Higher Dimensional Class Field Theory: The variety case. Linda M. Gruendken. A Dissertation. Mathematics

Transcription

1 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 Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2011 Florian Pop Supervisor of Dissertation Jonathan Block Graduate Group Chairperson Dissertation Committee Florian Pop, Samuel D. Schack Professor of Algebra, Mathematics David Harbater, Christopher H. Browne Distinguished Professor in the School of Arts and Sciences, Mathematics Robin Pemantle, Merriam Term Professor of Mathematics

2 HIGHER DIMENSIONAL CLASS FIELD THEORY: THE VARIETY CASE COPYRIGHT 2011 LINDA MEIKE GRUENDKEN

3 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

4 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 P n k is an open subset in projective space. Suppose that C X is the Wiesend idèle class group of X, π ab 1 (X) the abelianised fundamental group, and ρ X : C X π ab 1 (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 < C X 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 ρ X is injective and has dense image. We obtain a oneto-one correspondence of open geometrically bounded subgroups of C X with open subgroups of π ab 1 (X). Furthermore, we show that for an étale cover X X with maximal abelian subcover X X, the reciprocity morphism induces an isomorphism C X /N C X Gal(X /X). iv

5 Contents 1 Introduction 1 2 Preliminaries Basic Facts and Generalities Fundamental groups Covering Data The Wiesend Idèle Class Group Definitions and some Functiorial Properties The Reciprocity Homomorphism The Base Cases of the Induction Argument: Reciprocity for Regular Schemes of Dimension One and Zero From the Main Theorem to the Key Lemma Introducing induced covering data Geometrically bounded covering data v

6 4.3 Realisable subgroups of the class groups The Key Lemma Wildly ramified Covering Data A Review of Artin-Schreier-Witt Theory Artin-Schreier Theory Artin-Schreier-Witt Theory Covering data of Prime Index Affine n-space A n k The Case of Open Subsets of P n k Covering data of Prime Power Index Tamely ramified covering data revisited A review of Kummer Theory Tamely ramified covering data of cyclic factor group vi

7 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 onedimensional 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 π ab 1 (X). It is therefore natural to ask for a generalisation of class field theory to arithmetical schemes of higher dimensions. 1

8 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 F p in terms of generalised Jacobians. Finally, Parshin and Kato, followed by several others, proposed getting a higher dimensional Artin reciprocity map using algebraic K-Theory and cohomology theories. Although promising, these approaches become quite technical, and the heavy machinery involved makes the results very complicated and difficult to apply in concrete situations. It was G. Wiesend [21] who had the idea to reduce the higher dimensional class field theory to the well developed and known class field theory for arithmetical curves. He defines an idèle class group C X in terms of the arithmetical curves and closed points contained in X, and gets a canonical homomorphism ρ X : C X π ab 1 (X) in the hope of establishing properties similar to that of the Artin reciprocity map. Wiesend s work was supplemented by work by M. Kerz and A. Schmidt, where more details of Wiesend s approach were given [5]. Most notably, in the so-called flat case of an arithmetical scheme X over Spec Z (c.f. Definition 2.1.2), they were able to prove surjectivity of the canonical homomorphism, and they provided a concrete description of the norm subgroups in C X. The focus of this thesis is on the still-open regular variety case, where X is a 2

9 regular arithmetical variety over some finite field k of characteristic p. In this case, Kerz and Schmidt have already proved a higher dimensional reciprocity law for the abelianised tame fundamental group π ab,t 1 (X), which classifies all the tame étale abelian covers of X. A generalised Artin map can be defined between ρ X,t : C t X π ab,t 1 (X), where C t X 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 P n k an open subvariety, then we prove: Theorem Let X P n k be an open subvariety, and let ρ X : C X π ab 1 (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 C X and open subgroups N of π1 ab (X); it is given by ρ 1 (N) N. The reciprocity morphism ρ X is a continuous injection with dense image in π ab 1 (X). 2) A subgroup of C X is a norm subgroup iff it is open, of finite index and geometrically bounded. 3 X

10 3) If X X is an étale connected cover, X X the maximal abelian subcover, then N C X = N C X and the reciprocity map gives rise to an isomorphism C X /N C X Gal(X /X). This thesis is organised as follows: We begin by introducing notation and reviewing basic facts from the theories of arithmetical schemes, fundamental groups and covering data. We then review Wiesend s definition of the idèle class group and the reciprocity homomorphism, giving the right definitions as to guarantee commutativity of all relevant diagrams. (Unfortunately, in all the published works so far, the given morphisms do not make all the diagrams commute as stated.) We also review the results of classical class field theory as the base case of our induction argument in the proof of the Main Theorem 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 is reduced to the Key Lemma Theorem is shown as a corollary to the Main Theorem The Key Lemma is shown for open subvarieties X P n k in two steps. In Chapter 5, we analyse the behaviour of index-p m 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

11 Chapter 2 Preliminaries 2.1 Basic Facts and Generalities In this thesis, we shall be concerned with arithmetical schemes X over Spec Z: Definition X is said to be an arithmetical scheme over Z if it is integral, separated and of finite type over Spec Z. We assume that all schemes are arithmetical, unless otherwise stated, and distinguish two cases: Definition If the structural morphism X Spec Z has open image, X will be called flat. If this is not the case, then the image of X Spec Z is a closed point p Spec Z, and X is a variety over the residue field of the point k = k(p) = F p. We now collect some general facts and tools to be used in later chapters: 5

12 Fix a field k, and let K k be a field over k. Then let deg tr K denotes the transcendence degree of a field K. For a scheme X, let dim X denote the Krull dimension of a scheme X as a topological space (cf. [9, Definition 2.5.1]). Definition Let X be an arithmetical scheme with structural morphism f : X Spec Z, and let f(x) be the closure of the image of f in Spec Z. Denote the function field of X by K(X). Then the Kronecker dimension d of X is defined as dim X = deg tr K(X) + dim(f(x)). Remark For example, both Spec Q and Spec Z have Kronecker dimension one, while Spec F p has Kronecker dimension zero. An arithmetical variety of Kronecker dimension one is given by Spec F p [t]. Definition 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 Let X be any scheme. An étale cover of X is a finite étale cover Y X, and a pro-étale cover of X is the projective limit of étale covers of X. Let k be an arbitrary field, and consider a subvariety X P n k. If C X is a curve, i.e. a one-dimensional integral closed subscheme of X, then C P n k is quasi- 6

13 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 Let C be a geometrically connected projective curve. Then the arithmetic genus g a (C) is defined as g a (C) = 1 χ k (O X ), where χ k (O X ) denotes the Euler-Poincaré characteristic of the structural sheaf O X. Let C be a quasiprojective curve, with regular compactification C. Then the arithmetic genus is defined by setting g a (C) := g a (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 g C. Definition If k is an arbitrary field, and X P n k is a projective variety of dimension d, the degree of X over k is defined as the leading coefficient of the Hilbert polynomial p X (t), multiplied by d!. It is denoted by deg k X. Fact Let k be an arbitrary field, X P n k 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 g C g. In particular, this holds for all regular curves C = C X. Note: The converse of this is not true: If C P 2 k is the completion of V (x y n ) A 2 k P2 k, then C has degree n, but g C = 0 as C is rational. Fact 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

14 Proposition (Chebotarev Density Theorem). Let Y X be a generically Galois cover of connected normal schemes over Z. Let Σ be a subset of G = G(Y X) that is invariant under conjugation, i.e. gσg 1 = Σ for all g G. Set S = {x X : F rob x Σ}. Then the Dirichlet density δ(s) is defined and equal to δ(s) = Σ / G. Proof. See [16]. Lemma (Completely Split Covers). Let X be a connected, normal scheme of finite type over Z. If f : Y X is a finite étale 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 F rob x = 1 for all Frobenius elements above x, this follows directly from the Chebotarev Density Theorem Let S = {x : F rob x = 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

15 Definition An étale cover f : Y X is called locally trivial if it is completely split. Lemma (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 x 1,..., x n be closed points of X with pairwise different images in Z. Then there exists a curve C X such that the points x i are contained in the regular locus C reg 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 Let X be a regular, pure-dimensional, excellent scheme, X X a dense open subscheme, Y X an étale cover and Y the normalization of X in k(y ). Suppose that for every curve C on X with C = C X, the étale cover Y C X C extends to an étale cover of C. Then Y X is étale. Proof. See [5, Proposition 2.3] for a proof. 9

16 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 étale covers of X, together with morphisms of schemes over X form a category of X-schemes, which we denote by Et X. Now let Ω be an algebraically closed field, and fix s : Spec Ω X, a geometric point of X. If Y X is an element of Et X, consider the geometric fiber Y X Spec Ω, and let F ib s (Y ) denote its underlying set. Any morphism Y 1 Y 2 in Et X induces a morphism Y 1 X Spec Ω Y 2 X Spec Ω. Applying the forgetful functor, we get an induced set-theoretic map F ib s (Y 1 ) F ib s (Y 2 ). Thus, F ib s ( ) defines a set-valued functor on the categeroy Et X, 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 The fundamental group of X with geometric basepoint s is the automorphism group of the fiber functor F ib s associated to s, and is denoted by π 1 (X, s). Lemma The fundamental group is profinite and acts continuously on F ib s (X). Proof. See [19, Theorem 5.4.2]. 10

17 Proposition The functor F ib s induces an equivalence of the category of finite étale covers of X with the category of finite continuous π 1 (X, s)-sets. Under this correspondence, connected covers correspond to sets with transitive action, and Galois covers to finite quotients of π 1 (X, s) Proof. Cf. [19, Thm ]). Now let s, s : Spec Ω X be two geometric points of X. Definition A path from s to s is an isomorphism of fiber functors F ib s F ib s. Whenever such a path p exists, the fundamental groups are (non-canonically) isomorphic as profinite groups via conjugation by p: π 1 (Y, s) π 1 (Y, s ) [19, 5.5.2]. Proposition If X is a connected connected scheme and s are s two geometric points of X, then there always exists a path p from s and s. Proof. See [19, Cor ]. Thus, for a connected scheme X, the fundamental group π 1 (X) is well-defined up to conjugation by an element of itself. In particular, the maximal abelian quotient π ab 1 (X) does not depend on the choice of geometric basepoint [19, Remark 5.5.3]. Definition The maximal abelian quotient π ab 1 (X) is called the abelianised fundamental group of X. Now let X, Y be schemes with geometric points s and s. 11

18 Definition 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. Spec Ω s Y f s X Proposition In the situation above, a morphism that is compatible with s and s induces a morphism on fundamental groups f : π 1 (Y, s) π 1 (X, s ). Proof. See the remarks following Remark 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 : π 1 (X, s) p p 1 π 1 (X, s ) π 1 (Y, s ) Remark In the situation above, whether basepoints are already compatible or not, the preimage f 1 (N) of any normal subgroup N of π 1 (X, s ) is always well 12

19 defined. We shall thus drop the geometric points from our notation in later chapters, when dealing with covering data. Remark While the fundamental group functor π 1 (X, s) is not representable in Et X, the category of finte étale covers of X, it is pro-representable and thus representable in the larger category of profinite limits of finite étale 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-étale covers of X. In particular, fixing a universal cover with geometric basepoint x amounts to choosing a system of compatible basepoints for every pro-étale cover Y X. The following proposition summarizes some further properties of fundamental groups: Proposition 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 étale subcover. Then 1. The induced morphism f : π 1 (Y ) π 1 (X) is injective with open image N Y := f(π 1 (Y )), and the association Z f(π 1 (Z)) is one-to-one if we restrict to connected étale 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 N Y is normal in π 1 (X). If this is the case, the Galois 13

20 group G = Aut(Y/X) is isomorphic to the group π 1 (X)/N Y. 4. If g : W X is another étale 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 Y f W X g corresponds to the open subgroup N Y N Z. If f and g are Galois over X with groups G f and G g, then so is W X, and it has Galois group G, where G < G f G g is a subgroup projecting surjectively onto G f and G g. 5. There exists a minimal étale cover Z Y such that Z X is Galois. N Z is then the smallest normal subgroup of π 1 (X) contained in N Y. Proof. Apply Proposition and Propositions of [19]. Recall from Definition that a pro-étale cover f : Y X is the inverse limit of finite étale covers. Analogously to above, we then have the following proposition: Proposition 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-étale subcover. Then 14

21 1. The induced morphism f : π 1 (Y ) π 1 (X) is injective with closed image N Y := f(π 1 (Y )), and the association Z f(π 1 (Z)) is again one-to-one if we restrict to connected étale 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 N Y is normal in π 1 (X). If this is the case, the Galois group G = Aut(Y/X) is isomorphic to the group π 1 (X)/N Y. 4. If g : W X is another pro-étale cover, let w, y denote the geometric basepoints of W, respectively Y which are compatible with x. Let Z W X Y be the irreducible component which contains the geometric basepoint y w. Then we have a commutative diagam Z Y f and the cover f g : Z X corresponds to the open subgroup N Y N W. If f and g are Galois over X with groups G f and G g, then so is f g, say with Galois group G, then G < G f G g is a subgroup projecting surjectively onto G f and G g. W X g 5. There exists a minimal étale cover Z Y such that Z X is Galois. N Z is then the smallest normal subgroup of π 1 (X) contained in N Y. Notation Lastly, we establish and summarise some notation for later chapters: 15

22 1. We have already introduced the tilde notation for morphisms between fundamental groups which are induced by morphisms of schemes: If f : Y f X is a morphism, then f : π 1 (Y ) π 1 (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 i x : 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 i C : C C X denote the composition of the normalisation morphism with the inclusion of the curve in X. 4. Let i C : 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 ĩ C : π 1 ( C) π 1 (X) the induced morphism on the fundamental groups. Similary, for the inclusion i x : x X of a closed point, the induced morphism π 1 (x) π 1 (X) is denoted by ĩ x. 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

23 f y = f ĩ x and f C = f ĩ C denote the fiber products of f with the maps induced by i x and i C, respectively. Let D C X Y be the irreducble component determined by the fixed universal cover. Then we have a commutative diagram: π 1 ( D) ef C π 1 ( C) ei D ei C π 1 (Y ) ef π 1 (X) Similarly, let y x X Y be the closed point determined by the univesal cover, then we get the commutative diagram: π 1 (y) ef x π 1 (x) ei y ei x π 1 (Y ) ef π 1 (X) 2.3 Covering Data Let X be an arithmetical scheme, and recall from Notation 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 = (N C, N x ) C,x, where C and x range over the curves and closed points of X, respectively. For each C, respectively x, N C π 1 ( C) and N x π 1 (x) are taken to be normal subgroups of the fundamental groups of the normalisation of C and the 17

24 fundamental group of x, respectively. Since the subgroups N C, N x 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 x i x X i C 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 π 1 ( x) π 1 ( C) π 1 (x) i C i x π 1 (X). Definition A collection D = (N C, N x ) C,x of open normal subgroups N C < π 1 ( C), N x < π 1 (x) is a covering datum if the N C and N x satisfy the following compatibility condition: (*) For every curve C X, x X, and any x lying above x in x C, the preimages of N C and N x under the the canonical morphisms above agree as subgroups of π 1 ( x). Definition Let f : Y X, and recall the notations for induced morphisms on the fundamental groups set in and If D is a covering datum 18

25 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 N y = f 1 y (N x ). 2. For a curve C Y, we let N C = 1 f C (N f(c) ), where f(c) is the closure of the image of C in X, i.e. either a point or a curve. Definition 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 [π 1 ( C) : N C ], [π 1 (x) : N x ] 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 D C 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 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 f D C : Y D C D C, fc : Y C D C denote the covers of C defined by D and D. Also let f D x : x D, f D C : xd x denote the cover of x defined by D, respectively D. 19

26 We say that D is a subdatum of D if f D C is a subcover of f D C for all curves C X, and if f D x is a subcover of f D x for all x X. Definition A covering datum D is called trivial if N x = π 1 (x) for all closed points x X, and N(C) = π 1 ( C) for all curves C X. We say that D is weakly trivial if N x = π 1 (x) for all closed points x X. Definition Let D be a covering datum on a scheme X. Given an étale cover Y = Y N X corresponding to the open subgroup N π 1 (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) = N x for all closed points x X. 4. f realises D if N(x) = N x for all closed points x X and N(C) = N C 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 U i X has one of the above properties, we say that D has that property over U. Let f : X N X be a Galois étale cover corresponding to the open normal subgroup N π 1 (X), and recall Notations and For every 20

27 curve C X with normalisation C and every closed point x X, we define the pullbacks N(C) := ĩ 1 C (N), N(x) = ĩ 1 x (N). They correspond to the étale covers f C : X N C C and f x : X N x x, respectively. Then for any x C x, the diagram π 1 ( x) π 1 ( C) π 1 (x) π 1 (X) commutes since it is induced from the commutative diagram x x C X In particular, the pullbacks of N(C) and N(x) must agree in π 1 ( x) for any x and C, i.e. the datum (N C, N x ) C,x is a covering datum. Thus, the Galois étale cover f : X N X induces a covering datum of X, which we shall denote by D N. Notation Given a covering datum D on X, we let Y D C C, y x be the étale covers of connected schemes corresponding to N C π 1 ( C) and N x π 1 (x). Then given a morphism Y X, we have diagrams Y C Y D C Y C Y x y Y x Y D C C y x Notation Now let Z, Z be the connected components of Y C and Y C Y D C, respectively, and let z, z be points in the finite sets Y x and Y x y containing 21

28 the relevant geometric base points. Then we get the following diagrams of connected schemes: Z q C Z z q x z Y D C p C f C g C C y p x g x x f x Following the remarks on the correspondence between connected pro-étale 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 N C N(C) and N(x) N x of π 1 ( C) and π 1 (x), respectively. We also note that we can identify N(C) with f C (π 1 (Z)), N C with g C (π 1 (YC D )) and similarly for closed points. It is now possible to express the properties through of f in relation to D as inclusions/equalities of subgroups in π 1 ( C) and π 1 (x), but also as inclusions/equalities of subgroups in π 1 (Y C ), π 1 (YC D) and π 1(y), π 1 (Y x). Full characterisations are given as follows: Corollary Let D be a covering datum on a scheme X. 1. If f : Y X is a pro-étale cover, then TFAE: (a) f trivialises D. (b) N x N(x) in π 1 (x) for all closed points x X, and N C N(C) in π 1 ( C) for all curves C X. 22

29 (c) We have 1 f x (N x ) = π 1 (Y x) for all closed points x X and f 1 C (N C) = π 1 (Y C) for all curves C X. (d) The covers p C and p x 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 ). (b) f weakly trivialises D. (c) N x N(x) for all closed points x X. (d) We have 1 f x (N x ) = π 1 (y) for all closed points x X. (e) The covers p x are trivial for all closed points x X. 3. For a pro-étale cover f : Y X, TFAE: (a) f realises D. (b) We have f 1 x (N x ) = π 1 (Y x), g 1 (N(x)) = π 1 (y) for all closed points x x X and 1 f C (N C) = π 1 (Y C), g 1 C (N(C)) = π 1(YC D ) for all curves C X. (c) The covers p C and q C are trivial for any curve C X and the covers p x, q x are trivial for any closed points x X. 4. Analogously to 3), TFAE: 23

30 (a) f weakly realises D. (b) We have f 1 x (N x ) = π 1 (Y x), g 1 (N(x)) = π 1 (y) for all closed points x x X. (c) The covers p x, q x are trivial for any closed points x X. Lemma Let X be an arithmetical scheme, and let D be a covering datum on X. An étale 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 and : f weakly trivialises D if and only if the morphisms p x are trivial covers of arithmetical schemes. So now consider the covers p C. If x is a point of C x, then the morphism p ex induced by base changing from x to x is again trivial. Thus, p C is locally trivial. By Lemma , a locally trivial finite cover of arithmetical schemes is trivial, so p C is a trivial cover for any curve C X, as claimed. Similarly, if f weakly realises D, then we know that all the morphisms p x and q x are trivial for all closed points. As above, this implies that p C and q C are locally trivial and thus trivial. Definition Let X be an arithmetical scheme, and let D be a covering datum on X. Then D is called effective if it is realised by a pro-étale cover Y X. If this is the case, the associated cover Y N is called the realisation of the covering datum 24

31 D. If the realisation of D is a finite cover, we say that D has a finite realisation. Remark Note that if D has a finite realisation f : X N X, then there exists a curve C X such that the canonical morphism i C : C X (cf. Definition ) and the induced morphism ĩ C : π 1 ( C) π 1 (X) induces an isomorphism π 1 ( C)/N(C) π 1 (X)/N. In particular, the degree of the realisation f is equal to the index of the covering datum. Proof. X N corresponds to the open normal subgroup N π 1 (X). We define N(C) := ĩ 1 C (N) for any curve C X, and likewise N(x) := ĩ 1 x (N) for closed points x. Then we have natural inclusions π 1 ( C)/N(C) π 1 (X)/N and π 1 ( x)/n(x) π 1 (X)/N for all x and C. As N(x) = N x and N(C) = N C, this implies that the index of D is bounded by deg(f). Now let C be an irreducible curve as guaranteed by Lemma , then f C : Y C C has the same Galois group as f. Therefore π 1 ( C)/N(C) π 1 (X)/N is an isomorphism, and we have deg(f) = [π 1 ( C) : N C ]. Thus, D must have index exactly equal to deg(f), as claimed. Proposition Let X be a normal, arithmetical scheme and D = (N C, N x ) (C,x) a covering datum on X. 25

32 1. Then D has at most one finite realisation. 2. Let Y 1, Y 2 be two étale covers of X weakly realising D over an open subset U. Then Y 1 = Y If there exists an open subscheme U i X such that the pullback i (D) can be weakly realised by an étale 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 X N1, X N2 be two realisations of D, and let N i, i = 1, 2 be the corresponding open normal subgroups. Then N 1 N 2 correspons to a completely split cover of Y 1, so by Lemma , it is an isomorphim. It follows that N 2 N 1, and thus N 2 = N 1 by symmetry. 2. Let N 1, N 2 be the open subgroups of π 1 (X) associated to the two realisations Y 1, Y 2, and let N i (x) := ĩ 1 x (N i ) denote their pullbacks to π 1 (x) for any closed point x. Let Y = Y 1 Y 2. If N 1 (x) = N 2 (x) for all x U, then U Y U Y 1 is completely split, and thus an isomorphism by Lemma For V an irreducible component of Y 1 U, this means that V Y 1 V is an isomorphism of connected schemes. In particular, the Galois closure of this cover is just V Y 1 itself, and is connected. Now if Y Y 1 were not an isomorphism, then the Galois closure Y Y 1 would 26

33 have nontrivial Galois group as well. Since V Y 1 is connected, the Galois groups of the two vertical covers in V Y Y are isomorphic. V Y 1 Thus this would imply that V Y 1 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 étale covers Y C of the full curve C in X. By Proposition , Y X is an étale cover; we let N π 1 (X) be the corresponding open normal subgroup. Recall the notations set in ), and let N(C) := ĩ 1 ec to π 1 ( C), respectively π 1 (x). (N), N(x) := ĩ 1(N) denote the pullback x Now let C be a curve on X with C U. Then the preimages ĩ 1 ex (N(C)) and ĩ 1 ex (N C) in π 1 ( 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 N C of π 1 ( C) coincide, and so N(x) = N x for every regular point x of C. By Lemma , every point is contained in the regular locus of a curve meeting U, so we get N(x) = N x for all closed points x X. By Lemma , we conclude that Y = Y N is a realisation of D. 27

34 Remark In general, a realisation of a covering datum is automatically finite if the covering datum is of bounded index and tame. It is always étale by Lemma 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 étale covers, but only pro-étale. Theorem Let X be a regular arithmetical scheme, and let D = (N C, N x ) 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 Let X be a regular arithmetical scheme, and let D = (N C, N x ) C,x be a covering datum on X. If there exists an open subscheme U i 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 and Lemma Proof of Theorem We first show that it suffices to prove the Theorem under the additional assumption that f is étale. Claim If the covering datum D is trivialised by a finite cover f : Y X, then there also exists a finite étale 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 étale. (Such an X exists by the purity of the branch locus, cf. [9, Section 8.3, Ex. 28

35 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 < π 1 (X ) giving a weak realisation of D over the open subscheme X, which must be a full realisation by Proposition 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 , by Proposition , it suffices to find a subgroup N of π 1 (X) which gives a weak realisation over a dense open subset U, i.e. an open normal subgroup N such that ĩ 1 x (N) = N x for all x U. In particular, we can replace X by any open dense subset. Using Lemmas and , 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 , there exists a curve C X such that D = C Y is irreducible and such that f C : D C is again a Galois étale cover with group G. By the Chebotarev Density Theorem , there are infinitely many n-tuples (x 1,..., x n ) of points in C such that G is the union of the conjugacy classes [F rob xi ]. Since the regular locus of C is of codimension one, it is finite, so by replacing those x i which are not contained in the regular locus of C, we may assume that x i C reg for all i. Since Y trivialises D, we have f C (π 1 ( D)) N C as subgroups of π 1 ( C). Let 29

36 N = ĩ C (N C ) be the image of N C in π 1 (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) = ĩ 1 x (N) for any x X, then we show that N(x) = N x for all x which g : X Z maps to points which are distinct from the images of S = {x 1,..., x n } 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 yields a curve C X containing x, x 1,... x n as regular points, and such that C Y is irreducible. Lemma Let C be a curve in X such that C Y is irreducible, let N = ĩ C (N C ) be the image of N C in π 1 (X), and N(z) = ĩ z (N) the pullback to π 1 (z). Then N(z) = N z for all regular points z of C. Proof. We first show that if we define N(C) := ĩ 1 C (N), then N(C) = N C. Indeed, ĩ C is easily seen to be injective by applying the second criterion of [19, Corollary 5.5.8] to the canonical morphism j = ĩ C : 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 corresponding to a surjective morphism B A = B/I, and D Spec A is a finite étale cover, then D is also affine, say D = Spec E. We thus have to show that if E = A[f 1,..., f n ] = A[x 1,..., x n ]/(g 1,..., g n ) is a finite étale algebra over A, where the g i are polynomials with coefficients in A, then E is the tensor product F B of some finite étale algebra F. But taking h i B[x 1,..., x n ] to be any lifts of the g i, 30

37 then we can take F = B[x 1,..., x n ]/(h 1,..., h n ). Clearly, F is finite over B, and it is flat since Spec F Spec B is finite and surjective [9, Remark ]. Last, it can easily be seen that if F were not étale over B, then E would have to be ramified over B/I as well (e.g. by applying the criterion of [9, Example ]). 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 étale cover V of U such that we have a commutative diagram D U = 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 étale. Thus D is equal to the normalisation of C in k(d U ). Now compare this with D = C Y, a normal cover of C since the étaleness of Y X is preserved under base change to C. D is birational to D U 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 i C : 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 étale cover Y, and since D = D C, the claim follows by associativity 31

38 of the fibre product. In conclusion, we have that N(C) = ĩ 1 C (ĩ C (N C )) = N C. Now if z is a regular point of C, then there is a unique point π 1 ( z) of C lying above z C, and we have a natural isomorphism π 1 ( z) = π 1 (z) fitting into the following diagram: π 1 ( z) π 1 ( C) = π 1 (z) A diagram chase comparing N(z) to N z now easily proves the claim. Returning to the proof of Theorem , we have N(x i ) = N xi for all i. Let N(C ) = ĩ 1 C (N) denote the preimage under the natural map ĩ C : π 1 ( C ) π 1 (X), M its image in G, and M the image of N in G. If x i is the unique point of C lying above x i, we have a diagram: π 1 ( C) G π 1 ( C ) N C M, M N C π 1 ( x i ) N xi Noting that the image of N xi in G must be the same for both triangles, we have, in particular, that F rob xi M iff F rob xi M for all i. Since the x i were picked such that the {F rob xi } 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 N x = N (x) by Lemma Now since N(x) and N (x) both map to M = M when composing the canonical morphism ĩ x : π 1 x π 1 (X) with the projection 32

39 π 1 (X) G, they must be equal. Thus we have N x = N(x) as claimed. This finishes the proof of the theorem. 33

40 Chapter 3 The Wiesend Idèle Class Group In this chapter, we let X be an arithmetical scheme, and define the Wiesend idèle class group for both the flat and variety case (cf. Definition 2.1.2). We define the reciprocity homomorphism and establish some functorial properties. 3.1 Definitions and some Functiorial Properties Let X be an arithmetical scheme. As before, we let X denote the set of closed points of X and let C X be a curve. If X is in the flat case, then we have two possibilities: If the image of the structural morphism C Spec Z is a point p Spec Z then C is called vertical. Otherwise, C Spec Z has dense image. Then the regular compactification P (C) of C is isomorphic to some order of Spec R, where R O K 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

41 to Spec O K. Definition Let C X be a curve inside an arithmetical scheme X. If the structural morphism of C factors through Spec F p 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 ]) the following: Fact 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 ν x it defines. For a scheme X, let the idèle group (after Wiesend) be the abelian topological group group defined by J X = Z.x K(C) ν x X C X ν C with the direct sum topology. Note that this is a countable direct sum of locally compact abelian groups (cf. [21, Section 7, Remarks after 1st Definition]). A typical 35

42 element is of the form ((n x.x) x, (t C,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 If X is an arithmetical scheme with idèle group J X, then J X is Hausdorff but not necessarily locally compact. If C X is a horizontal curve on X, let C arch denote the set of archimedean places. If C X is vertical, set C arch =. Then the subgroup J 1 X = C X ν C arch is the connected componenet of the identity. K(C) ν Definition Let f : X Y be a morphism of arithmetical schemes. We define the induced morphism f J : J X J Y 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 f J (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 F p n = k(y). For ν a valuation on K(C), let k(ν) = O ν /m ν 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 f J as f J (t) = ν(t)[k(ν) : k(y)].y. 36

43 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 J Y has a factor K(D) ω, so and we can define f J : K(C) ν K(D) ω by the norm N K(C)ν/K(D) ω. 2c) Lastly, if ω has a center z on D, then J Y does not contain the summand K(D) ω, but instead the discrete summand Z.z, so we define f J (t) = ν(t) [k(ν) : k(z)]. z for all t K(C) ν. Finally, define f J : J X J Y by setting the image of f J equal to the sum of the images of the components, as they were defined above. Then J is a continuous homomorphism by definition. Remark The identity morphism X X induces the identity J X J X. 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 idèle groups with induced morphism f J. We shall use the functorial properties of the induced morphisms repeatedly. Among other things, they imply that if the diagram C f 2 D f 1 X f 4 Y f 3 37

44 of schemes over S is commutative, then so is the diagram J C f 2 J J D f 1 J J X f 4 J J Y f 3 J of induced morphisms on the idèle groups. For a curve C X with normalisaton C, the composition i C of the canonical maps C C X thus induces a morphism (i C ) J : J ec J X (3.1.1) There also exists a natural inclusion map j C : K(C) J ec given by componentwise inclusion: t ((ν x (t). x), (i ν (t)) ν ), where ν x is the discrete valuation associated to the closed point x C, and i ν : K(C) K(C) ν is the inclusion of the function field in its completion at ν. Composing j C with i C, and taking the direct sum over all curves C X, we get a map j = Σ C (i C j C ) : C X K(C) J X. (3.1.2) Definition Let X be an arithmetical scheme of Kronecker dimension dim(x) 2, or a regular curve. Then the Wiesend idèle class group of X is 38

45 defined to be the quotient of J X by the image of j: C X = J X / im(j) = Z.x K(C) ν / x X C X ν C C X im(i C j C ) together with the quotient topology. For an arithmetical scheme of dimension zero, i.e. a point x, we define analogously C x = J x = Z.x. Remark For an arithmetical scheme X with class group C X, let D X denote the connected component of the identity. Then D X is equal to the closure of the image of the subgroup J 1 X in C X defined in (also see [21, Section 7]). Proposition For a morphism f : X Y, the induced map f J on the ideal class groups descends to a continuous homomorphism f : C X C Y on the class groups. Proof. To show this, we have to prove that im(j) is contained in the kernel of the canonical map f J : J X J Y C Y. Since im(j) = Σ C X im(i C j C ), this amounts to showing that each (i C j C )(K(C) ) is is contained in the kernel of f J. There are several cases to consider: 1) f(c) = y is a closed point of Y. Then for t K(C), we have (f J C j C )(t) = f J C (((i ν (t)) ν C, (ν x (t)) x C e) (3.1.3) = ν(t)[k(ν) : k(y)].y + ν x (t)[k(x) : k(y)].y ν C x e C = ( ν(t)[k(ν) : k(y)]).y ν V K(C) = (deg(div(t)).y = 0 (3.1.4) 39

46 where V K(C) denotes the set of places on K(C). We obtain a canonical diagram K(C) j C i J C ec J X (f C ) J f J 0 Z.y can. J Y, where the first square commutes by the above computation, and the second square commutes as the middle vertical map is just f J i C restricted to J ec. Thus, the outer rectangle also commutes, and the image of K(C) is in the kernel of f J, as required. 2) f(c) is dense inside a curve D of Y. There exists a unique f : C D induced by f such that the following diagrams commute: C C X!h D f C D Y. Since the maps j C : K(C) J C and j D : K(D) J D factor through J ec J C and J ed J D, respectively, we get diagrams f K(C) j C J ec J C J X N K(C)/K(D) h J (f C ) J K(D) j D J ed J D J Y. We show that the left square is commutative: Indeed, we show that h J j C = j D N K(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 = ν ey for some 40 f J