Therefore

π1(C)/N(C) → π1(X)/N

is an isomorphism, and we have deg(f) = [π1(C) : 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 etale covers of X weakly realising D over an open subset U .

Then Y1 = Y2.

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

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

realisation. The realisation of D is given by Y ′, 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-

tions Y1, Y2, and let Ni(x) := i−1x (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

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 ′ //

Y ′

V // Y1

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

groups as well, contrary to assumption.

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

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

induces on curves C ′ = C ∩ U of U extend to etale covers YC of the full

curve C in X. By Proposition 2.1.15, Y ′ −→ X is an etale cover; 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−1eC (N), N(x) := i−1x (N) denote the pullback

to π1(C), respectively π1(x).

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

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

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

NC of π1(C) 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

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 etale by 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

etale covers, 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

be a covering datum on X. If there exists an open subscheme Ui→ 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 etale.

Claim 2.3.17. If the covering datum D is trivialised by a finite cover f : Y −→ X,

then there also exists a finite etale cover f ′ : Y ′ −→ X trivialising D.

Proof. Let X ′ ⊂ X be an open dense subset such that f |X′ : Y ′′ = X ′×X Y −→ X ′

is etale. (Such an X ′ exists by the purity of the branch locus, cf. [9, Section 8.3, Ex.

28

2.15].) Then the restriction D|X′ of D to X ′ is trivialised by f |X′ , and thus effective

by the results of the previous paragraphs. In particular, there exists a subgroup

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 ′′ 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

U , i.e. an open normal subgroup N such that i−1x (N) = Nx for all x ∈ U . In partic-

ular, 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 etale cover 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 [Frobxi ]. Since the

regular locus of C is of codimension one, it is finite, so by replacing those xi which

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

Since Y trivialises D, we have fC(π1(D)) ≤ NC as subgroups of π1(C). Let

29

N = iC(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

N(x) = i−1x (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

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

C ′ × Y is irreducible.

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

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

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

Proof. We first show that if we define N(C) := i−1C (N), then N(C) = NC . Indeed,

iC is easily seen to be injective by applying the second criterion of

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

First we consider the case where C is normal, i.e. C 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 −→ SpecA is a finite

etale cover, then D is also affine, say D = SpecE. We thus have to show that if

E = A[f1, . . . , fn] = A[x1, . . . , xn]/(g1, . . . , gn) is a finite etale algebra over A, where

the gi are polynomials with coefficients in A, then E is the tensor product F ⊗B of

some finite etale algebra 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 −→ SpecB is finite and surjective [9, Remark 4.3.11]. Last, it

can easily be seen that if F were not etale over 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

etale cover 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

D′ = C×Y , a normal cover of C since the etaleness of Y −→ X is preserved under

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 C is not necessarily

a subscheme of X. We have iC : C −→ C −→ X, and are given a cover D of C.

Since C 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 etale cover Y , and since D = D×C, the claim follows by associativity

31

of the fibre product.

In conclusion, we have that N(C) = i−1C (iC(NC)) = NC . Now if z is a regular

point of C, then there is a unique point π1(z) of C lying above z ∈ C, and we have

a natural isomorphism π1(z) ∼= π1(z) fitting into the following diagram:

π1(z) //

∼=

π1(C)

π1(z)

;;wwwwwwwww

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

N(C ′) = i−1C′ (N) denote the preimage under the natural map iC′ : π1(C ′)→ π1(X),

M ′ its image in G, and M the image of N in G. If xi is the unique point of C lying

above xi, we have a diagram:

π1(C) // G π1(C ′)oo NC// M,M ′ NC′

oo

π1(xi)

OO::vvvvvvvvv

ccHHHHHHHHH

Nxi

OObbFFFFFFFFF

;;xxxxxxxxx

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

have, in particular, that Frobxi ∈ M iff Frobxi ∈ M ′ for all i. Since the xi were

picked such that the Frobxi are all the conjugacy classes of G, this implies that

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

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 ix : 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 Idele Class Group

In this chapter, we let X be an arithmetical scheme, and define the Wiesend idele

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 ∈ SpecZ 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 SpecR,

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 SpecOK .

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 C. 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 C 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 C 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 idele group (after Wiesend) be the abelian topological

group group defined by

JX =⊕x∈|X|

Z.x⊕⊕C⊂X

⊕ν∈C∞

K(C)×ν

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 idele group JX , then JX is

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

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

Carch∞ = ∅. Then the subgroup

J 1X =

⊕C⊂X

⊕ν∈Carch∞

K(C)×ν

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 D, 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 D, 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 idele groups with induced morphism fJ .

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

Among other things, they imply that if the diagram

C

f1

f2// D

f3

Xf4

// Y

37

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

JCf1J

f2J

// JDf3J

JXf4J

// JY

of induced morphisms on the idele groups.

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

maps

C −→ C −→ X

thus induces a morphism

(iC)J : J eC −→ JX (3.1.1)

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

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

j = ΣC (iC∗ jC) :∑C⊂X

K(C)× → JX . (3.1.2)

Definition 3.1.6. Let X be an arithmetical scheme of Kronecker dimension

dim(X) ≥ 2, or a regular curve. Then the Wiesend idele class group of X is

38

defined to be the quotient of JX by the image of j:

CX = JX/ im(j) =

⊕x∈|X|

Z.x⊕⊕C⊂X

⊕ν∈C∞

K(C)×ν

/∑C⊂X

im(iC∗ jC)

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

image of the subgroup J 1X 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 fJ : 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 fJ .

There are several cases to consider:

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

(fJ |C jC)(t) = fJ |C(((iν(t))ν∈C∞ , (νx(t))x∈ eC) (3.1.3)

=∑ν∈C∞

ν(t)[k(ν) : k(y)].y +∑x∈ eC

νx(t)[k(x) : k(y)].y

= (∑

ν∈VK(C)

ν(t)[k(ν) : k(y)]).y

= (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

K(C)×

jC // J eC iC∗ //

(f |C)J

JXfJ

0 // Z.y can. // JY ,

where the first square commutes by the above computation, and the second square

commutes as the middle vertical map is just fJ iC∗ restricted to J eC . Thus, the

outer rectangle also commutes, and the image of K(C)× is in the kernel of fJ , as

required.

2) f(C) is dense inside a curve D of Y . There exists a unique f : C −→ D

induced by f such that the following diagrams commute:

C

∃!h

// C

f |C

// X

f

D // D // Y.

Since the maps jC : K(C)× −→ JC and jD : K(D)× −→ JD factor through

J eC −→ JC and J eD −→ JD, respectively, we get diagrams

K(C)×

NK(C)/K(D)

jC // J eC //

hJ

JC(f |C)J

// JXfJ

K(D)×jD // J eD // 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 eD. We have a direct summand for every discrete valuation ω on K(D). If ω

does not have a center on D, then the ω-summand is K(D)×ω , and if ω = νey for some

40

center y ∈ D, then the ω-summand is Z.y. Let pω and pey denote the corresponding

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

jC(t) = ((νx(t))x∈| eC|), (iν(t))ν∈C∞) ,

jD(s) = ((νy(s))y∈| eD|), (iω(s))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 D, any ν on K(C) restricting to ω on

K(D) does not have a center on C. 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

(pω hJ jC)(t) =∑

ν:ν|K(D)=ω

hJ ((iν(t)) =∑

ν:ν|K(D)=ω

NK(C)ν/K(D)ω(t)

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]):

NK(C)/K(D)(t) =∑

ν:ν|K(D)=ω

NK(C)ν/K(D)ω(t) for all t ∈ K(C)×ν .

2) Now consider ω-summands for ω = νey with center y ∈ D. Then K(C)×ν maps

into the ω-summand Z.y iff we have ν|K(D) = ω. Furthermore, we get a contribution

from the discrete x-summand Z.x if and only if f(x) = y, which is equivalent to the

condition νex|K(D) = νey. We have

(pω jD NK(C)/K(D)(t) = (pey jD)(NK(C)/K(D)(t)) = vey(NK(C)/K(D)(t)) . y

and

(pey F jD)(t) =∑

ν∈C∞ : ν|K(D)=νeyν(t) [k(ν) : k(y)] . y +

∑ex:f(ex)=ey νey(t) [k(x) : k(y)] . y

=

∑ν∈C∞ : ν|K(D)=νey

ν(t) [k(ν) : k(y)] +∑

νex : ex∈ eC, νex|K(D)=νeyνey(t) [k(x) : k(y)]

. y

=

∑ν : ν|K(D)=νey

ν(t) [k(ν) : k(y)]

. y

Thus, showing that F jC and jDNK(C)/K(D) agree on the y-summand amounts

to showing that

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

∑ν : ν|K(D)=νey

ν(t)[k(ν) : k(y)]

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 idele group, respectively the Wiesend idele class group of X.

Then JX is generated by the images of J eC under the canonical maps iC∗, and CX

is generated by the images of C eC under iC∗.

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 = SpecA

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 SpecA is thus an integral

curve C.

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

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

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

set of generators of mp modulo mp2, i.e. not a coordinate system of Ap. Thus x is

a regular point of SpecB.

3.2 The Reciprocity Homomorphism

In this section, we show the existence of a homomorphism ρX : CX −→ πab1 (X),

which will be called the reciprocity homomorphism. This morphism is a generalisa-

tion 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

of the idele group: ψ : JX −→ πab1 (X), and then show that ψ factors through CX .

We define as follows:

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

πab1 (X) associated to x.

2) For a curve C ⊂ X, recall that C denotes the normalisation of C, and iC

the canonical homomorphism iC : C −→ X (cf. 2.2.13). Also recall from 3.1.1 that

44

(iC)J : J eJ −→ JX denotes the induced morphism on idele groups .

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

ρC : J eC −→ πab1 (C) is the reciprocity homomorphism associated to the normal

curve C (cf. [12, Ch. ]).

Then ψ is trivial on the image of j :∑

C K(C)× −→ JX , so it descends to a

homomorphism ρX : CX −→ πab1 (X).

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

homomorphism ρX : CX −→ πab1 (X) defined above is called the reciprocity homo-

morphism of X.

Let f : X −→ Y , and recall from Section 2.2 that f denotes the induced

morphism on fundamental groups. Recall also that the abelianised fundamental

groups πab1 (X), πab1 (Y ) are well-defined without reference to geometric basepoints.

Notation 3.2.2. Given f , f as above, we let f : πab1 (X) −→ πab1 (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

CXf∗

//

ρX

CYρY

πab1 (X)f

// πab1 (Y )

commutes.

45

Proof. It is most convenient to show this at the level of the idele groups, i.e. to

show that the diagram

JXfJ

//

ρX

JYρY

πab1 (X)f

// πab1 (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

are finite fields, πab1 (x) = π1(x)), Cx = Z. x and similarly for y. The reciprocity

morphisms are

ρx : Z. x −→ π1(x)

ρy : Z. y −→ π1(y) .

Here, ρx : Z. x −→ πab1 (x) maps 1. x 7→ Frobx, where Frobx = FrobFqn denotes the

Frobenius associated to the finite field k(x). Similarly, Froby = FrobFq , so we have

Frobnx = Froby in π1(y); thus the upper square in the following diagram commutes.

Z.x //

Z.y

1.x //

n.y

π1(x) //

π1(y)

Frobx

// Frobx = Frobny

πab1 (X) // πab1 (Y ) Frobx // Frobx = Frobny

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

under the canonical homomorphism π1(x) −→ πab1 (X). We shall keep this notation

46

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

functoriality of π1 and πab1 (cf. Section 2.2 or [19, Ch. 5]).

2) Now let C ⊂ X be a curve, and recall the canonical morphism

iC : C −→ C −→ X (cf. 2.2.13). Recall also the notation for induced mor-

phisms on the idele and idele class 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

fJ on the idele groups 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

πab1 (C)

iC∗

(f |C)∗// π1(y)

iy

πab1 (X)f∗

// πab1 (Y )

commutes. Moreover, by the definition of the Artin map on idele class group with

restricted ramification (cf. [12, Section]), the square

K(C)×ν//

ρC

Z.y

ρy=Frob(.)y

πab1 (C)(f |C)∗

// 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 D, 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:

K(C)×νN //

ρC

K(D)×ω

ρD

πab1 (C)(f |C)∗

// πab1 (D)

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

N(t) = u′πfn, where u′ is a unit of OK(D), so ρD(N(t)) = Frobfnk(ω), where Frobk(ω)

is the generator of Gal(Fp/k(ω)) inside Gal(Fp/Fp).

On the other hand, ρC(t) = Frobnk(ω). Viewing πab1 (C) and πab1 (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

Frobk(ν) gets mapped to Frobfk(ω). So (f |C)∗(Frobk(ν)) = Frobfk(ω) and the above

diagram commutes, as claimed.

2.2.2) Now assume that ω has a center on the normalisation D, 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

K(C)×νfJ

//

ρC

Z.yρD

πab1 (C)(f |C)∗

// π1(y) .

48

Indeed, let π be a uniformiser of K(C)ν . For t ∈ K(C)×, write t = uπn, where

u is a unit of K(C)ν . We have fJ (t) = ν(t)[k(ν) : k(y)]. y, which gets mapped to

Frobyn[k(ν):k(y)] in π1(y).

On the other hand, we have ρC(t) = Frobk(ν)n, so since Frobk(ν) = Frob

[k(ν):k(y)]y ,

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 Z, the procyclic group generated by

the Frobenius automorphism Frobx 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→ Frobx. We get the following

Proposition 3.3.1. Let x be a zero-dimensional arithmetical scheme. Then ρx

induces an exact sequence

0 −→ Cxρx−→ 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 ' Zx

the idele class group of X.

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

open subgroups N of the fundamental group π1(x); it is given by ρ−1x (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 etale connected 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,

and open subgroups of π1(x) are of the form < Frobnx >, where the bar denotes the

topological closure. Then clearly, the association

N = < Frobnx > 7→ ρ−1x (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 ≤

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

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 SpecOK 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 con-

nected etale covers 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 etale cover 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-idele class

group ofK(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

0 −→ CCρC−→ πab1 (C) −→ Z/Z −→ 0 .

If C0C and πab,01 (C) denote the degree-0 parts of the class and fundemental group,

respectively, then we obtain surjectivity of the reciprocity morphism restricted to

C0C, and thus an isomorphism

C0C ' πab,01 (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 idele class group CC. Then there exists a

canonical exact sequence

0 −→ DC −→ CCρC−→ πab1 (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

open subgroups N of πab1 (C); it is given by ρ−1C (N) ← N . The inverse cor-

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

closure.

If H < CC corresponds to N < πab1 (C), then

CC/H ' πab1 (C)/N .

53

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

3. If D −→ C is an etale connected cover, D′ −→ C the maximal abelian sub-

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

phism

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 Z/Z.

For the second statement, we note that for an etale cover 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 idele class 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 subgroupNCD correspond to a unique abelian

etale cover 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

open subvarieties of Pnk .

We begin begin by introducing induced covering data, the link between covering

data considerations and subgroups of the Wiesend idele class 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.

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

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

and image of the reciprocity morphism ρX : CX −→ πab1 (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

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

for open subvarieties X ⊂ Pnk 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 sub-

groups of the Wiesend idele class 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 cov-

ering 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

be an open subgroup of the Wiesend idele class group. Let HC := i−1C∗(H) and

Hx := i−1x∗ (H) denote the preimages in C eC and Cx ' Z.x, respectively. By the

reciprocity law of the base cases (cf. Section 3.3), the images ρ eC(HC) = NC and

ρx(Hx) = Nx are open subgroups of πab1 (C), respectively π1(x). Let NC denote the

56

subgroup of π1(C) corresponding to NC ≤ πab1 (C).

Definition 4.1.2. Let X be a regular arithmetical scheme, and let H < CX be an

open subgroup of the Wiesend idele class 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 x is a point of x× C, then the commutative square

x

// C

x // C

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

Cex

// C eC

π1(x)

// π1(C)

Cx // CC π1(x) // π1(C) .

Let Hex denote the preimage of H in Cex. Then Hex is the preimage of HC and of

Hx under the natural morphisms above. By the functoriality of the reciprocity

morphism, see the diagram below. Thus, ρex(Hex) = Nex is actually equal to the

preimage of Nx and NC in π1(x). In particular, these two concide whenever x is a

point on C, and the compatibility condition for a covering datum is satisfied.

57

Cxρx

Cexρex

oo // C eCρ eC

π1(x) π1(x) //oo π1(C) .

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. IfH < CX of finite index, then induced covering datumDH 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-

sation πab1 (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

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

CX/H ' πab1 (X)/N .

58

Proof. Let H ′C be the preimage of N in CX , and H ′C , H ′x the preimages of H in

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

πab1 (C) // πab1 (X)

C eC iC∗ //

ρ eCOO

CX

ρX

OO

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 H ′C = HC for all curves C ⊂ X. Since the images of C eCgenerate 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 iC : π1(C) −→ π1(X). If N / π1(X) is an

open subgroup, let NC := i−1C (N) denote its preimage in π1(C). By Lemma 2.1.14,

there exists a curve C ⊂ X such that the injection π1(C)/NC → π1(X)/N is an

isomorphism. Let N NC denote the images of N , respectively NC in the abelianised

fundamental groups, and recall from above that ρ−1X (N) = H. Then we obtain a

diagram of injections

CX/H ρX // πab1 (N)

C eC/HC ρ eC //

?

OO

πab1 (C)/NC

'

OO

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

59

the Artin map ρ eC induces an isomorphism

ρ eC : C eC/HC ' πab1 (C) .

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:

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

(which exists by Fact 2.1.10), say U ⊂ Pnk . Let Ank ⊂ Pnk be the canonical

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

U → Ank ⊂ Pnk . Let U ⊂ Pnk be the closure of U in Pnk .

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

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

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

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

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 D′ be a covering datum on X, and consider the

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

etale cover fC : Y DC −→ C, thus deg fC = [π1(C) : NC ]. Notice that Y DC −→ C

is smooth, and let gY DC denote the geometric genus of (the completion of) C.

Let kC be the field of constants of Y DC . Finally, let degkC RC denote the

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]):

gYC = (deg fC)(gC − 1) + 1 +1

2(degkC RC) (4.2.1)

= [π1(C) : NC ](gC − 1) + 1 +1

2(degkC RC)

where RC is the branch locus of fC inside the compactification YDC of Y DC .

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

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

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

in the covering datum is solely determined by degkC RC , the degree of the

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

gYC ≤ m(gC − 1) + 1 +1

2(degkC RC)

61

Definition 4.2.2. Let Ui→ Pnk 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 degkC C ≤ d, the cover

Y DC −→ C defined by D satisfies gY DC ≤ δ(d)

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

is a condition only involving the regular curves C in U with C ⊂ U ⊂ Pnk 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 idele class group. We say that the subgroup H is geomet-

rically 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 etale over 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

above the points xj: If Giy is the i-th ramification group at y, a point lying over

some xj, we have

degRC =n∑j=1

∑y∈f−1

C (xj)

(∞∑i=0

|Giy| − 1

)[k(y) : kC ] .

62

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

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

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

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

Then since Giy is a subgroup of GC for all j and i, the sum

∑∞i=0 |

(Giy| − 1

)is

bounded above by mj · (|GC | − 1). Note also that the set SC given by

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

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

degkC RC ≤ n ·m ·M · (|GC | − 1) . (4.2.2)

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

curve C once the embedding i : U → Pnk 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.

Proposition 4.2.6. Let U ⊂ Pnk 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,

degk(U\U

)· degk C =

n(C)∑j=1

multj(xj)[k(xj) : k] ≥ n(C) (4.2.3)

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

n(C) ≤ degk C · degU\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.

Definition 4.2.7. Let U ⊂ Pnk 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

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

64

Suppose that C has degree degkC C ≤ d. The set of ramification numbers of D is

defined to be the set of the ramification numbers m(y,C) of points y ∈ YDC 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

points y ∈ Y DC\Y DC .

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 degkC ≤ d, and let Y DC −→ C be the

cover induced by D. Putting together Equation (4.2.2) and Proposition 4.2.6, we

have

degkC RC ≤ N(d) ·mC ·M(d) · (|GC | − 1) ≤ N(d) ·mC ·M(d) · (h− 1) (4.2.5)

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,

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

Then

gY DCn= deg fC(gC − 1) + 1 +

1

2degkC RC

n→∞−→ ∞ .

65

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

closed points y lying above C\C in Y DC −→ C, then

degkC RC ≤ N(d) ·m ·M(d) · (|GC | − 1) .

and thus

gY DCn= deg fC(gC − 1) + 1 +

1

2degkC RC

≤ h(gC − 1) + 1 +N(d)mM(d) (h− 1)

2

whenever C is a curve with degkC C ≤ d.

Now consider a Galois etale cover Y −→ U . As above, let U ⊂ Pnk denote

the closure of U in projective space, and C the closure of a regular curve C ⊂ U .

If Y = UN corresponds to the open normal subgroup N / π1(U), let DN be the

corresponding covering datum.

Proposition 4.2.9. Let U ⊂ Pnk be a regular quasi-projective arithmetical variety,

and let f : Y −→ U be the etale cover corresponding to the open normal subgroup

N / π1(U). Then the induced covering datum DN 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 etale over 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

fC is etale over C, we have RC ⊆ Y \Y . Noting that degk RC = [kC : k] degkC RC ,

66

we get

degk RC = (degk C) · (degk R) ≤ d [kC : k] (degk R) .

for any curve of degree degkC C ≤ d. Then

gYC = (deg fC)(gC − 1) + 1 +1

2degkC RC

≤ (deg f)(g(n, d)− 1) + 1 +degk RC

2[kC : k]

≤ (deg f)(g(n, d)− 1) + 1 +d

2(degk R)

is bounded by constants depending only on invariants of U , invariants of the given

etale cover 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.

Proof. Let XN −→ X be the finite realisation of D, then DN = D. The covering

datum DN is geometrically bounded by the proposition.

Now we define the notion of a geometrically bounded covering data for any reg-

ular 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 onX. ThenD is geometrically bounded onX if there exists a quasi-projective

regular open subscheme Ui→ 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

covering datum that is effective with finite realisation, let U ′i′

→ X be another

quasi-projective regular open subscheme. Apply Lemma 2.3.13.3 to U ′ and U ∩U ′,

then the pullback i′∗(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 cov-

ering datum on X. If D is effective with a finite realisation, then D is geometrically

bounded.

Proof. Let Ui→ X be a quasi-projective regular open subscheme, and let Y −→ X

be the etale cover 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

correspondence of these subgroups with open subgroups of finite index in πab1 (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 N1, N2 be two open

subgroups of πab1 (X). Then the following are equivalent:

1) N1 ⊂ N2

2) ρ−1X (N1) ⊂ ρ−1

X (N2)

3) (ρX iC∗)−1(N1) ⊂ (ρX iC∗)−1(N2) for all curves C ⊂ X

4) (ρX ix∗)−1(N1) ⊂ (ρX ix∗)−1(N2) 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

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

subgroups of CX and open subgroups N of the abelianised fundamental group πab1 (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: IfH < CX is an open subgroup of

the class group which is finitely realisable, say by fN : XN −→ X, where N /π1(X),

map H 7→ N , where N is the image of N in πab1 (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

ρ−1X (N) = H.

Conversely, if N is an open normal subgroup of πab1 (X), ρ−1X (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−1X (N) in C eC and Cx, respectively, then by reciprocity for curves and

points, we have ρ−1eC (N(C)) = HC for all curves C ⊂ X, ρ−1x (N(x)) = Hx for all

closed points x ∈ |X|.

70

C eC ρ eC //

iC∗

πab1 (C)

eiC

Cxρx

//

ix∗

πab1 (x)

eix

CXρX // πab1 (X) CX

ρX // πab1 (x)

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

closed points x and we have Dρ−1X (N) = DN . Since N realises DN by definition, this

shows that N realises Dρ−1X (N).

We can now state the Main Theorem of this thesis. Let X be a regular arith-

metical variety, and recall from that an open subgroup H ≤ CX induces a covering

datum DH .

Main Theorem 4.3.4. Let k be a finite field, and let X ⊂ Pnk 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-

rically bounded, and the converse will be shown for open affine subsets of Pnk 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 idele class group CX . Recall from 3.1.9

that a subgroup is a norm subgroup iff H = f∗(NCY ) for some etale cover Y −→ X.

Corollary 4.3.7. Let X ⊂ Pnk 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 idele class group and open sub-

groups N of πab1 (X); it is given by ρ−1X (N) 7→ N . Then ρX is a continuous injection

with dense image in πab1 (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 : X ′′ −→ X is an etale connected cover, X ′ −→ X the maximal abelian

subcover, then NCX′′ = NCX′ and the reciprocity map gives rise to an isomorphism

CX/NCX′′'−→ 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 etale cover

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 particu-

lar, 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∗(CX′′) 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

CX/N (f∗) ' πab1 (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-

metic varieties by Remark 3.1.7. As πab1 (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-

tame or curve-tamely ramified if the covers Y DC −→ C 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 −→ C must have degrees prime to p, or

be etale covers.

Definition 4.4.3. A cover YC −→ C 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:

Key Lemma 4.4.4. Let k be a finite field of characteristic p, and let X ⊂ Pnk 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 arith-

metical 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.

Proof. Let C ⊂ X be a regular curve as in Notation 4.2.1, and let fC : Y DC −→ C be

the cover of C defined by D. Recall that we let m(C,y) denote the ramification num-

ber 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 idele class group. Let H be an open subgroup

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.

IfH is finitely realisable, then the realisation f : X ′ −→ X is a finite cover which,

in particular, trivialises DH . Recall that iC : C −→ X denotes the composition of

the normalisation with the inclusion morphism, and that iC∗ : denotes the induced

morphism on the Wisend idele class groups. Let HC := i−1C∗(H), HiC := i−1

C∗(Hi)

denote the preimages in C eC , and set N(C) := ρ eC(HC), Ni(C) := ρ eC(HiC).

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

have ρ−1X (Ni) = Hi. Then N :=

⋂ni=1 Ni is open and normal as a finite intersection

of open normal subgroups of π1(X). Let YN be the corresponding etale cover of X.

Then we have

ρ−1X (N) = ρ−1

X (∩iNi) = ∩iρ−1X (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 cover-

ing 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.

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

is always a subcover of Y DHeC −→ C.

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 ofDH . SinceDH 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-etale covers of the corresponding

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

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

Specifically, in both cases we construct several pro-etale covers 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,

finite over X. This way, we obtain an etale cover of X ⊂ Ank 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 etale cover, which finishes the proof

of Part 1) of the Key Lemma.

We then construct several pro-etale covers 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

X. This way, we obtain an etale cover of X ⊂ Ank which trivialises the covering

datum, and by Proposition 2.3.13 this means that D is also effective with a fi-

nite realisation. Thus, D is realised by an etale cover, 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 Nii∈I be a family of closed subgroups of π1(X) such

that each of the associated pro-etale covers 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:

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

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

Since Ni is closed, it follows that

Nx ⊇ < i−1x (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 A be a universal cover of A, i.e. the integral closure of A inside the maximal

separable algebraic extension K/K of the function field in which A is unramified.

Let ℘ : A −→ A be the map sending x 7→ xp − x then ker℘ = Z/p, then we

obtain a short exact sequence of π1(SpecA)-modules

1 −→ Z/pZ −→ A℘−→ A −→ 1 .

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

short exact sequence

0 −→ A/℘(A) −→ π1(SpecA′) −→ Z/pZ −→ 0

Thus we get a canonical isomorphism

Ψ : A/℘(A)m'−→ Hom(π1(SpecA),Z/pZ) .

Note that a group homomorphism f : π1(SpecA) −→ Z/p has kernel of index

p if and only if its image is non-trivial. Conversely, any normal index-p subgroup

of π1(SpecA) gives rise to such a homormisphm (i.e. the induced canonical pro-

jection). Thus, we see that Hom(π1(SpecA),Z/p) and thus A/℘(A) classifies all

Z/p-covers of SpecA.

80

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

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

f ′ = f + hp − h for some h ∈ A. Since ap = a for all a ∈ Fp, we then have that

af ′ = af + ahp − ah = af + (ah)p − (ah) ∼ af , as required. Recalling the natural

Fp-vector space structure of Hom(π1(SpecA),Fp), it follows immediately that Ψ is

an Fp-linear map.

Remark 5.1.1. Let a ∈ Fp, f ∈ A, Hf (z) = zp − z − f . Since ap = a for all a ∈ Fp,

we have Haf (z) = aHf (a−1z), so (Hf ) = (Haf ) as ideals in A[z]. Thus f and af

define the same cover of SpecA, which we denote by Y[f ].

Conversely, if Y[f ] = Y[f ′] is a non-trivial Z/p-cover of SpecA, then Nf = ker(πf )

and Nf ′ = ker(πf ′) are equal. Thus the surjection πf : π1(SpecA) Fp gives rise

to an isomorphism πf : π1(SpecA)/Nf ′ ' Fp.

π1(SpecA)/Nf ′πf

//

πf ′ &&MMMMMMMMMMMMFp·a

Fp

Then, since πf ′ also gives such an isomorphism, πf ′ π−1f is an automorphism of

the additive group Fp, and as such representable by an element a ∈ F∗p. Therefore,

πf ′ = 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 SpecA:

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

M =⊕

i∈I Fp.[fi] of A/℘(A) and exponent-p etale covers of SpecA:

M 7→ YM ↔ NM := ∩f∈MNf .

M is finitely generated of rank n if and only if YM is a (Z/p)n-extension of Ank .

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:

1. If fii∈I is any generating set for M , then NM =⋂i∈I Nfi : Indeed, if

f =∑m

i=1 ai.fi with ai ∈ Fp, then by the additive property of Ψ, Nf ⊂ ∩iNfi ,

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 ′, let I be a basis of

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 ′ - see [8, Theorem III.5.1]). Then clearly,

NM =⋂f∈I Nf ⊃

⋂f∈I′ Nf = NM ′ as claimed.

3. For any two submodules M , M ′, we have

a) NM+M ′ = NM ∩NM ′ , and

b) NM∩M ′ =< NM , N′M >

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- etale cover 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

function field K(Y[f ]) := K(C)[z]/(Hf (z)), where Hf (z) = zp − 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).

1. φ is etale at x if and only if there exists f ′ ∈ [f ] such that νx(f′) ≥ 0. If this

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 ′ ∈ [f ] is such that

νx(f′) < 0.

83

3. If φ is totally ramified above x, then let f ′′ ∈ [f ] be such that

νx(f′′) = minf ′∈[f ]νx(f ′). Then gcd(νx(f

′), p) = 1, and the ramification

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)].

Corollary 5.1.4. Let k be any finite field, and let φ : Y [f/ga] −→ P1k be a Z/pZ-cover

of curves that is etale over the open subscheme D+(g) ⊂ P1k. If

P1k\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

1. φ is etale at xj if and only if νj(f/ga) ≥ 0, and totally ramified iff νj(f/g

a) <

0.

2. If νj(f/ga) < 0 is relatively prime to p, then we have mj = −νj(f/ga).

3. One can always find a representative of f ′/ga′

of [f/ga] such that for all j,

νj(f′/ga

′) 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 character-

istic p, and A a finitely generated k-algebra with separable closure A.

In the following sections, we shall either set A = k[x1, . . . , xn], or

A = AG := k[x1, . . . , xn]G, the localisation of k[x1, . . . , xn] at the set Ga : a ∈ Z

Recall that the m-th Witt polynomial Wm ∈ A[X0, . . . , Xm] is defined as

Wm(X0, . . . , Xm) = Xpm

0 + pXpm−1

1 , . . . , pmX0 .

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 ,

S1 =1

p(Xp

0 + Y p0 − S

p0) +X1 + Y1 =

1

p(Xp

0 + Y p0 − (X0 + Y0)p) +X1 + Y1 ,

and in general

Sm =1pm

(Wm−1(X0, . . . , Xm−1) +Wm−1(Y0, . . . , Ym−1)−Wm(S0, . . . , Sm−1)) +Xm + Ym .

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:

Letting X = (X0, . . . , Xm−1) and Y = (Y0, . . . , Ym−1) be two elements of Am,

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].

Remark 5.1.6. Since char(A) = p, we have [f0, . . . , fm−1]p = [fp0 , . . . , fpm−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):

Notation 5.1.7. For f = [f0, . . . , fm−1] ∈ Wm(A), k ≤ m, let f (k) denote the trun-

cated 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.

Lemma 5.1.9. degXj σk = degYj σk ≤ pk−j.

86

Define a homomorphism ℘ : Wm(A) −→ Wm(A) by setting

℘([a0, . . . , am]) = [a0, . . . , am]p [a0, . . . , am], where denotes the subtraction op-

eration of Witt vectors. By Remark 5.1.6, we have

ker℘ = [a0, . . . , am] : api = ai for all i

= [a0, . . . , am] : ai ∈ Fp for all i

= Wm(Fp)

' Z/pmZ .

Then we have an exact sequence of π1(Spec A)-modules

0 −→ Z/pm −→ Wm(A)℘−→ Wm(A) −→ 0 .

Applying the long exact sequence of etale cohomology, since Wm(A) is affine,

the connecting homomorphism induces an isomorphism

Ψ : Wm(A)/℘(Wm(A))'−→ Hom(π1(SpecA),Z/pmZ) . (5.1.1)

The isomorphism Ψ makes it possible to associate to each Witt vector f an

exponent-pm cyclic cover of SpecA:

Let f = [f0, . . . , fm] ∈ Wm(A). The homomorphism

Ψ(f) : π1(SpecA) −→ Z/pmZ

has image pm−sZ/pmZ ' Z/psZ, where s ≤ m, and corresponds to the normal

subgroup kerψf of index ps, which in turn defines a Galois Z/psZ-cover of SpecA,

and denoted by Y[f ].

87

Definition 5.1.10. Let f = [f0, . . . , fm] ∈ Wm(A). The cover Y[f ] −→ SpecA

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

Y[f ] −→ SpecA is a proper Z/pmZ-cover if and only if f0 6∈ ℘(A).

Noting that Hom(π1(SpecA),Z/pmZ) has a natural structure as a Z/pmZ-

module, we can define a natural Z/pmZ-module structure on Wm(A)/℘(Wm(A))

so that Ψ becomes a Z/pmZ-module homomorphism:

For a ∈ Wm(Fp), f ∈ Wm(A), we have ap = a, and define a · [f ] := [af ].

Now if f ′ is another representative of [f ], then f ′ = f ⊕ hp h for some

h ∈ Wm(A). Then

af ′ = 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

M =⊕

i∈IWm(Fp).[fi] of Wm(A)/℘(Wm(A)) and cyclic ps etale covers of SpecA,

s ≤ m:

M 7→ YM ↔ NM := ∩f∈MNf .

88

M is finitely generated of rank n if and only if YM is a finite cover of SpecA.

Proof. For f ∈ Wm(A)/℘(Wm(A), let Y[f ] −→ SpecA 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

yp0 − y0 = f0,

yp1 − y1 = −σ1(yp0,−y0) + f1,

. . . ,

ypm−1 − ym−1 = −σm−1(yp0, . . . , ypm−2,−y0, . . . ,−ym−2) + fm−1 .

The first equation determines a Z/pZ-cover Y[f0] of SpecA, 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 SpecA (see the graphic below).

Notation 5.1.13. Let f ∈ Wm(A) be a Witt vector of length m, and recall that f (k)

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 SpecA, 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

Z/pmZ

Z/pm−1Z

Y[f (m−2)]

Z/pZ

...

Z/pZ

Y[f0]

Z/pZ

SpecA

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

Hf (k+1)(y0, . . . , yk−1) = ypk − yk + σk(yp0, . . . , y

pk−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 ∈ F×p we have

Hf0(a−10 y0) = a−1

0 Ha0f0(y0). Similarly, using Lemma 5.1.8, if ak−1 6= 0, we have

Hf (k)(a−1k−1y0, a

−1k−1y1, . . . , a

−1k−1fk−1)

= (a−1k−1)pypk−1 − a

−1k−1yk−1 + σk((a−1

k−1)py0, . . . , (a−1k−1)py0,−a−1

k−1y0, . . . ,−a−1k−1yk−1)− fk−1

= a−1k−1

(ypk−1 − yk−1 + σk(yp0 , . . . , y

pk−1,−y0, . . . ,−yk−1)− ak−1fk−1

)= a−1

k−1H(af)(k)(y0, . . . , yk−1)

Therefore, whenever a ∈ (Z/pmZ)×, 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 ]

for all a ∈ Wm(Fp). Conversely, if the Z/pmZ-covers Y[f ], Y[f ′] are equal for two

90

elements f , f ′ ∈ Wm(A)/℘(Wm(A)), we must have

Ψ(f) = Ψ(f ′) : π1(SpecA) Z/pmZ .

Then Nf = ker Ψ(f) and Nf ′ = ker Ψ(f ′) are equal, and Ψ(f) gives rise to an

isomorphism Ψf : π1(SpecA)/Nf ′ ' Z/pmZ.

π1(SpecA)/Nf ′Ψf //

Ψf ′ ''PPPPPPPPPPPZ/pmZ

·a

Z/pmZ

Then, since Ψ(f ′) also gives such an isomorphism, Ψ(f ′) Ψ(f)−1 is an automor-

phism of the additive group Z/pmZ, and as such representable by an element

a ∈ (Z/pmZ)×. Therefore, Ψ(f ′) = a ·Ψ(f). As we showed Ψ to be an Z/pmZ-linear

map, aΨ(f) corresponds to [af ], so we have [f ′] = [af ].

Thus we get a one-to-one correspondence between Z/pmZ-submodulesWm(Fp).[f ]

of Artin-Schreier-Witt space Wm(A)/℘(Wm(A)), and Z/psZ-covers of SpecA, 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 imme-

diately, as do the statements about general submodules when taking direct limits

(see the proof of Proposition 5.1.2 for details).

Remark 5.1.14. Y[f ] has degree pm over SpecA 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 letAG denote the localisation

of k[x1, . . . , xn] at the set Ga : a ∈ N. Define subsets of Artin-Schreier-Witt space

Wm(AG)/℘(Wm(AG)) by

M∆,m(AG) =⟨[(

F0

Ga0, . . . ,

Fm−1

Gam−1

)]: pm−j−1aj ≤ ∆ for all j,

pm−j−1(degxi

Fj − aj degxiGj)≤ ∆ for all i

⟩.

M i∆,m(AG) =

⟨[(F0

Ga0, . . . ,

Fm−1

Gam−1

)]: pm−j−1aj ≤ ∆ for all j,

pm−j−1(degxi

Fj − aj degxiGj)≤ ∆ for all j 6= i

⟩.

Proposition 5.1.17. Let M∆,m(AG), M i∆,m(A)G) be the subsets of Artin-Schreier-

Witt space defined above. Then M∆,m(AG) and M i∆,m(AG) are Z/pm-submodules of

Wn(AG)/℘(Wn(AG)).

Proof. The addition in Wm(AG)/℘(Wm(AG)) is inherited from the addition of Witt

Vectors, so

»„F0

Ga0, . . . ,

Fm−1

Gam−1

«–+

" F ′0

Ga′0, . . . ,

F ′m−1

Ga′m−1

!#

=

»„F0

Ga0+

F ′0

Ga′0,F1

Ga1+

F ′1

Ga′1

+ σ1(F0

Ga0,F ′0

Ga′0

), . . .Fm−1

Gam−1+

F ′m−1

Ga′m−1

+ σm−1(. . . ,Fm−2

Gam−2, . . . ,

F ′m−2

Ga′m−2

)

!#

92

Now Aj be the power of G in the denominator of

σj(. . . ,Fj−1

Gaj−1 , . . . ,F ′j−1

Ga′j−1

). By Lemma 5.1.9, Aj ≤ maxl<jpj−lal, pj−la′l. Since we

assumed pm−l−1al, pm−l−1a′l < ∆ for all l, this implies that pm−j−1Aj ≤ ∆.

Now let aj be the power of G in the denominator of the j-th coordinate in the

sum, then aj is bounded by aj ≤ maxaj, a′j, Aj ≤ ∆pm−j−1 , as required.

Similarly, since by assumption degxj(Fl/Gal),

degxj(F′l /G

a′l) ≤ ∆/pm−l−1

for all j and l, Lemma 5.1.9 implies that

degxj(σj(. . . ,Fj−1

Gaj−1, . . . ,

F ′j−1

Ga′j−1)) ≤ max

l<jpl∆/pm−l−1 = ∆/pm−j−1

for all j (or for all j 6= i).

Thus, M∆,m(AG) and M i∆,m(AG) are additive subgroups. It’s clear that the

action of Z/pm on Wm(A)/℘(Wm(A)) leaves the degree of numerators and denom-

inators of any element FGa

unchanged, so M∆,m(AG), M i∆,m(AG) are submodules.

Definition 5.1.18. For m = 1, we obtain the vector subspaces

M∆(AG) := M∆,1(AG) =⟨[F/Ga] : a ≤ ∆,degxi

F ≤ a degxiG+ ∆ for all i

⟩⊂ AG/℘(AG)

M i∆(AG) = M i

∆,1 =⟨

[F/Ga] : a ≤ ∆,degxjF ≤ 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∆,m(A) :=⟨[F0, . . . , Fm−1] : pm−s−1 degxi Fs ≤ ∆ for all i, s

⟩M i

∆,m(A) :=⟨

[F0, . . . , Fm−1] : pm−s−1 degxj Fs ≤ ∆ for all s, for all j 6= i⟩,

both of which are Z/pm-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

of prime index p. We show that for X = Ank and, more generally, for any X ⊂ Pnk

an open subset, a geometrically bounded covering datum is realisable.

5.2.1 Affine n-space Ank

We begin by considering the affine n-space X = Ank . Let A = k[x1, . . . , xn] and let D

be a geometrically bounded covering datum of cyclic index p on X = SpecA. 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

bounded covering datum of index p on Ank . Then there exists an etale cover of An

k

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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, . . . , xi, . . . , xn] and let φi : Ai −→ A, φi(xj) = xj

be the natural inclusions, then the induced morphism

Φi : Ank −→ An−1

k

are called the projection fibrations onto the coordinate hyperplanes.

Fix a projection fibration Φi. A closed point ω ∈ An−1k corresponds to a maximal

ideal mω = (g1, . . . , gi, . . . , gn), where gj = gj(x1, . . . , xi, . . . , 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

residue field of ω, the fiber Cω,i of Φi above ω is isomorphic to Spec k(ω)[xi] ' A1k(ω).

As before, we let Y DCω,i −→ Cω,i denote the cover induced by the subgroup

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

F , the base change of the cover YF over Cω → An 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 poly-

nomial of k(x1, . . . , xi, . . . , 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.

Proof. Write k(ω) = k(θ1, . . . , θi, θn), where θj+1 is such that

k(x1, . . . , xi, . . . , xj+1)/(g1, . . . , gi, . . . , gj+1) = k[x1, . . . , xi, . . . , xj ]/(g1, . . . , gi, . . . , gj)(θj+1

).

Then any a ∈ k(ω) can be written as a =∑aIθ

I , where I = (α1, . . . , αi, . . . , αn−1)

is a multi-index such that∑αj ≤ [k(ω) : k]. If we now define an element of

A by Fa(x1, . . . , xi, . . . , xn) :=∑aI x

I , then p(Fa) = a in k(ω). Moreover, given

f(xi) = adxdi + . . . + a0 ∈ k(ω)[xi], we can define F (x1, . . . , xn) = Fadx

di + . . . 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

M∆(A) and M i∆(A).

Lemma 5.2.4. Let A = k[x1, . . . , xn] and fix a positive integer ∆. Let Mi ⊂M i∆(A)

be vector subspaces in Artin-Schreier space A/℘(A). If we set M = ∩Mi, then

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M ⊂ M∆(A), i.e. every equivalence class in M contains an element G such that

degxi G ≤ ∆.

Proof. As M ⊂ Mi ⊂ M i∆(A) for all i, every equivalence class of M contains an

element Fi such that degxi(Fi) ≤ D. We have Fi = Fj + hp − h for some h ∈ A.

Write

Fi =∑α

aαxαi ,

Fj =∑α′

bα′xα′

i ,

where the coefficents aα and bα′ are polynomials in k[x1, . . . , xi, . . . , xn], and consider

the terms of Fi that are of highest xi-degree D. Note that for any non-constant

h ∈ k[x1, . . . , xn], the monomials of highest xi-degree in hp − h contain all xk to a

power divisible by p. Thus, if D is not divisible by p, or if aD contains a monomial

whose exponents are not all multiples of p, and Fj+hp−h is equal to Fi, then hp−h

must be of xi-degree strictly smaller than D. In particular, aDxDi is left unchanged

by subtracting hp−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

in aD, and such that pc divides D. Then we can write

aD = a′D(xpc

1 , . . . , xpc

n ) =∑I

aIxpcα1

1 . . . xpcαnn ,

where the sum goes over all multi-indices I = (pcα1, . . . , pcαn) of size n− 1. Then

97

let a′I be such that a′pc

I = a′I , and define h =∑

I a′Ix

D′ , where D = pcD′. Clearly,

hpc − h = (hp

c − hpc−1

) + (hpc−1 − hpc−2

) + . . .+ (hp − h) ∈ ℘(k[x1, . . . , xn])

and F ′i = Fi + (hpc −h) ∼ Fi has degxi(F

′i ) < d. Repeating the process if necessary,

we thus obtain F ′i of xi-degree D′ such that either p does not divide D′, or such

that aD′ 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

degxi(Fj) = degxi(F′i ) ≤ 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.

Proposition 5.2.5. Let k be a finite field, and let Φi : Ank −→ An−1

k be the ith

coordinate fibration. Let D be a geometrically bounded covering datum of index p

on Ank , and let Y DCω,i −→ Cω,i be the cover of Cω,i defined by D. Then for all ω and i,

there exists an Artin-Schreier representative fω,i ∈ k(ω)[xi] such that Y[fω,i] = Y DCω,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 DCω,i = Y[fω,i]

for some fω,i ∈ k(ω)[xi]. Let s = degxi(fω,i). Whenever s > 0 and p|s, the

highest term is of f is of the form axps′

i . We may then make a change of variables

z 7→ z − a′xs′i , where a′p = 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

Recall that Y[fω,i] denotes the normalisation of A1k(ω) in

K(fω,i) := k(ω)(xi)[z]/(zp − z − fω,i(xi)) .

The regular compactification of A1k(ω) is P1

k(ω); let Y [fω,i] denote the normalisation

of P1k(ω) in K(fω,i). We have a commutative diagram

Y[fω,i]//

fCω,i

Y [fω,i]

fCω,i

A1k(ω)

// P1k(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:

Indeed, the degree of Cω,i as a subvariety of Pnk 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 degkCω,iCω,i = 1

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

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P1k(ω)\A1

k(ω), then [k(y∞) : k(x∞)] = 1. Note also that k(x∞) = k(ω), and that we

have gCω,i = gA1k(ω)

= 0 for all fibers Cω,i, Cω,i\Cω,i = P1k(ω)\A1

k(ω). Then Hurwitz’s

formula (cf. 4.2.1) gives

gY[fω,i]= p(gCω,i − 1) + 1 +

1

2degk(ω) RC

= p(gCω,i − 1) + 1 +1

2

∞∑i=0

(|Gy∞| − 1) [k(y∞) : k(ω)]

= 1− p+1

2

my∞∑i=0

(|Gy∞| − 1)

=(my∞ − 1)(p− 1)

2

By Lemma 5.1.4, we have

gY[fω,i]=

(s− 1)(p− 1)

2(5.2.1)

Since D is geometrically bounded, Definition 4.2.2 gives a constant δ = δ(1)

such that gYCω,i ≤ δ for all ω ∈ An−1k , for all coordinate fibrations i. Thus, s is

bounded by

s ≤ 2δ

p− 1+ 1 ,

which proves the Proposition as claimed.

Proof of Proposition 5.2.1. Following the notation established above, let

Y DCω,i −→ Cω,i denote the induced by the covering datum D, and for each ω and i,

let fω,i ∈ Aω,i be an Artin-Schreier representative of Y DCω,i .

Recall that Cω,i ' A1k(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

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the fibers have degree degkC Cω,i = degk(ω) A1k(ω) = 1 over their field of definition

kC = k(ω). Thus, by Proposition 5.2.5, there exists a positive integer ∆ such that

deg(fω,i) ≤ ∆ for all ω,i.

From Lemma 5.2.3, we may now find lifts Fω,i ∈ π−1([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

of M∆(A) and M i∆(A) from Definition 5.1.18. Then Mi ⊂ M i

∆(A) for all i and

YMi−→ An trivialises all NCω,i by construction.

The following proposition shows that YMiis 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 −→ X ′ be a fibration with regular fibers, D a covering

datum on X. If Y −→ X is a pro-etale cover 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

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 = 〈Mi〉i, 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) = 〈[F ] : degxi F ≤ ∆ for all i〉.

Any class in M∆(A) has a representative of total degree ≤ n∆, so M∆ is finitely

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generated. Thus M is of finite rank, so YM −→ Ank trivialises D by Theorem 2.3.10.

Then D is realisable with a finite realisation by Theorem 2.3.15.

5.2.2 The Case of Open Subsets of Pnk

Let k be a finite field. In this section, we consider open subsets X ⊂ Pnk , and

show that bounded covering data of prime index p on such X are finitely realisable

(Theorem 5.2.11).

Since Pnk is Noetherian, we can write X as a finite union of principal open

subsets: X = ∪mj=1D+(Gj), where for any j, Gj ∈ k[X0, . . . , XN ] is a homogeneous

polynomial and defines a hypersurface of Pnk . We let A = k[x1, . . . , xn], and consider

the open affine subset Ank = D+(X0) ' SpecA of Pnk . The intersection of a simple

open D+(G1) with An is given by D(G), where G ∈ A is the dehomogenisation of

G1.

Let AG denote the localisation of A at the set Ga : a ∈ N. Then

SpecAG ' D(G) ⊂ Ank .

Definition 5.2.8. Let φi : k[xi] → A be the inclusion, then we shall call the

induced morphism Φi : Ank −→ A1

k the projection fibration onto the i-th coordinate

axis.

If ω ∈ A1k is a closed point with residue field k(ω) = k[xi]/(hi(xi)), let Aω,i denote

the ring k(ω)[x1, . . . , xi, . . . , xn]. Then the fiber above the closed point ω ∈ A1k is

102

given by Cω,i ' SpecAω,i ⊂ Ank .

Recall from Section 5.1.1 that the surjection pω,i : A Aω,i induces a surjection

of Artin-Schreier spaces

πω,i : A/℘(A) Aω,i/℘(Aω,i)

such that πω,i([F ]) = f for any F ∈ p−1ω,i(f).

Now let gω,i = 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

p′ω,i : AG Aω,ig

given by sending F/Ga 7→ f/ga. Composing with the natural projection, we get a

surjection p′ω,i : AG Aω,ig /℘(Aω,ig ) whose kernel contains ℘(AG). Thus we get a

surjection of Artin-Schreier spaces

π′ω,i : AG/℘(AG) Aω,i/℘(Aω,i) (5.2.2)

such that π′ω,i([F/Ga]) = f/ga for any F ∈ p′−1

ω,i (f).

Lemma 5.2.9. Given [f/ga] ∈ Aω,ig /℘(Aω,ig ), there exists a preimage

[F/Ga] ∈ π−1i ([f/ga]) which has a representative such that degxj F = degxj f for all

j 6= i.

Proof. We prove the claim by constructing F ∈ p−1i (f) such that degxj F = degxj f

for all j 6= i. Then π′ω,i([F/Ga]) = f/ga so F/Ga 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

can write any a ∈ k(ω) as a =∑m−1

k=0 αkθk, where m = [k(ω) : k] and αk ∈ k. Now

let Fa(xi) =∑m−1

i=0 αkxki , then Fa maps to a under the surjection k[xi] k(ω).

Writing f as f(x1, . . . , xi, . . . , xn) =∑

I aIxIxki , where the sum goes over multi-

indices I of size n − 1, and xI = xα11 . . . xi . . . x

αn−1n for the multiindex

I = (α1, . . . , αn−1), we may define F by F (x1, . . . , xn) =∑

I FaIxIxki . 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 spaceAG/℘(AG) classifies p-exponent

covers of SpecAG. Also recall the special vector subspaces M∆(AG), M i∆(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

Ga : a ∈ Z, and fix a positive integer ∆. Let Mi ⊂ M i∆(AG) be vector sub-

spaces of the associated Artin-Schreier space AG/℘(AG). If we set M = ∩Mi, then

M ⊂ M∆(AG), i.e. every equivalence class in M contains an element F/Ga such

that a ≤ ∆, and degxj F ≤ a degxj(G) for all j.

Proof. As M ⊂Mi for all i, any equivalence class contains an element Fi/Gai 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 ′/Gb such that

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FiGai

+F ′p

Gpb− F ′

Gb=

FjGaj

Let ai = maxai, bp, then aj ≤ ai and the above is equivalent to

FiGeai−ai + F ′pGeai−bp − F ′Geai−p = FjG

eai−aj (5.2.3)

Bringing the terms containing F ′ to the other side, we have equality of xi-degrees

as follows:

degxiFi = maxp degxi

F ′+(ai−bp) degxiG,degxi

F ′+(ai−p) degxiG,degxi

Fj+(ai−aj) degxjG .

So if degxi F′ ≤ b degxi G, then this implies that

degxi Fi ≤ maxai degxi G, degxi Fj + (ai − aj) degxi G .

and therefore

degxi Fi ≤ ai degxi G .

If this is the case, we can take F/Ga = Fi/Gai and are done. So now assume that

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

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

Fi = α0xα11 . . . xαn

n + (terms of lower i-order, same xi-degree) + (terms of lower xi-degree) ,

G = β0xβ11 . . . xβn

n (terms of lower i-order, same xi-degree) + (terms of lower xi-degree) ,

F ′ = α0xα′11 . . . x

α′nn + (terms of lower i-order, same xi-degree) + (terms of lower xi-degree) ,

and we must have that αl−aiβl = p (α′l − bβl) for all l. In particular, p must divide

αl − aiβl for all l.

Thus we can write αl−aiβl = pNl, and let N ′l = Nl−βl. Then setting b′ = baipc,

we have b′p ≤ ai. Let F ′ = γ′0xN ′11 . . . x

N ′nn , where γ′0 is such that α0 = γ′p0 β

(bp−ai)0 .

Then we can replace FiGai

by

Fi + F ′pGai−bp − F ′Gai−b

Gaii

,

thereby replacing the term of highest i-order in the numerator F ′i by a lower-order

term. For all j 6= i, one now checks that the xj-degree of F ′i is still bounded by

the xj-degree of the denominator: degxj F′i ≤ a′i degxj G. Also, we have chosen b′

so that b′p ≤ ai,k, so that the power of G in the denominator remains unchanged:

a′i = 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

F ′i

Ga′i

such that p 6 |αl − aiβl for some l, or such that degxi F′i is less than that of the

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denominator. In the latter case, we get that degxi F′i ≤ ai degxi G, and can take

F ′i/Ga′i as the required representative. In the former case, we again compare F ′i/G

a′i

to the representative Fj/Gaj :

F ′iGeai−a′i + F ′pGeai−bp − F ′Geai−b = FjG

eai−aj .

The condition p 6 |αl−a′iβl ensures that no term of F ′pGa′i−b can match the term

of highest xi-degree of FiGeai−ai ; thus degxi F

′i = degxi Fj ≤ aj degxi G and we may

take F ′i/Ga′i 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

be an open subset of Pnk , and D a geometrically bounded covering datum of prime

index p on X. Then there exists an etale cover Y −→ X realising D.

Proof. We have Pnk ' Proj k[X0, . . . , Xn]. As above, we write X = ∪mj=1D+(Gj),

where for any j, Gj is a homogeneous polynomial in X0, . . . , XN and defines a

hypersurface of Pnk . By Propositions 2.3.13 and 2.3.15, it will suffice to find a

trivialisation of D over D+(G1), an open subscheme.

Now let Φi : Ank −→ A1

k be the projection onto a coordinate axis, as defined in

Definition 5.2.8. For a closed point ω ∈ A1k, let Hω,i ' An−1

k(ω) denote the fiber above

ω, and set H ′ω,i = Hω,i ∩X.

Then H ′ω,i is the fiber of Φi|X : X −→ A1k, the restriction of Φi to X.

107

Construction 5.2.12 (Induction Construction). The fiber H ′ω,i of Φi|X above a closed

point ω ∈ A1i is of dimension n− 1 and isomorphic to D(g) ⊂ An−1

k(ω), where g = gω,i

is the image of G under the canonical surjection pω,i : A Aω,i, which is given by

”modding out by (hi(xi))”. Let Dω,i denote the restriction of D to H ′ω,i. Then Dω,i

is a geometrically bounded covering datum of index bounded by p on H ′ω,i.

Thus we may use induction on the dimension n of X to prove Theorem 5.2.11.

The induction hypothesis is that Dω,i has a finite realisation Yω,i −→ H ′ω,i, and the

base case is given by n = 2, where the fibers of Φi are curves on which Dω,i will be

trivially realisable. Indeed, we have Hω,i ' A1k(ω), H

′ω,i ' D(g) ⊂ A1

k(ω) and Dω,i is

just the p-exponent cover Y DH′ω,i−→ H ′ω,i cover of regular curves induced by D.

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

5.1.1), we have Yω,i = Y[fω,i/g

aω,iω,i ]

for some element fω,i/gaω,iω,i ∈ Aω,ig .

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

datum of index p = char(k) on X = D+(G) ⊂ Pnk , and identify Ank with V+(X0).

Let Φi be the projection fibration onto the ith coordinate axis, let Hω,i be the fiber

above a closed point ω ∈ A1k. Set H ′ω,i = Hω,i ∩ X ⊂ An

k , and let Dω,i denote the

restriction of D to H ′ω,i. If Dω,i is effective with a finite realisation Yω,i −→ C ′ω,i,

then there exists a positive integer ∆ such that

1. Yω,i = Y[fω,i/gaω,i ] with aω,i ≤ ∆ for all ω,i.

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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

realisation is a proper Z/pZ-cover Y[fω,i/g

aω,iω,i ]−→ H ′ω,i. We want to find a curve

C ⊂ H ′ω,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

the projection fibration onto the jth coordinate hyperplane Φj : An−1k(ω) −→ An−2

k(ω).

Let Cη,j ⊂ An−1k(ω) be the fiber above a closed point η ∈ An−2

k(ω).

Define Aη,j := k(η)[x1, . . . , xi, xj, . . . , xn], then we have a surjection

Aω,i Aη,j .

Denote the images of f , g by fη, gη. Now let Aω,ig , Aη,jgη denote the localisations at

the sets gb : b ∈ Z and gbη : b ∈ Z, respectively, then we can extend the above

surjection to a surjection

pη,j : Aω,ig Aη,jgη , which maps

109

f

ga7→ fη

g aη

.

Then since Y[f/ga] −→ H ′ω,i realises Dω,i on H ′ω,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]

Cη,j // H ′ω,i .

Now write f/ga = f ′/ga11 · · · g

all , 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

of codimension one xj in Pn−1k(ω)\D+(g) which correspond to finite primes of An−1

k(ω).

Let νxj be the valuation associated to gj and xj, then νxj(f/ga) = −aj ≤ −∆.

Claim 5.2.14. Let f′:= pη,j(f

′), gr = pη,j(gr) for r = 1, . . . , l denote the images of

f ′ and gr in Aη,j. Then f is relatively prime to gr for all r.

Proof. Since f ′ and gr are relatively prime in Aω,i, there exist polynomials

hr,h′r ∈ Aω,i such that hr f

′ + h′r gr = 1. Letting hr, h′r denote the images of

hr, h′r in Aη,j, we have hr f

′j + h

′r gr = 1, as claimed.

Now we have pη,j(fga

) = f ′

ga11 ···g

all

= fga

. Then if y is a closed point of Cη,j\Cη,j

lying in the closure V (gr) of xr, we thus have νy(f/ga) = νy(f

′/ga1

1 · · · gall ) ≤ −ar

By Lemma 5.1.4, if y′ denotes a point of lying above y we then have

mCη,j ,y′ = −νy(f/ga) = aj > ∆, which contradicts the uniform bound on rami-

fication numbers of Dω,i.

110

Proof of 5.2.13.2. Let Hω,i ' Pn−1k(ω) denote the closure of H ′ω,i in Pnk , and let H

′ω,i

denote the closure of H ′ω,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.

So if Yω,i −→ H ′ω,i realises Dω,i then the normalisation Y ω,i of H′ω,i is etale at

H′ω,i\H ′ω,i ' V+(X0) ∩H ′ω,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

degxj fω,i > aω,i degxj gω,i + ∆ in Y[fω,i/g

aω,iω,i ]

.

Now let Φ′j : An−1k(ω) −→ An−2

k(ω) be the projection fibration onto the jth hyperplane.

For η ∈ An−2k(ω) a closed point, let Cη,j = Spec k(η)[xi] ' A1

k(η) be the fiber above η,

and note that Cη,j ' P1k(η). Denote Cη,j ∩X by C ′η,j. Let Aη,j = k(η)[xj], then we

have the canonical surjection

pη,j : Aω,i Aη,j .

Since ∪ηCη,j = An−1k(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 .

We have two cases: If the left-hand side is non-empty, there exists an η ∈ An−2k(ω)

such that(Cη,j\Cη,j

)∩X = P1

k(η)\A1k(ω)∩X is non-empty; let x∞ denote its unique

111

point. Note that since x∞ ∈ X, Y [f/ga] −→ Hω,j must be etale at 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 etale at X∞ ∈ Cη,j.

We have Cη,j ' Spec k(η)[xj], so we let νx∞ = ν1/xj denote the valuation asso-

ciated to point x∞ at infinity, then

ν1/xj(f/ga) = − degxj(f/g

a) = − degxj(f/ga)

= a degxj g − degxj f < −∆ < 0

Then by Lemma 5.1.3, Y [f/ga] −→ Cη,j is not etale at x∞, a contradiction.

Now consider the case where Y [f/ga] −→ Hω,j is not necessarily etale at 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.

But by Lemma 5.1.4, mC,x∞ = − degxj(f/ga) > ∆, a contradiction.

Returning to the Proof of Theorem 5.2.11, by Lemmas 5.2.13 and 5.2.13, there

exists a positive integer ∆ such that each cover Yω,i −→ H ′ω,i has an Artin-Schreier

representative fω,i/gaω,iω,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.

By Lemma 5.2.9, we may always choose a lift [Fω,i/Gaω,i ] of [fω,i/g

aω,iω,i ] such that

degxj Fω,i = degxj fω,i for all j 6= i. So fix i and let Mi =< [Fω,i/Gaω,i ] : ω ∈ A1

k > be

112

the vector subspace of AG/℘(AG) generated by these lifts. Then the associated cover

YMi−→ X trivialises D on each of the fibers H ′ω,i by construction. By Lemma 5.2.7,

it follows that YMiweakly 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.

Recall Definition 5.1.18 of the vector subspaces M∆(AG), M i∆(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 etale cover 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 etale above X by Lemma 2.1.15,

so all induction assumptions are satisfied.

5.3 Covering data of Prime Power Index

In this section, we let X ⊂ Pnk 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

be an open subset of Pnk . Let D a geometrically bounded covering datum of cyclic

index pm on X. Then there exists an etale cover of Y −→ X realising D.

Remark 5.3.2. Setting X = Ank ⊂ Pnk , the theorem equals includes the case where

113

D is a covering datum of cyclic index pm on Ank .

As before, we let Φi : Ank −→ A1

k denote the projection fibration onto the ith

coordinate axis. Denote by Hω,i ' An−1k(ω) the fiber above a closed point ω ∈ A1

k, and

by H ′ω,i = Hω,i∩D(G) ' D(g) the intersection of the fiber with X. If G denotes the

dehomogenisation of G at X0, then g = gω,i is the image of G under pω,i : A Aω,i.

Then as in the Induction Construction 5.2.12, we assume by induction on n

that the restriction Dω,i of D to H ′ω,i is realisable. As such, it is either trivial

or a Z/ps-cover of C ′ω,i for some s ≤ m; in either case it is represented by some

[f ] = [ f0ga0, . . . , fm−1

gam−1 ] ∈ Wm(Ag). As in 5.2.12, the base case is given by the covering

datum itself.

Lemma 5.3.3. Let X = D+(G) be a simple open subset of Pnk , let D be a geometri-

cally bounded covering datum of cyclic prime power index pm on X, and let Dω,i be

the restriction of D to H ′ω,i. Assume that Dω,i is realised by an Artin-Schreier-Witt

cover Yω,i = Y[f ], where [f ] = [ f0ga0, . . . , fm−1

gam−1 ] ∈ Wm(AG)/℘(AG).

Then there exists a postive integer ∆ such that

1. pm−s−1as ≤ ∆ for all s and

2. pm−s−1(

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

covering datum on X, then there exists a positive integer ∆′ such that whenever Dω,i

is realisable by a Z/pmZ-cover Y[f ′] −→ H ′ω,i for some element [f ′] = [f ′0

ga′0, . . . ,

f ′m−2

ga′m−2

]

of Wm−1(AG)/℘(Wm−1(AG)), we have that

pm−s−2a′s ≤ ∆′ for all s, and

pm−s−2(

degxj f′s − a′s degxj g

)≤ ∆′ for all s, for all j 6= i . (5.3.1)

Recall that for f ∈ Wm(AG), k = 1, . . .m − 1, we let f (k) denote the truncated

Witt vector of length k, and that we have a tower of succcessive Z/pZ-covers

Y[f ] −→ Y[f (m−1)] −→ . . . −→ Y[f0] −→ H ′ω,i . (5.3.2)

Then applying (5.3.1) to the truncated Witt vector [f (m−1)], we get

pm−s−2as ≤ ∆′ for all s, and

pm−s−2(

degxj fs − as degxj g)≤ ∆′ for all s, for all j 6= i . (5.3.3)

Now let Φ′j : An−1k(ω) −→ An−2

k(ω) be the projection fibration onto jth coordinate

hyperplane of Hω,i ' An−1k(ω). Let η ∈ An−2

k(ω) be a closed point, recall that we set

Aω,i = k(ω)[x1, . . . , xi, . . . , xn], and define Aη,j = k(η)[xj]. Let g = gη,j is the image

of G under the combined surjections:

pη,j pω,i : A Aω,i Aη,j ,

let Cη,j ' Spec k(η)[xj] = A1k(η) denote the fiber above η and set

C ′η,j := Cη,j ∩X ' D(g).

115

For an element f ∈ Aω,i, we set f := pη,j(f), and note that degxj f = degxj f .

For f ∈ Wm(AG), we set f := pη,j(f) = [ f0

ga0, . . . ,

fm−1

gam−1 ]. Then the pullback of

the tower (5.3.2) to C ′η,j is given by

Y[f ] −→ Y[f (m−1)] −→ . . . −→ Y[f0] −→ C ′η,j , (5.3.4)

where f (s) denotes the truncated Witt vector of length s associated to f .

Recall the definition 4.2.5 of the ramification number m(C′η,j ,z)of Y[f ] −→ C ′η,j

at a closed point z ∈ Y[f ], then m(C′η,j ,z)is equal to the last lower jump of the

ramification filtration Giz associated to z (cf. [14, 2.4]). Then by [14, Lemma

3.1], m(Y[f ],z)is equal to the unique lower jump of the ramification filtration Gi′

z

of Y[fω,i]−→ Y

[f(m−1)ω,i ]

, i.e. the ramification number m′(C′η,j ,z)of z in the uppermost

Z/pZ-subcover of the tower (5.3.4).

Recall from the proof of 5.1.12 that Y[fω,i]−→ Y

[f(m−1)ω,i ]

is defined by

ypm−1 − ym−1 = −σm−1(yp0, . . . , ypm−2, y0, . . . , ym−2) +

fm−1

gam−1,

where the ys are the elements generating K(Y[f

(s)ω,i]

) over K(Y[f

(s−1)ω,i ]

).

Now let zl−1 ∈ Y[f(l)ω,i]

be the image of z under Y[fω,i]−→ Y

[f(l)ω,i]

for l = m, . . . , 1,

and let x ∈ C ′η,j be the image of y under Y[fω,i]−→ C ′η,j. By Lemma 5.1.3, we have

m′(C′η,j ,z) = −νzm−2(σm−1(yp0, . . . , ypm−2, y0, . . . , ym−2) +

fm−1

gam−1) . (5.3.5)

Recall thatD is geometrically bounded, and that we defined ∆ to be the constant

from Lemma 4.2.2, then ∆ is such that m′(C′η,j ,zm−2) = m(C′η,j ,z)≤ ∆. Since σj is

homogeneous of degree pj, and νzm−2(σm−1(yp0, . . . , ypm−2, y0, . . . , ym−2)) < 0, we have

116

νzm−2(σm−1(yp0, . . . , ypm−2, y0, . . . , ym−2)) = min

l<m−1pm−1νzm−2(y

pl ), p

m−1νzm−2(yl)

= pm minl<m−1

νzm−2(yl)

= pm minl<m−1

1

e(zm−2/zl)νzl(yl)

≥ pm minl<m−1

νzl(yl)

≥ pm minl<m−1

pνzl(flgal

)

≥ pm+1 minl<m−1

νxl(f lgal

)

= pm+1 minl<m−1

1

e(zl/x)νx(

f lgal

)

≥ pm+1 minl<m−1

νx(f lgal

)

By the induction assumption, we have νx(flgal

) ≥ − ∆′

pm−l−2 . This implies that

νzm−2(σm−1(yp0, . . . , ypm−2, y0, . . . , ym−2)) = pm+1 min

l<m−1νx(

flgal

)

≥ −pm+1 maxl<m−1

∆′

pm−l−2

≥ −pm+1∆′

Thus by (5.3.5),

−νzm−2(fm−1

gam−1) = max−νzm−2(σm−1(yp0, . . . , y

pm−2, y0, . . . , ym−2)),m′(C′η,j ,z)

= maxpm+1∆′,m(Y[f ],z)

≤ maxpm+1∆′,∆ .

117

and therefore,

−νx(fm−1

gam−1) = −e(zm−2/x)νzm−2(

fm−1

gam−1)

≤ pm−1νzm−2(fm−1

gam−1)

≤ maxp2m∆′,∆ .

Let us define ∆′′ := maxp2m∆′,∆. For x the point at infinity of Cη,j, we then

obtain degxj(fm−1/gam−1) = degxj(fm−1/g

am−1) ≥ ∆′′. Since ∆′′ > p∆′, we have

from 5.3.1 that pm−s−1(

degxj fs − as degxj g)≤ ∆′′ as well, which proves 5.3.3.2.

To show 5.3.3.1, assume that fm−1

gam−1 is written in lowest terms: gm−1 6 | fm−1.

Then write fm−1

gam−1 = f ′

gα11 ···g

αll

such that gr is irreducible and relatively prime to f ′ for

all r. Then we havefm−1

gam−1 = f′

gα11 ···g

αll

∈ Aη,j. Now let x ∈ Cη,j\C ′η,j be a point in

V (gr), then

am−1 ≤ αr = νgr

(fm−1

gam−1

)≤ νx

(fm−1

gam−1

)≤ ∆′′

and we have pm−s−1as ≤ 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-

Schreier-Witt representative [f ] = [ f0ga0, . . . , fm−1

gam−1 ] ∈ Wm(AG)/℘(AG) such that for

118

all ω, for all i, we have

pm−s−1a′s ≤ ∆′ for all s, and

pm−s−1(

degxj f′s − a′s degxj g

)≤ ∆′ for all s, for all j 6= i (5.3.6)

for some fixed positive integer ∆′.

Now let Fω,i,s =Fω,i,sGaω,i,s

be the representative of a preimage of [ fsgas

] defined

by Lemma 5.2.3, and set Fω,i = (Fω,i,0, . . . ,Fω,i,m−1) ∈ Wm(AG)). Then since

degxjFω,i,sGaω,i,s

= degxjfω,i,sgaω,i,s

, and the exponents as of G in the denominators remain

unchanged, [Fω,i] ∈M∆,m(AG) for all ω ∈ A1k.

Now let Mi := 〈Wm(Fp).[Fω,i]〉ω be the Wm(Fp)-module generated by the Fω,i,

then Mi ⊂M i∆′,m(AG). By construction, the associated pm- exponent cover YMi

−→

X trivialises the covering datum D on the fibers H ′ω,i of Φ|X . By Lemma 5.2.7, YMi

thus weakly trivialises D over X ∩Ank . The cover YM associated to M = ∩iMi then

weakly trivialises D over X ∩ Ank by Theorem 5.0.8.

By Lemma 5.3.4 (see below), M ⊂ MD,m(AG) is finitely generated. Then

Lemma 2.3.10 implies that YM trivialises D over X ∩ Ank , 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 etale above 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

given vector subspaces Mi ⊂ M i∆,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

(F0, . . . ,Fm−1) such that if Fs = FsGas

,

pm−s−1(

degxj Fs − as degxj G)≤ p∆ for all j and

pm−s−1as ≤ p∆ for all j and s , .

Since M ⊂ Mi ⊂ M i∆,m(AG) for all i = 1, . . . , n, [F] has representatives

Fi = (Fi,0, . . . ,Fi,m−1) such that if Fi,s =Fi,sGai,s

, then

pm−s−1ai,s ≤ ∆ and

pm−s−1(

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

and 5.2.2: We found F = FGa

such that [F] ∈MD,1 realises D.

So assume that any Z/pm−1Z-cover can be realised by an element ofM∆,m−1(AG).

In particular, for [F] ∈ M , the Z/pm−1Z-cover Y[F

(m−1)i ]

has a representative

(H0, . . . ,Hm−2) ∈M∆,m−1(AG).

Then we have (H0, . . . ,Hm−2) F(m−1)i ∈ ℘(Wm(A)), so we can replace Fi by

F′i = Fi ⊕ (H0, . . . ,Hm−1) F(m−1)i = (H0, . . . ,Hm−2,F

′m−1). Here,

F′m−1 = Fi,m−1 + σm−1(H0, . . . ,Hm−2,Fi,0, . . . ,Fi,m−2) =F ′m−1

Ga′m−1.

120

By construction of Fi, for all j 6= i, we have Fi,m−1 =Fi,m−1

Gai,m−1 such that

ai,m−1 ≤ ∆ and p(

degxj Fi,m=1 − ai,m−1 degxj G)≤ ∆.

Writing Hs = HsGbs

, we can apply the induction assumption to get

(degxj Hs − bs degxj G

)≤ ∆

pm−s−2for all j, for all s = 0, . . . ,m− 2 ,

bs ≤∆

pm−s−2for all s = 0, . . . ,m− 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∆ .

It follows that F′i = [F ′0/Ga′0 , . . . , F ′m−1/G

a′m−1 ] satisfies

pm−s−1a′s ≤ p∆ for all s and

pm−s−1(

degxj F′s − a′s degxj

)≤ p∆G for all j 6= i for all s ,

so that [F′i] ∈Mp∆,1 ⊂ A/℘(A).

Now we compare F′i to F′j: We must have F′i F′j ∈ ℘(Wm(A)). But

F′i F′j = (0, . . . , 0,F′i,m−1 − F′j,m−1), so this implies that [F′i,m−1] = [F′j,m−1]

in A/℘(A). By Lemma 5.2.10, this implies that F′i,m−1 ∈Mp∆,1(AG).

Thus there exists an element Hm−1 ∈ [Fi,m−1] such Hm−1 = HGA

, where

degxj H − A degxj G ≤ p∆ and A ≤ p∆. Then F′i ∼ (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:

Theorem 6.0.5. Let k be a finite field of characteristic p, and let X ⊂ Ank 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 etale covers (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 k′ containing

the m-th roots of unity, and consider A′ = A ⊗ k′. Let A′ be a universal cover of

A′, i.e. the integral closure of A in the maximal seperable algebraic extension of

the function field K = K(A) in which A′ does not ramify. Consider the short exact

sequence of π1(SpecA′)-modules

1 −→ µn −→ A′× n−→ A′

×−→ 1 .

The long exact sequence of cohomology then gives rise to a short exact sequence

0 −→ A′×/(A′×)m −→ Hom(π1(SpecA′), µm) −→ Pic(A′) −→ 0 ,

where Pic(A′) denotes the Picard group of the ring A′ (see [11, Prop. 4.11] for

details). In particular, if Pic(A′) = 0, we get a canonical isomorphism

Ψ : A′×/(A′×)m '−→ Hom(π1(SpecA′), 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 A′ can be assumed to have trivial

Picard group Pic(A′) = 0. From now on, we only consider ring A′ satisfying this

hypothesis.

123

Definition 6.1.2. We call Km(A′) := A′×/ (A′×)m

the Kummer space of level m of

A′, and note the natural µm-module structure on K(A′) given by ζrn ·[a] = [a]r = [ar].

Now let ∆a = ar : r ∈ Z be the subgroup of K(A′) generated by an element a

of A′×. Note that we have ∆a = ∆a′ for two cyclic subgroups of order n if and only

if gma = a′rg′m for some elements g, g′ ∈ A′×, 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

etale cover corresponding to Na, then Ya is equal to the normalisation of SpecA′ in

the field K( m√

∆a).

If Y[a] = Y[a′] is a Z/nZ-cover of SpecA′, then Na = ker(Ψ(a)) and Na′ =

ker(Ψ(a′)) are equal, and of full index n. In particular, Ψ(a) also factors through

Na giving rise to an isomorphism Ψ(a) : π1(SpecA′)/Na′ ' µm. Thus we get

π1(SpecA)/Na′Ψ(a′)

//

Ψ(a) ''NNNNNNNNNNNNµm

r

µm

Then Ψ(a) Ψ(a)−1 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(a′). Since

Ψ is compatible with the µm-module structure, this implies that [ar] = [a′], which

is equivalent to ∆a = ∆a′ for the associated cyclic subgroups of order n in K(A′).

Conversely, if ∆a = ∆a′ ⊂ A′×/ (A′×)n

are cyclic subgroups of order m, then

clearly K( m√

∆a) = K( m√

∆[a′]), so Y[a] = Y[a′].

124

Thus Ψ induces a one-to-one correspondence of simple µn-submodules of order

k dividing m with Z/kZ-covers of SpecA′:

∆a 7→ Y[a] ↔ Na < π1(SpecA′)

Here, Y[a] corresponds to the normal subgroup Na of index k in π1(SpecA′).

More generally, if M ⊂ Km(A′) is a µm-module, then we can associate to M the

normalisation YM of SpecA′ in K( n√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∈MNa.

If M is finite, then NM is a finite intersection of normal subgroups with finite

index, and thus also of finite index. If this is the case, then YM −→ SpecA′ is a

(finite) etale cover.

Conversely, if Y −→ SpecA is an etale Galois 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 SpecA in the compositum of the function fields

K( m√

∆ai). Writing M = 〈[ai] : i = 0, . . . , r〉 ⊂ 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 A′ be any finitely generated algebra over a finite field k′

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 SpecA′:

M 7→ YM ↔ NM := ∩a∈MNa .

M is finitely generated of rank r if and only if YM is an exponent m-etale cover of

Ank′ 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:

1. For M ⊂ Km(A′), the associated pro-etale cover YM is the normalisation of

K in K( n√M), the field obtained by adjoining all m-roots of elements in M .

2. If aii∈I is any generating set for M ⊂ Km(A′), then NM =⋂i∈I Nai : Indeed,

if f =∏m

i=1 frii with ri relatively prime to m, then since Ψ is multiplicative,

Na ⊂ ∩iNai , so NM ⊂⋂i∈I Nai . The other inclusion is trivial.

3. For any two submodules M , M ′, we have

a) MM ′ ↔ NM ∩NM ′ , and

b) M ∩M ′ ↔< NM , N′M >

126

We now define several important submodules of Kummer space Km(A′) for the

free k′-aglebra A′ = k′[x1, . . . , xn] and its localisation AG = k′[x1, . . . , xn]G at some

element G ∈ A.

For A = k[x1, . . . , xn], we let as above k′ be a finite extension of k containing

the mth roots of unity, and A′ = k′[x1, . . . , xn]. We define a submodule of Kummer

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

set f r ∈ A : degxj ≤ D for all j, r < m relatively prime to m.

We also define M iD(A′) =< [f ] : degxj f ≤ D for all j 6= i >, an (infinitely

generated) µm-submodule of K(A′), and show:

Proposition 6.1.4. In the above notations, the intersection of the spaces M iD(A′)

over all i equal to MD′(A′) for some positive integer D′, and thus finite.

Proof. By assumption, each equivalence class [F ] ∈MD′(A′) contains an element Fi

of M iD(A), i.e. such that degxj Fi ≤ D for all j 6= i. If Gm|Fi for some polynomial

G, without loss of generality replace Fi by F ′i = Fi/Gm. Then degxj F

′i ≤ degxj Fi

for all j, and thus degxj Fi ≤ D for all j 6= i. Then we have F ′i = F rjH

m for some

polynomial H ∈ A′ and some r < m that is relatively prime to m. But F ′i is not

divisible by any n-th power of a polynomial, so H must be constant, and we have

degxi F′i ≤ r degxi Fj ≤ mD. Thus we have F ′i ∈MmD(A).

The other inclusion is trivial by definition.

Let A′G denote the localisation of A′ at some element of A′, and consider the

127

Kummer space K(AG). Define µm-submodules

MD(AG) =< [F/Ga] : a ≤ D, degxj F ≤ a degxj G for all j > and

M iD(AG) =< [F/Ga] : 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:

Proposition 6.1.5. In notations as above, the intersection of the spaces M iD(A′G)

over all i contained in MD′(A′G) for some positive integer D′, and thus finite.

Proof. By assumption, each equivalence class [F/Ga] ∈MD′(A′G) contains elements

Fi/Gai ∈ M i

D(AG), i.e. such that Fi and G are relatively prime, and such that

ai ≤ D and degxj Fi ≤ ai degxj G for all j 6= i. If Hm|Fi for some polynomial H,

without loss of generality replace Fi/Gai by F ′i/G

ai , where F ′i = Fi/Hm. Now we

may assume that F ′i is not divisble by the m-th power of any polynomial, and since

degxj F′i ≤ degxj Fi for all j, it is still true that degxj Fi ≤ ai degxj G for all j 6= i.

Now we have

F ′iGai

=

(FjGaj

)r (H

Gb

)mfor some polynomial H ∈ A′ and some r < m that is relatively prime to m.

Write HGb

in lowest terms, then we have two cases: If b > 0, then H and G

are relatively prime and ai = raj + mb. Multiplying through by Gai , we have

F ′i = F rjH

m. But F ′i is not divisible by any n-th power of a polynomial, so then H

must be constant, and we haveF ′iGai

=F rjGraj

. As the Fl and G are relatively prime

128

by assumption, this implies that ai = raj ≤ mD. Taking degrees on both sides,

we also get that degxi F′i = r degxi Fj ≤ raj degxi G = ai degxi G. Thus we have

F ′i ∈MmD(AG), as required.

If b = 0, then write H = GcH ′ with H ′ relatively prime to G, then

F ′i = F rj (GcH ′)m As before, this implies that GcH ′ is contant, so c = 0 and H ′

is constant and we get that F ′i ∈MmD(AG) as above.

6.2 Tamely ramified covering data of cyclic factor

group

In this section, we shall show that for X = Ank or X ⊂ Pnk 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].

Theorem 6.2.1. Let k be a finite field of characteristic p, and let X ⊂ Pnk 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

over an affine open subset of the form D(G) ⊂ Ank ' D+(X0). We again make use

129

of the fibrations Φi : Ank −→ A1

k, let ω denote a closed point of A1k with residue field

k(ω) ' k[xi]/(hi(xi)), and Cω,i the fiber of Φi over ω.

As before, we denote X ∩ Cω,i by C ′ω,i, and assume by induction that Dω,i is

realisable by a Galois cover Y[fω,i/g

aω,iω,i ]

whose equivalence classe [fω,i/gaω,iω,i ] ∈ Km(AG)

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 etale at the point at infinity of a

curve D(g) ⊂ A1k. 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)[x1, . . . , xi, . . . , xn], and AG Aω,igω,i . They

clearly induce surjections of Kummer spaces analogously to those of Artin-Schreier

space:

pω,i : Km(A) Km(Aω,i) and

πω,i : Km(AG) Km(Aω,ig ) .

Then the lemma below is proven entirely analously to Lemma 5.2.9 of the Artin-

Schreier case:

Lemma 6.2.2. Given [f/ga] ∈ Km(Aω,ig ), there exists a preimage [F/Ga] ∈ π−1ω,i([f/g

a])

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

Mi = 〈[Fω,i/Gaω,i ] : ω ∈ A1k〉. Then as before, Mi corresponds to a pro-etale cover

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-

alised by an element [F/Ga] ∈ 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

X = Ank since we then have X = D+(G), G(X0, . . . , Xn) = X0.

131

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