Knots in Number Theory

T.D. van Mulligen

Master thesis, September 2017

Thesis supervisor: Dr. R.I. van der Veen Second supervisor: Dr. G.S. Zalamansky

Mathematisch Instituut, Universiteit Leiden Contents

1. Introduction ...... 1

2. Topological preliminaries ...... 5 2.1 Knots...... 5 2.2 Covering spaces...... 7

3. Algebraic preliminaries ...... 13 3.1 Proﬁnite groups...... 13 3.2 Aﬃne schemes...... 15 3.3 Finite étale coverings...... 17 Interlude on Galois categories...... 19 3.4 Étale fundamental groups...... 20

4. The linking number and the Legendre symbol ...... 23 4.1 The linking number...... 23 4.2 The mod 2 linking number for primes...... 26

5. Decomposition of knots and primes ...... 31 5.1 Decomposition of knots...... 31 5.2 Decomposition of primes...... 37

6. Homology groups and ideal class groups ...... 43 6.1 Homology groups and the Hurewicz theorem...... 43 6.2 Ideal class groups and Artin reciprocity...... 44 ii Contents

7. The Alexander and Iwasawa polynomials ...... 47 7.1 Diﬀerential modules...... 47 7.2 The Alexander polynomial...... 55 7.3 The Alexander polynomial as a characteristic polynomial...... 61 7.4 Complete diﬀerential modules...... 63 7.5 The Iwasawa polynomial...... 66 7.6 The Iwasawa polynomial as a characteristic polynomial...... 69

Bibliography ...... 71 CHAPTER 1

Introduction

It had long been the consensus that arithmetic topology sprung from the mind of Barry Mazur in the 1960s. However, in 2011 a set of unpublished notes by Mazur, titled “Remarks on the Alexander Polynomial”, was discovered, which was dated 1963/1964. In its introduction, Mazur states:

“Mumford has suggested a most elegant model as a geometric interpretation of the above situation: Spec p is like a one-dimensional knot in Spec which is like a simply connected three-manifold.”/

The dimension they talk of is the étale cohomological dimension, and indeed, as a result of Artin- Verdier duality the étale cohomological dimension of Spec k is 3, where k is a ring of integers, O O while it is 1 for a ﬁnite ﬁeld Spec ¿q [Mor12, p. 40 and 2.42]. This suggests a connection between knots, which are embeddings

S1 , M −−→ of a 1-dimensional object into a 3-manifold M, and primes, which can be viewed as embeddings

Spec k p , Spec k, O / −−→ O where p is a prime ideal of k and k p a finite field. O O / The suggestion by Mumford has inspired a fruitful analogy between knots and primes and this field is now known as arithmetic topology. Some of the analogous concepts are listed in table 1.1. In this thesis we intend to explore the merits of this analogy. In chapters 2 and 3 we will give some background information on the topological and the algebraic side of things, including the analogy between the topological fundamental group π1 and the étale fundamental πet´ group 1 . This will further motivate the analogy between knots and primes, since we will show that 2 1. Introduction

Knots Primes 1 S Spec ¿p 3 Spec 1 3 Knot S , Prime Spec ¿p , Spec → → Tubular neighborhood VK Field of p-adic numbers p

q∗ q 1 2 Mod 2 linking number lk2 L, K Legendre symbol (q 1 q) ( ) ( p ) ∗ (− )( − )/ q∗ p lk K, L lk L, K Quadratic reciprocity: p q ( ) ( ) ( ) ( ∗ ) First homology group H1 X Ideal class group H k ( ) ( ) Hurewicz theorem Artin reciprocity

Alexander module AK Complete Alexander module Aψ

Knot module H1 X Iwasawa module H ( ∞) ∞ Alexander polynomial ∆K t Iwasawa polynomial f T ( ) ( ) Tab. 1.1: Some analogous concepts in the theory of knots and primes.

et´ et´ π Spec ¿p ˆ and π Spec 1, 1 ( ) 1 ( ) where ˆ indicates the proﬁnite completion of . This is analogous to the case of knots where we have 1 3 π1 S and π1 R 1. ( ) ( )

In chapter 4 we will start our exploration of arithmetic topology by studying the linking number (for knots) and the Legendre symbol (for primes). Although these two concepts don’t seem connected at first, we will see that they are actually defined very similarly and that they imply similar results on the decomposition of knots and primes in coverings. Chapter 5 will expand on this and discusses decomposition in more generality while considering subcovers and ramification of knots and primes. In chapter 6 we will give an argument for the analogy between the first homology group and the ideal class group. This chapter serves as a stepping stone for chapter 7, where the Alexander polynomial for knots is compared to the Iwasawa polynomial for primes. We build the theory to compute the Alexander polynomial from the ground up, after which we will show that the same construction very much applies to the Iwasawa polynomial. Although chapters 2 and 3 serve to review some background knowledge, not everything can be covered. Some knowledge of algebraic topology (particularly homology), knot theory (particularly the knot group and the Wirtinger presentation), geometry (particularly affine schemes), algebraic number theory (particularly ramification of primes in number fields and possibly some class field theory) and profinite groups will be useful. Every chapter is written so that the first half deals with knots and the second half deals with primes. Furthermore, the results in every chapter are as much as possible laid out in such a way 3 that putting the halves of a chapter side by side would instantly show the similarities between knot theory and number theory. Ideally, the reader will experience multiple instances of déjà vu over the course of reading this text. 4 1. Introduction CHAPTER 2

Topological preliminaries

In order to deﬁne and recognize analogous concepts in the theory of knots and primes, we will need some basic familiarity with knots and covering spaces.

2.1 Knots

We know knots from everyday life. Mathematical knots, however, need to be closed. The way to model this is to define a knot as an embedding of a circle in 3-space. Definition 2.1.1. Let M be a 3-manifold. A knot is an embedding S1 , M. The manifold M is 1 → called the ambient space and we call X 1 : M S the knot complement. We denote a tubular S \ neighborhood of K by VK and define the knot exterior as XK : M V K. \ Usually, we will take M S3. See figure 2.1 for some basic knots. To streamline notation, when we talk of a knot K, we mean the image f S1 M of the embedding f : S1 , M. Of course, topologically, any two knots will always( be) homeomorphic. ⊆ However, we→ obviously don’t want every embedding S1 , 3 to be considered the same. Therefore, we need a different notion of equivalence between knots.→

Unknot Trefoil knot Figure-eight knot Cinquefoil knot Three-twist knot

Fig. 2.1: Some basic knots.

Deﬁnition 2.1.2. Let K, L M be two knots. We say that K and L are equivalent if there exists an ⊆ ambient isotopy taking K to L, i.e., there exists a continuous map F : M 0, 1 M such that F0 is × [ ] → the identity on M, Ft is a homeomorphism for every 0 t 1 and F1 K L. ≤ ≤ ( ) 6 2. Topological preliminaries

See ﬁgure 2.2 for an ambient isotopy between two representations of the trefoil knot.

t 0 t 0.25 t 0.5 t 0.75 t 1

Fig. 2.2: Ambient isotopy on the trefoil knot. The dashed lines indicate the deformation of the knot.

Since we are going to take a closer look at the way different knots interact with each other, we need to define the notion of a link. Ãn 1 Definition 2.1.3. Let M be a 3-manifold and n 1. We call an embedding i1 S , M an Ãn 1 ≥ → n-component link. Here i1 S is the disjoint union of n circles.

A knot is a link with 1 component and two links L1 and L2 are equivalent if there exists an ambient isotopy taking L1 to L2. One of our goals is to construct covering spaces using knots. We are going to do this by cutting open multiple copies of the ambient space and gluing them together in a certain way. For this we need the following definitions. Definition 2.1.4. A knot is polygonal if it consists of a finite set of straight line segments. We call a knot tame if it is equivalent to a polygonal knot.

All knots in this text are assumed to be tame. Figure 2.3 illustrates why the trefoil knot is a tame knot.

Fig. 2.3: A polygonal representation of the trefoil knot.

We move on to the deﬁnition of a Seifert surface. Deﬁnition 2.1.5. Let K be a knot. A Seifert surface is a compact, connected, oriented surface S such that ∂S K, where ∂S is the boundary of S.

It’s a known fact that every tame knot has a Seifert surface [Rol03, Chap. 5, Sect. A, Theorem 4]. See figure 2.4 for a Seifert surface of the trefoil knot. We end this section with a definition of the knot group. 3 Definition 2.1.6. Let K be a knot in S . The knot group GK is defined as the fundamental group of the 3 3 knot complement in S , i.e., GK : π1 S K . ( \ ) In the next section we’re going to take a look at covering spaces. Using covering spaces, we will be able to obtain information about the knot group. 2.2. Covering spaces 7

+ −

Fig. 2.4: Seifert surface of the trefoil knot, with + and indicating the orientation. −

2.2 Covering spaces

The theory of covering spaces is important for the study of the fundamental group. Let us first define what a covering space is. Definition 2.2.1. Let X be a topological space. A covering map is a continuous surjective map 1 Ã p : Y X such that for every x X there is an open neighborhood Ux of x such that p− Ux i I Vi → ∈ ( ) ∈ for some index set I and p Vi : Vi Ux is a homeomorphism for every i I. The space Y is called the | → 1 Ã ∈ covering space. An open subset U X such that p− U i I Vi is called evenly covered by p. 1 ⊆ ( ) ∈ 1 The cardinality #p− x is called the degree of the covering. If #p− x n < , then we speak of a finite cover and in particular( ) of an n-fold cover. ( ) ∞

Let’s take a look at some examples of covering spaces. Example 2.2.2 (Coverings of the circle). We view S1 ¼ as the unit circle in the complex plane. 1 1 n ⊆ 1 2πxi Deﬁne pn : S S , z z for n 1 and p : S , x e . These are coverings of the → 7→ ≥ ∞ → 7→ circle. I

One thing coverings allow us to do, is lift a homotopy in X to its covering space Y. Theorem 2.2.3. Let p : Y X be a covering map and let H : Z 0, 1 X be a homotopy → × [ ] → and H˜ 0 : Z 0 Y a lift of H0 in Y, i.e., H0 p H˜ 0. Then there exists a unique homotopy × { } → ◦ H˜ : Z 0, 1 Y extending H˜ 0 and lifting H, i.e., H p H˜ . × [ ] → ◦ In particular, taking Z to be a singleton, theorem 2.2.3 tells us that a path in X can be lifted to its covering space. Corollary 2.2.4. Let p : Y X be a covering map and let γ : 0, 1 X be a path in X. Let x : γ 0 1 → [ ] → ( ) and choose a point y p− x . Then there exists a unique path γ˜ : 0, 1 Y such that γ p γ˜ and γ˜ 0 y. ∈ ( ) [ ] → ◦ ( ) 1 Using the path-lifting property, we can define an action of π1 X on a fiber p x . ( ) − ( ) Definition 2.2.5. Let p : Y X be a covering and let γ : 0, 1 X be a loop in X with endpoints x X. → [1 ] → ∈ Denote with γ the class of γ in π1 X . For every y p− x , there exists a unique lift γ˜ y : 0, 1 Y [ ] ( ) ∈ ( ) 1 [ ] → of γ such that γ˜ y 0 y. Then we define γ y : γ˜ y 1 h− x . This is called the monodromy ( ) 1 [ ]· ( ) ∈ ( ) 1 action of π1 X on h− x . The induced map ρx : π1 X Aut p− x is called the monodromy permutation( representation) ( ) . ( ) → ( ( )) Proposition 2.2.6. The monodromy action is well-defined. 8 2. Topological preliminaries

Proof. Firstly, note that a constant path in X can only be lifted to a constant path in Y, since the elements of a fiber can be separated by disjoint open sets and a path has to be continuous. Secondly, for two loops γ, γ0 : 0, 1 X with γ γ0 π1 X a path homotopy H : 0, 1 0, 1 X between γ and γ can[ be] lifted → to a homotopy[ ] [ ]H˜ ∈: 0,( 1 ) 0, 1 Y between γ˜ [and]γ˜ × [ ] → 0 [ ] × [ ] → 0 by theorem 2.2.3. Since H˜ 1 is a lift of the constant map to γ 1 γ 1 , we have γ˜ 1 γ˜ 1 . ( ) 0( ) ( ) 0( ) Also note that every element γ of π1 X actually defines an automorphism on the fiber 1 1 [ ] ( ) 1 p− x : denote with γ− the reverse path of γ, i.e., we define γ− : 0, 1 X, t γ 1 t . If ( ) 1 1 [ ] → 7→ ( 1− ) γ y γ y0 for some y, y0 p− x , then let γ˜ y γ˜ − be the concatenation of γ˜ y and γ˜ − . Note y0 y0 [ ]· [ ]· 1 ∈ ( ) 1 1 that γ˜ y γ˜ − 1 y0. Then p γ˜ y γ˜ − γ γ− constx , where constx is the constant y0 y0 ( )( ) [ ( )] [ 1 ] [ ] map on x. Then we find y constx y p γ˜ y γ˜ − y y0. This proves injectivity. For any y0 [ ]· [ ( )] · 1 1 1 1 1 1 1− y p− x let γg− y be the lift of γ− with γg− y 0 y. Denote y0 : γg− y 1 . Then γg− y is the ∈ ( ) 1 1 ( ) ( ) lift of γ with γg1− 0 y and γg1− 1 y. Therefore, γ y y. This proves surjectivity. − y ( ) 0 − y ( ) [ ]· 0 There is a correspondence between covering spaces and subgroups of the fundamental group. Before we state this correspondence, we’re going to expand our vocabulary of covering spaces a little. Definition 2.2.7. Let p : Y X be a covering. →

(1) A covering p0 : Y0 X is called a subcovering of p if there exists a continuous f : Y Y0 such that p p f. Then f→is called a morphism of coverings. → 0 ◦ (2)A deck transformation or covering transformation is a homeomorphism h : Y Y such that p p h. The group of deck transformations is denoted by Aut Y X . → ◦ ( / ) (3) The covering p is called Galois or Normal if for every x and every pair y, y p 1 x , there 0 ∈ − ( ) exists a deck transformation h Aut Y X such that h y y0. We then denote the group of deck transformations by Gal Y X . ∈ ( / ) ( ) ( / ) (4) The space Y is called a universal covering space if it is simply connected, i.e., if π1 Y 1. ( )

It is known that a morphism Y Y0 of coverings over X is itself a covering if Y, Y0 and X are connected and X is locally connected→ [Ful95, Exercise 11.12]. Furthermore, if Y is connected and 1 p : Y X is a covering, then P is Galois if and only if for every x and every pair y, y0 p− x there→ is a unique h Aut Y X such that h y y [Die10, p. 65]. Since these conditions certainly∈ ( ) ∈ ( / ) ( ) 0 hold for connected covers of 3-manifolds (e.g. connected covers of the knot complement XK in S3), we will make regular use of these facts.

Example 2.2.8. We continue our example of the circle. Let pn and p be deﬁned as in exam- ∞ 1 2πxi n ple 2.2.2. Since is simply connected, p is a universal cover. Deﬁne hn : S , x e / ∞ → 7→ for n 1 and note that p pn hn. This shows that every pn is a subcover of p . I ≥ ∞ ◦ ∞ We now state the Galois correspondence for coverings. Theorem 2.2.9. By a connected covering we mean a covering p : Y X such that Y is connected. → The map p : π1 Y π1 X induced by such coverings is injective and gives rise to a bĳection ∗ ( ) → ( ) conn. cov. p : Y X isom. subgroups of π1 X conjugacy { → }/ −→ { ( )}/ p : Y X p π1 Y . ( → ) 7−→ ∗( ( )) This correspondence has the following properties: 2.2. Covering spaces 9

(1) A covering p0 : Y0 X is a subcover of p : Y X if and only if p π1 Y is a subgroup of p0 π1 Y0 up to conjugacy. → → ∗( ( )) ∗( ( ))

(2) A covering p : Y X is Galois if and only if p π1 Y is a normal subgroup of π1 X . We then have → ∗( ( )) ( ) Gal Y X π1 X p π1 Y . ( / ) ( )/ ∗( ( )) Furthermore, if p : Y X is a Galois cover we have a bĳection → conn. subcov. of p : Y X Gal Y X subgroups of Gal Y X conjugacy { → }/ ( / ) −→ { ( / )}/ between connected subcovers of p and subgroups of Gal Y X . ( / ) Example 2.2.10. Let X be a topological space and suppose it has a universal covering X˜ → X. Since π1 X˜ 1, theorem 2.2.9(2) tells us that p is a Galois cover with Gal X˜ X ( ) ( / ) π1 X p π1 X˜ π1 X . Since p π1 X˜ is a subgroup of any subgroup of π1 X , by theorem( 2.2.9)/ ∗((1)( any)) connected( ) cover of∗(X is( a)) subcover of the universal cover. ( ) A connected and locally path-connected space X has a universal covering if and only if X is semilocally simply connected, i.e., every point x X has a neighborhood U such that the ∈ homomorphism π1 U, x π1 X, x is trivial [Kos80, Theorem 22.1]. In particular, this holds for link complements, so( every) → connected( ) cover of a link complement is a subcover of its universal cover. I

Example 2.2.11. We continue our example of the circle. Deﬁne pn and p as in example 2.2.2. Since p is a Galois cover, theorem 2.2.9 tells us ∞ ∞ 1 1 1 π1 S π1 S p π1 Gal S . ( ) ( )/( ∞)∗( ( )) ( / ) 1 1 Note that the only subgroups of are of the form n for n 0. Since every covering pn : S S is Galois with ≥ → 1 1 1 Gal S S z ζnz n π1 S n, ( / ) h 7→ i / ( )/ 1 by theorem 2.2.9(2) we get pn π1 S n for every n . Therefore, p and pn give all ( )∗( ( )) ∈ ∞ possible connected coverings of the circle. I

As seen in the last two examples, we can get information on the structure of the fundamental group by looking at the universal cover. Next, we want to deﬁne the so-called inﬁnite cyclic cover of a knot complement. For this we will need some information about the knot group. Here is a key fact that we will use. 3 ab ab Lemma 2.2.12. Let K S be a knot. We have GK , where GK is the abelianization of the knot group. ⊆

Proof. This proof will make use of some homology theory. Let V : VK be a tubular neighboor- hood of K. Set U : S3 K, the knot complement of K. Since U V S3, we can apply the Mayer-Vietoris sequence\ ∪

3 3 ... H2 S ; H1 U V; H1 U; H1 V; H1 S ; ... (2.1) −→ ( ) −→ ( ∩ ) −→ ( ) ⊕ ( ) −→ ( ) −→ The ﬁrst homology group of the torus T is isomorphic to (this can be calculated using the homology of CW-complexes, which we will not go into here).⊕ Since U V is homotopy equivalent ∩ 1 to the torus, we have H1 U V; H1 T; . Likewise, we get H1 V; H1 S ; , so (2.1) reduces to the exact( sequence∩ ) ( ) ( ) ( )

0 H1 U; 0. −→ ⊕ −→ ( ) ⊕ −→ 10 2. Topological preliminaries

From the isomorphism H1 U; it follows that H1 U; is a ﬁnitely generated projective -module [Eis04⊕, Appendix( 3, Proposition) ⊕ A3.1], so it is( free[Corollary) 8.5]Rot-Adv. H U H U πab U This gives us 1 ; . By the Hurewicz theorem we have 1 ; 1 , from which ab ( ) ( ) ( ) GK follows.

We can now deﬁne the inﬁnite cyclic cover.

3 3 Deﬁnition 2.2.13. Let K S be a knot and XK : S K the knot complement. By theorem 2.2.9 there ⊆ \ exists a connected cover p : X XK such that p π1 X GK, GK which is uniquely deﬁned up ∞ → ∗( ( ∞)) [ ] to isomorphism over XK. We then have

ab Gal X X π1 X p π1 X GK . ( ∞/ ) ( )/ ∗( ( ∞))

We call X the inﬁnite cyclic cover of X. By theorem 2.2.9, there exists a subcover Xn XK such that ∞ → Gal Xn XK n. We call this cover Xn the n-fold cyclic cover of XK. ( / ) / Using Seifert surfaces, we can give an explicit construction of the inﬁnite cyclic cover.

3 Example 2.2.14. Let K S be a knot. Let SK be a Seifert surface of K. Let Y be the space obtained ⊆ 3 by cutting the knot complement XK : S K along SK XK. Figure 2.5 illustrates this cut. We \ ∩ can also see how Y contains two surfaces homeomorphic to SK XK, indicated in ﬁgure 2.5 by + ∩ S and S−.

cut along S + S S− S

K K

Fig. 2.5: Cutting along a Seifert surface S.

+ + Now let Yi be a copy of Y for every i , where we denote Si : S Yi and Si− : S− Yi. X ∈ Ã Y ⊆S+ S ⊆ Then we get by taking the disjoint union i i and identifying i with i−+1 for every ∞ ∈ i . The canonical projection p : X XK now gives an infinite cyclic cover. ∈ ∞ → We can construct an n-fold cyclic cover Xn of XK likewise. We simply let Yi be a copy of Y + for every i n and define Si and Si− as above. We then obtain Xn by taking the disjoint Ã ∈ /Y S+ S i n union i n i and identifying i with i−+1 for every . I ∈ / ∈ / Finally, we give the definition of a ramified covering. Definition 2.2.15. Let M and N be n-manifolds, where n 2, and let f : N M be continuous. ≥ → Define SN : y N f is not a homeomorphism in a neighborhood of y and SM : f SN . Let k k { ∈ | } ( ) D be the k-dimensional unit disk. We say that f : N M is a covering ramified over SM if the following⊆ conditions are satisfied: →

1) The map f N SN : N SN M SM is a covering map. | \ \ → \ 2) For every y SN, there are a neighborhood V of y, a neighborhood U of f y , homeomorphisms φ : V 2 n 2 ∈ 2 n 2 ( ) → D D ψ U D D e e y > f n 2 φ ψ f − and : − and an integer 1 such that e idD − , × e → × 2 ( ) ( × )◦ ◦ where fe z : z (considering D in the complex plane). The integer e is called the ramiﬁcation index of (y.) 2.2. Covering spaces 11

We call f N SN the covering associated to f. When f N SN is a Galois covering, we call f a ramiﬁed Galois covering| \ . | \

The concept of ramiﬁed coverings serves a very useful purpose in the context of knots and links.

3 3 Example 2.2.16. Let K be a knot in S and let SK be a Seifert surface of K. Let Y be S cut + along S K. Then we obtain two surfaces S and S− homeomorphic to SK (cf. example 2.2.14). + + If we take n > 0 copies of Y, we can construct a space Mn by identifying S : S Yi with i ⊆ S S Y + i n p M S3 i−+1 : − i 1 for all . The canonical projection : n is almost a covering ⊆ ∈ / 1 → map. To be exact, p 1 : M p K X is the n-fold cyclic cover. We will show that Mn p− K n − K the map p is actually| a covering\ ( ) ramified\ ( over) →K. 1 Let y p− K and set x : p y K. Let VK be a tubular neighborhood of K. Take ∈ ( ) ( ) ∈ 2 1 1 U to be a neighborhood of x homeomorphic to D D VK through ψ. Set V : p− U 1 2 ×1 ⊆ ( ) and Vi p− U Yi for all i. We have V D D through a homeomorphism φ, where ( ) ∩ ×1 the solid torus segment V has to wrap around p− K in n copies of Y. Therefore, if we set 2 2 n ( ) fn : D D , z z , then → 7→ φ V D2 D1 × p fn idD1 ψ × U D2 D1 × commutes, so p is a ramified (Galois!) covering over K with ramification index n. See figure 2.6 for an illustration where n 3. This construction is called the Fox completion of the covering 1 p Mn p 1 K : Mn p− K XK. I | \ − ( ) \ ( ) →

1 1 1 Y1 p K Y2 p K Y3 p K − ( ) − ( ) − ( ) y y y φ V1 V V V 1 S 2 S 3 S 1− 2− 3− S+ S+ S+ 1 2 3 V3 V2

p f3 id 1 × D K

x ψ

U U

Fig. 2.6: A covering Y Y Y S3 ramiﬁed over K with ramiﬁcation index e 3. 1 ∪ 2 ∪ 3 → 12 2. Topological preliminaries CHAPTER 3

Algebraic preliminaries

In this chapter we’ll gain some familiarity with concepts and constructions such as p-adic numbers, étale morphisms and aﬃne schemes.

3.1 Proﬁnite groups

To define the étale fundamental group, we’ll need to know what a profinite group is. We start with the definition of a topological group. Definition 3.1.1. A topological group is a group G with a topology such that the group operation is continuous.

A profinite group is constructed using inverse systems of finite groups. We’ll need the following definitions.

Deﬁnition 3.1.2. Let I, be a partially ordered set. Let Ci i I be a collection of objects in a category ( ≤) ( ) ∈ and for every i j let fij : Cj Ci be a morphism. Then Ci i I, fij i j is an inverse system if itC satisﬁes the following≤ properties:→ (( ) ∈ ( ) ≤ )

1) For every i I we have fii idC . ∈ i 2) For every i j k we have fik fij fjk. ≤ ≤ ◦ Inverse systems give rise to inverse limits.

Definition 3.1.3. Let Gi i I, fij i j be an inverse system of groups. We define the inverse limit of this inverse system as follows:(( ) ∈ ( ) ≤ ) Ö lim Gi : xi i I Gi for all i j we have fij xj xi . {( ) ∈ ∈ | ≤ ( ) } i i I ←−− ∈ We will most commonly deal with inverse systems of groups or rings, specifically profinite groups and rings. 14 3. Algebraic preliminaries

Deﬁnition 3.1.4. We call a group G a proﬁnite group if we have

G lim Gi ←−−i for some inverse system Gi i I, fij i j of finite groups. Similarly, a ring R is called a profinite ring if we have (( ) ∈ ( ) ≤ ) R lim Rj ←−−j for some inverse system Rj j J, gij i j of finite rings. (( ) ∈ ( ) ≤ )

Profinite groups G lim Gi and profinite rings R lim Rj are often endowed with the i j Î Î topology induced by the product←−− topology on i I Gi or ←−−j J Rj, which turns them into topological groups and topological rings, respectively.∈ ∈ The canonical example of a profinite group or ring are the p-adic integers. These arise as the inverse limit of an inverse system obtained by modular arithmetic. Firstly, note that has an obvious partial order. Secondly, for every m n we have canonical quotient maps n m ≤ qmn : p p . Since for every l m n we have qln qlm qmn, we find that the quotient/ groups→ / pn form an inverse system.≤ ≤ We can now define the◦p-adic integers. / Definition 3.1.5. Let p be a prime. The ring of p-adic integers p is the inverse limit of the n ∈ inverse system p n , qmn m n : (( / ) ∈ ( ) ≤ ) n p : lim p . / ←−− Let’s see what the p-adic integers look like explicitly. m Example 3.1.6. Elements of p are sequences a1, a2, a3, ... of integers where an am mod p ( ) ≡ for all m n. The ring homomorphism , p sends an integer n to the element n, n, n, ... ≤ → ( ) ∈ p. This homomorphism is not surjective, however. For example,

n 1 Õ− pi 1, 1 + p, 1 + p + p2, 1 + p + p2 + p3, ... n ( ) i0 ∈ is well-deﬁned in p, but is not in the image of , p. Now let’s observe some more facts of → p. Addition and multiplication in p is done pointwise, i.e., we have

a1, a2, a3, ... + b1, b2, b3, ... : a1 + b1, a2 + b2, a3 + b3, ... ( ) ( ) ( ) and

a1, a2, a3, ... b1, b2, b3, ... : a1b1, a2b2, a3b3, ... . ( )·( ) ( ) This is well-deﬁned because addition and multiplication of integers respects the congruence relation in modular arithmetic. This means that 1, 1, 1, ... is the multiplicative identity in p. ( ) Then what does p× look like? It turns out that the units of p are easy to characterize.

Let x1, x2, x3, ... p. To find out if x1, x2, x3, ... is a unit, we have to find out if any ( ) ∈ ( ) multiple of xi is congruent to 1 for all i. The answer is neatly given by the following equivalences: i i nxi 1 mod p for some n gcd xi, p 1 p - xi x1 , 0. ≡ ∈ ⇐⇒ ( ) ⇐⇒ ⇐⇒ Therefore, x1, x2, x3, ... p× if and only if x1 , 0. I ( ) ∈ 3.2. Affine schemes 15

Lastly, we mention the profinite completion of a group. Let G be a group and consider the set of normal subgroups N C G such that G N is finite. Then the quotient groups, together N / with the canonical quotient maps qMN : G M g N for M N, form an inverse system of finite groups. We define the profinite completion/ → as its/ inverse limit.⊆

Deﬁnition 3.1.7. Let G be a group and deﬁne the set

: N C G G N is ﬁnite , N { | / } which is naturally partially ordered by inclusion. Then the proﬁnite group

Gˆ : lim G N / N ←−−∈N is called the proﬁnite completion of G.

Example 3.1.8. Consider the group . Every subgroup of is of the form n, so is the set of all sugroups of . We get N ˆ lim n / ←−−n and elements of ˆ are sequences a1, a2, a3, ... where an n and where an am mod m ( ) ∈ / ≡ if m n, e.g., 0, 1, 2, 3, ... ˆ . I | ( ) ∈

3.2 Aﬃne schemes

In this section we’ll briefly explore affine schemes. Since an affine scheme is defined as the spectrum of some commutative ring with a certain topology defined on it, they are integral to the study of primes. We start with the definition of the Zariski topology on the spectrum of a ring.

Proposition 3.2.1. Let R be a commutative ring with an ideal I R. Deﬁne VI : p Spec R I p . Ð ⊆ { ∈ | ⊆ } The collection of closed sets I R ideal VI deﬁnes a topology on Spec R. We call this the Zariski topology. ⊆ { }

Proposition 3.2.1, as well as the following lemma, is easy to prove, but can be found in [Mum99, Chap. II, Sect. 1].

Lemma 3.2.2. Let R be a commutative ring and f R. Deﬁne Df : p Spec R f < p . Then ∈ { ∈ | } Df f R forms a basis for the Zariski topology on Spec R. ( ) ∈

In the following, let Rf indicate the localization of the ring R at the multiplicatively closed 2 3 n set f, f , f , ... , , i.e., Rf consists of fractions r f , where r R and n . For a more rigorous treatment,{ see [Eis04} , Chap. 2, Sect. 1]. / ∈ ∈

Lemma 3.2.3. Let R be a commutative ring and endow Spec R with the Zariski topology. Deﬁne the n partial order f g Df Dg. Let f, g R and suppose Df Dg. Then f gh for some n ≤ ⇔ ⊆ ∈ a ⊆ahm ∈ and h R and we can deﬁne a map φfg : Rg Rf, gm fnm . Then Rf f R, φfg f g is an ∈ → 7→ (( ) ∈ ( ) ≤ ) inverse system of rings. 16 3. Algebraic preliminaries

Proof. First, suppose Df Dg. Then for every p with g p we have f p. It is a basic fact in commutative algebra that⊆ the radical of an ideal I is the intersection∈ of all prime∈ ideals containing p n I, so f g . This gives us f gh for some n and h R. To check that Rf f R, φfg f g ∈ ( ) n1 ∈ n∈2 n1n2(( ) ∈n2 ( ) ≤ ) is an inverse system, let f1 f2 f3, with f f2h1 and f f3h2 (so f f3h h2). Then ≤ ≤ 1 2 1 1

i i n2i i n2i n1n2i i φf f φf f a f φf f ah f ah h f φf f a f . ( 2 3 ◦ 1 2 )( /( 1)) 2 3 ( 2/ 2 ) 2 1 / 1 1 3 ( / 3) Also, we clearly have φff idRf , so Rf f R, φfg f g is an inverse system of rings. (( ) ∈ ( ) ≤ ) Now we are ready to deﬁne the structure sheaf on a spectrum. To this end, note that a topological space X can be viewed as a category with the open sets as its objects and inclusion maps between open sets as its arrows. This way, we can view Spec R as a category.

Deﬁnition 3.2.4. Let R be a commutative ring and endow Spec R with the Zariski topology. The structure sheaf on Spec R is the functor

Spec R : Spec R CRing O −→ U lim Rf 7−→ ←−−f R Df∈ U ⊆

i : U1 , U2 lim Rf lim Rf, xf Df U2 xf Df U1 , ( → ) 7−→ → ( ) ⊆ 7→ ( ) ⊆ ←−−f R ←−−f R Df∈ U2 Df∈ U1 ⊆ ⊆ where CRing is the category of commutative rings.

The idea behind this deﬁnition is that the structure sheaf on Spec R is the unique sheaf that sends open sets Df to Rf.

Deﬁnition 3.2.5. An aﬃne scheme Spec R, Spec R is a spectrum Spec R of some commutative ring R equipped with the Zariski topology together( withO its structure) sheaf.

We will rarely make explicit use of the Zariski topology and the structure sheaf. Instead, we will make extensive use of the following duality between the category of aﬃne schemes and the category of commutative rings.

Theorem 3.2.6. The functor

Spec : CRing Aff −→ R Spec R 7−→ 1 f : R S Spec S Spec R,p f− p ( → ) 7−→ ( → 7→ ( )) defines a duality between the category CRing of commutative rings and the category Aff of affine schemes.

There are some very useful consequences of this duality. For one, for all R, S CRing there is a one-to-one correspondence between ring homomorphisms R S and morphisms∈ of schemes Spec S Spec R. Also, since is an initial object in CRing, i.e.,→ for every commutative ring R there→ is exactly one ring homomorphism R, by the above duality Spec is a terminal object in Aﬀ, i.e., for every aﬃne scheme Spec→R there is exactly one morphism of schemes Spec R Spec . → 3.3. Finite étale coverings 17

Example 3.2.7. Let’s take a look at Spec and how it interacts with Spec ¿p. For every prime p there is a unique ring homomorphism q : ¿p. Therefore, there is also a unique ∈ → morphism of schemes i : Spec ¿p Spec . Since every ﬁeld has only one prime ideal, namely →1 the zero ideal 0, we have i 0 q 0 p . Figure 3.1 illustrates the way Spec ¿p injects into ( ) − ( ) ( ) Spec . I

Spec ¿ Spec ¿ Spec ¿ Spec ¿ Spec ¿p 2 3 5 7 ···

Spec 2 3 5 7 p ( ) ( ) ( ) ( ) ··· ( ) Fig. 3.1: Primes in Spec .

3.3 Finite étale coverings

In order to deﬁne the étale fundamental group, we need to know what a ﬁnite étale algebra is.

Definition 3.3.1. Let K be a field and let A be a K-algebra. We call A finite étale if A is a finite product În of finite separable extensions, i.e., A i1 Li for some n, where every Li K is a finite separable field extension. /

Next we have the notion of a ﬁnite étale morphism.

Definition 3.3.2. Let f : Spec B Spec A be a morphism of affine schemes. We call f (or the corresponding ring homomorphism A→ B) a finite étale morphism or a finite étale covering if B is → a finitely generated, flat A-module and for any prime p Spec A, the fiber B A κ p is a finite étale κ p -algebra. Here κ p : Frac A p is the residue field∈ of p. We call B a finite⊗ étale( ) A-algebra if there( ) exists a finite étale( ) morphism( A/ ) B. →

As the terminology implies, ﬁnite étale covers are the algebraic analogue of ﬁnite topological coverings. To see this, let p : Y X be an n-fold covering and U X an evenly covered open 1 Ãn → 1 ⊆1 1 set, i.e., p− U i1 Vi. Then the inverse p− U and its maps p− U U and p− U Y give the pullback( ) of U , X Y: ( ) ( ) → ( ) → → ← Ãn i1 Vi Y p p U X .

Now suppose we have a finite étale covering f : Spec B Spec A and let p Spec A. The canonical map A κ p induces an embedding Spec κ p →, Spec A that sends∈ the only prime 0 of κ p to p Spec→ A(. It) turns out that the pullback of( Spec) →κ p , Spec A Spec B is given ( ) ∈ ( ) → ← by Spec B A κ p with maps Spec B A κ p Spec κ p and Spec B A κ p Spec B. Since ⊗ ( ) ⊗ ( ) → În( ) ⊗ ( ) → Spec B Spec A is finite étale, we have B A κ p Li, where Li κ p is a finite separable → ⊗ ( ) i 1 / ( ) 18 3. Algebraic preliminaries

În Ãn extension for every i. Since Spec i1 Li i1 Spec Li, we get the following pullback diagram:

Ãn i1 Spec Li Spec B f Spec κ p Spec A ( ) . We can see that, analogous to the topological case, Spec κ p is covered by the n primes of În ( ) i1 Li. Example 3.3.3. Let A B be a ring homomorphism. If A is a field, then its only prime is 0 → ( ) and we have B A κ p B A A B. Therefore, a A B is finite étale if and only if B is a ⊗ ( ) ⊗ În → finite étale A-algebra, i.e., if and only if B i1 Li for some n, where every Li A is a finite separable field extension. In particular, a field extension B A is finite étale if and/ only if it is / finite separable. I Example 3.3.4. Let d be a squarefree integer. Let’s take a look at the map √d and why it is not finite étale. Let p be a prime number. We have → [ ]

√d p √d p √d , [ ] ⊗ /( ) [ ]/ [ ] so if p √d is a prime ideal, the fiber √d p is the finite separable field extension [ ] [ ] ⊗ /( ) √ √ Îr √ ¿p d . If we have a decomposition p d i1 pi of p d into pairwise distinct prime ideals,[ ] by the Chinese remainder theorem[ we] get [ ]

r Ö √d p √d pi, [ ] ⊗ /( ) [ ]/ i1 which is a finite product of finite separable extensions of ¿p. Îr ei Now suppose p ramifies in √d , i.e., we have a decomposition p √d p of [ ] [ ] i 1 i p √d into prime ideals where ej > 1 for some 1 j r. Choose a non-zero element [ ] Îr ≤ ≤ xi pi for every 1 i r. Then xi √d p √d √d p is a non-trivial ∈ ≤ ≤ i 1 ∈ [ ]/ [ ] [ ] ⊗ /( ) nilpotent element, which means √d p can’t be a finite product of finite separable field [ ] ⊗ /( ) extensions. To summarize: if p ramifies in √d , then √d is not finite étale. Since a [ ] → [ ] prime p ramifies in √d if and only if p divides the discriminant of √d [Mar77, Theorem [ ] [ ] 24 and Theorem 34] and since the discriminant ∆ √d equals 4d, we find that there exists ( [ ]) some p such that √d p is not a finite étale κ p -algebra. Therefore, √d is not a finite étale algebra.[ ] ⊗ /( ) ( ) → [ ] More generally, it is known that if B is a finite étale cover, then Frac B Frac is an unramified extension. However, by Minkowski’s→ bound, for any non-trivial finite( )/ extension( ) K / there is a prime that ramifies in K. Therefore, there are no finite étale covers of . I

We close this section by defining a finite étale Galois cover. Definition 3.3.5. Let A be an integral domain. A finite étale cover Spec B Spec A is called a finite étale Galois cover of Spec A if Frac B Frac A is a finite Galois extension→ and if there exists a finite separable extension K Frac B such that( )/B is( the) integral closure of A in K. We then call B a finite Galois algebra over A/and denote( ) the automorphism group of B over A by Gal B A . Likewise, we write Gal Spec B Spec A : Aut Spec B Spec A . ( / ) ( / ) ( / ) 3.3. Finite étale coverings 19

For a ﬁnite étale Galois cover Spec B Spec, A we have → Gal Spec B Spec A Gal B A Gal Frac B Frac A . ( / ) ( / ) ( ( )/ ( ))

Interlude on Galois categories

Before we deﬁne the étale fundamental group, let’s take a look at how the topological fundamental group arises in the context of category theory. We need some rudimentary concepts for this. Deﬁnition 3.3.6. Let G be a topological group. A G-set is a set X with a group action of G on X which is continuous when X is equipped with the discrete topology.

Next we need to know what the automorphism group of a functor is. Definition 3.3.7. Let and be categories and let F : be a functor. An automorphism of F C D C → D is a collection of isomorphisms µC : F C F C C such that for every morphism f : C D in the diagram ( ( ) → ( )) ∈C → C F f F C ( ) F D ( ) ( ) µC µD F f F C ( ) F D ( ) ( ) commutes. The automorphisms on F form a group, the automorphism group of F, usually denoted Aut F . ( ) Now we can move on to Galois categories. Definition 3.3.8. A Galois category is a category together with a functor F : Set such that is equivalent to the category Aut F -set of finite AutCF -sets. The functor F is calledC → the fundamentalC functor. ( ) ( )

Toillustrate where this is going, let’s compute the automorphism group of a functor explicitly.

Example 3.3.9. Let CovX be the category of connected coverings of a topological space X with basepoint x X and let F : CovX Set be the functor that sends a covering p : Y X to the 1 ∈ → → ﬁber p x and that sends a morphism f : Y Y of coverings to its restriction f 1 . Now let − 0 p− x µ Aut (F )be an automorphism of F and let→f : Y Y be a morphism of coverings| ( p) : Y X ∈ ( ) → 0 → and p0 : Y0 X. Note that f is a covering map (deﬁnition 2.2.7) and therefore surjective. The following diagram→ commutes: f p 1 x p 1 x − ( ) ( 0)− ( ) µ µ Y Y0 1 f 1 p− x p0 − x ( ) ( ) ( ) . Y Y µ The important thing to realize here is that since 0 is a subcover of , the bĳection Y0 is deter- µ y p 1 x µ y f µ f 1 y mined by Y : for every element 0 − we have Y0 Y − . By theorem 2.2.9, we know that the universal cover∈ (X˜ )of (X)covers all coverings,( ) ( ◦ so◦ the)( entire) automorphism µ is determined by the bĳection µX˜ , which is in turn determined by a deck transformation g Gal X˜ X . Therefore, the automorphism group of F is isomorphic to Gal X˜ X , which is ∈ ( / ) ( / ) isomorphic to π1 X by theorem 2.2.9. I ( ) 20 3. Algebraic preliminaries

Looking at the chosen terminology, one could expect that the automorphism group of the fundamental functor is the fundamental group. Alas, since the category CovX does not only contain finite covers, it is not a Galois category. The category FCovX of finite connected covers of X, however, is Galois, and it turns out that it is equivalent to the category of finite πˆ1 X -sets, ( ) where πˆ1 X is the profinite completion of π1 X . ( ) ( ) Similarly to FCovX, the category FEtX of finite étale covers of an affine scheme X together with the fiber functor F : FEtX Set which we will define in the next section, is also a Galois category. Therefore, it is equivalent→ to the category of finite Aut F -sets. This automorphism group of F will be the étale fundamental group. ( )

3.4 Étale fundamental groups

We now want to define the étale fundamental group. In the case of topological spaces, there is an isomorphism π1 X Gal X˜ X between the fundamental group of a space X and the galois group of its universal( ) cover X˜( over/ ) X. This leads us to define the étale fundamental group of an affine scheme X : Spec A as the Galois group of its “universal étale covering” over X. Naturally, we need to find the appropriate notion of a universal cover for finite étale maps. For this we need the following results. Proposition 3.4.1. Let X be a topological space and p˜ : X˜ X a universal covering. Let x X. Then the fiber functor → ∈

Fx : CovX Set −→ Y X HomX x, Y ( → ) 7−→ ( ) Y Y0 HomX x, Y HomX x, Y0 ( → ) 7−→ ( ( ) → ( )) is represented by the universal covering, i.e., for any connected covering p : Y X there is a bĳection → 1 HomX X˜, Y p− x HomX x, Y . ( ) ( ) ( ) This characterizes a universal property of the universal covering of X. We can now translate this into algebraic terms. Theorem 3.4.2. Let A be a commutative ring and X : Spec A. Let x Spec Ω , Spec A be a geometric point, i.e., Ω is an algebraically closed ﬁeld. Then the ﬁber functor →

Fx : FEtX Set −→ Y X HomX x, Y ( → ) 7−→ ( ) Y Y0 HomX x, Y HomX x, Y0 ( → ) 7−→ ( ( ) → ( ) is prorepresentable, i.e., there exists an inverse system Xi i I, φij of ﬁnite étale Galois coverings of X such that (( ) ∈ ) lim HomX Xi, Y HomX x, Y . ( ) ( ) i I −−→∈

Although the algebraic analogue of the universal covering is the affine scheme X˜ : lim Xi, i the fiber functor in theorem 3.4.2 is not actually representable because X˜ is not generally←−− a finite étale cover of X. However, now that we have an analogue of the universal covering space X˜ T of a topological space XT , we can use the identity Gal X˜ T XT π1 XT (see example 2.2.10) to define the étale fundamental group of an affine scheme.( / ) ( ) 3.4. Étale fundamental groups 21

Deﬁnition 3.4.3. Let X, x and Xi i I, φij be as in theorem 3.4.2. Deﬁne X˜ : lim Xi. The étale (( ) ∈ ) i I fundamental group of X is ←−− ∈

et´ π X, x : Gal X˜ X : lim Gal Xi X . 1 ( ) ( / ) ( / ) i I ←−−∈ Example 3.4.4. Let F be a field. Then a finite étale Galois cover of Spec F corresponds to a finite Galois extension E F (see also example 3.3.3). Every finite Galois extension is contained in the separable closure Fsep/ of F, i.e., the maximal separable extension of F contained in an algebraic closure F. The extension Fsep F is Galois and we have / ! πet´ Spec F : lim Gal E F Gal lim E F Gal Fsep F 1 ( ) ( / ) / ( / ) E F finite Galois E F finite Galois / ←−− / −−→ [Ser79, p. 54, Corollary 1].

Let’s see what this means for finite fields. Let p be a prime. The finite extensions of ¿p are ¿pn with Gal ¿pn ¿p n. These extensions are naturally ordered by inclusion and can ( / ) / all be embedded in the separable closure ¿p. We find

et´ π Spec ¿p lim Gal ¿pn ¿p lim n ˆ . 1 ( ) ( / ) / ←−−n ←−−n

1 Recall that this is the analogue of π1 S . I ( ) Example 3.4.5. We can further generalize the argument in example 3.3.4 to any Dedekind domain A. Let F be the field of fractions of A and let A B be a finite étale Galois cover. Then B is the integral closure of some field K over A and→ as such, B is itself a Dedekind domain with Frac B K [Ser79, Chap. 1, Sect. 4]. Let p be a non-zero prime in A. Since B is a Dedekind ( ) Îr ei domain, we have a prime decomposition pB i1 Pi and by the Chinese remainder theorem we have r Ö ei B A κ p B P . ⊗ ( ) / i i1 From this we find that B is unramified. Therefore, we find that the finite étale Galois covers of A are exactly the unramified ring extensions A B. Let Bi i I be the collection of unramified → { | ∈ } ring extensions of A and let Ki : Frac Bi for every i I. Define F˜ : lim Ki. We have ( ) ∈ i −−→ et´ π Spec A lim Gal Bi A Gal F˜ F 1 ( ) ( / ) ( / ) ←−−i and we call F˜ the maximal unramified extension of F. I Example 3.4.6. We combine example 3.3.4 and example 3.4.5. Since there are no finite étale covers of , the maximal unramified extension of is ˜ and we find

πet´ Spec Gal 1. 1 ( ) ( / ) 3 Recall that this is the analogue of π1 1. I ( ) 22 3. Algebraic preliminaries CHAPTER 4

The linking number and the Legendre symbol

For a 2-component link K L S3, the linking number lk L, K measures how the two components are tangled up. Taking∪ the⊂ linking number mod 2, we( have) a function that assigns to a 2- component link the value 0 or 1. On this basic level, we can note that for primes p and q the p Legendre symbol 1, 1 does a similar thing. In this chapter we will see that the mod 2 ( q ) ∈ {− } linking number and the Legendre symbol are actually deﬁned analogously.

4.1 The linking number

Let K and L form a 2-component link and let p : X XL be the infinite cyclic cover of the ∞ ∞ → knot complement XL. Consider K as a loop in XL with endpoints x K. Then the monodromy ∈ permutation representation of GL induces a homomorphism ρ : GL Gal X XL . ∞ → ( ∞/ ) n Definition 4.1.1. Let τ be a generator of the Galois group Gal X XL . Then ρ K τ for some n . We define lk L, K : n and we call this the linking number( ∞/ of) K and L∞.([ We]) define the mod ∈ ( ) 2 linking number lk2 L, K ¿2 by lk L, K lk2 L, K mod 2. The sign of lk L, K depends on the ( ) ∈ ( ) ≡ ( ) ( ) choice of τ and the orientation of K as a loop in XL.

There are many ways to deﬁne the linking number [Rol03, Chap. 5, Sect. D]. The linking number is known to be symmetric, i.e., we have lk L, K lk K, L [Rol03, Chap. 5, Sect. D, Theorem 6]. For a particularly intuitive deﬁnition( of the) linking( ) number that showcases its symmetry, see [Ada05, p. 18]. We now turn to the following characterization of the mod 2 linking number.

Proposition 4.1.2. Let τ be a generator of the Galois group Gal X2 XL and let ρ2 : GL Gal X2 XL be the map induced by the monodromy permutation representation.( / We) have → ( / )

lk2 L,K ρ2 K τ ( ). ([ ])

Proof. Note that p2 : X2 XL is a subcover of p : X XL, so by theorem 2.2.9, we know that → ∞ ∞ → p π1 X p2 π1 X2 . Also note that ρ2 is the composite ( ∞)∗( ( ∞)) ⊆ ( )∗( ( )) q2 GL GL p2 π1 X2 ∼ Gal X2 XL , → /( )∗( ( )) → ( / ) 24 4. The linking number and the Legendre symbol

where the second map is the isomorphism from theorem 2.2.9 and where q2 is the natural quotient map. Likewise, we have q ∞ GL GL p π1 X ∼ Gal X XL . → /( ∞)∗( ( ∞)) → ( ∞/ ) Then the quotient map GL p π1 X GL p2 π1 X2 induces a commutative diagram /( ∞)∗( ( ∞)) → /( )∗( ( )) ρ2 GL Gal X2 XL ( / ) ρ ∞ r Gal X XL ( ∞/ ) , where r sends a generator τ Gal X XL to the generator τ Gal X2 XL . We now have ∈ ( ∞/ ) ∈ ( / ) lk L,K lk L,K lk2 L,K ρ2 K r ρ K r τ ( ) τ ( ) τ ( ). ([ ]) ( ∞([ ])) ( )

Example 4.1.3. We are going to compute the linking number of the Hopf link to prove that it is, in fact, linked. Let K L be the Hopf link in S3 comprised of two unknots K and L and let S be the Seifert surface of K∪. In other words, S is a disk with boundary K. We consider L as a loop in the knot complement XK with endpoints x. This is pictured in ﬁgure 4.1.

L K S

Fig. 4.1: Hopf link K L, with Seifert surface S of K. ∪

We construct the infinite cyclic cover of XK as we did in example 2.2.14. For every i , + ∈ let Yi be a copy of XK and cut it open along S. In every Yi we obtain two disks Si and Si− S i S+ S X Ð Y homeomorphic to . For every , we identify i with i−+1. Then : i i is the infinite ∞ ∈ cyclic cover of XK with covering map p : X XK. Its Galois group Gal X XK is generated ∞ ∞ → 1 ( ∞/Ð ) by the deck transformation τ : X X defined by τ Yi Yi+1. Let p− x i yi , where ∞ → ∞ ( ) ( ) ∈ { } S yi Yi. Figure 4.2 shows X with the lift L˜ of L in X with starting point y0. Here Yi Yi+1 ∈ S+ ∞ S ∞ ↔ indicates that i is identified with i−+1.

S S S Y 1 −1 Y0 0− Y1 1− Y2 − − L˜ S S S S

y0 y1 S+ S+ ˜ S+ 1 0 L 1 −

Fig. 4.2: The inﬁnite cyclic cover X of XK with the lift L˜ of L. ∞

Since the endpoint of L˜ is y1, we see that ρ L τ and we get lk K, L 1. Note how the linking number depends on our choice of τ and∞([ the]) orientation (clockwise( or) counterclockwise) on L. I 4.1. The linking number 25

Example 4.1.4. Although unlinked knots have linking number zero, the converse is not necessarily true. This example serves to illustrate that. Let K and L be two unknots that are linked together as in ﬁgure 4.3. This link is called the Whitehead link.

L K

Fig. 4.3: Whitehead link.

We construct the inﬁnite cyclic cover X of XK as in the previous example, using a Seifert 3 ∞ surface S of K to cut open copies of S . Let τ Gal X XK again be the element that sends ∈ ( ∞/ ) every Yi to Yi+1 and let L˜ once again be the lift of the loop L. The result is pictured in ﬁgure 4.4.

S S S Y 1 −1 Y0 0− Y1 1− Y2 − − L˜ S S L˜ S S

S+ S+ y S+ 1 0 0 1 −

Fig. 4.4: The inﬁnite cyclic cover X of XK with the lift L˜ of L. ∞

We see that the endpoint of K˜ is the same as its starting point y0, so ρ K idX and we ∞([ ]) ∞ ﬁnd lk K, L 0, even though the Whitehead link can’t be untangled. I ( ) The following corollary describes the connection between the mod 2 linking number and the decomposition of a knot in X2.

Corollary 4.1.5. Let K and L form a 2-component link with double cover p2 : X2 XL. If lk2 L, K 0, K K X p 1 K K K L K → ( K) X there exist knots 1, 2 2 such that 2− 1 2 and if lk2 , 1 there exists a knot 2 1 ⊆ ( ) ∪ ( ) ⊆ such that p− K K. In other words, we have: 2 ( ) ( K K L K 1 1 2 if lk2 , 0 p2− K ∪ ( ) ( ) K if lk2 L, K 1. ( )

1 Proof. Let x K and let p− x y1, y2 . For a loop γ in XL, let γ˜ y : 0, 1 X2 be the lift of γ ∈ 2 ( ) { } i [ ] → in X2 with starting point γ˜ y 0 yi. Since ρ2 : GL Gal X2 XL is induced by the monodromy i ( ) → ( / ) permutation representation, we have ρ2 γ yi : γ˜ y 1 . ([ ])( ) i ( ) ˜ ˜ ˜ ˜ Note that if Ky1 1 y1, then Ky1 0 Ky1 1 and Ky1 is actually a knot in X2. Then the ( ) 1 ( ) ( ) same holds for K˜ y and we have p− K K1 K2, where Ki : K˜ y and K decomposes into a 2 2 ( ) ∪ i 2-component link in X2. Conversely, if K decomposes into two knots K1 y1 and K2 y2 in X2, 3 3 then K1 K˜ y and K2 K˜ y by deﬁnition and we have K˜ y 1 y1. 1 2 1 ( ) ˜ ˜ ˜ ˜ If, however, we have Ky1 1 y2, then Ky2 1 y1 and Ky1 and Ky2 together actually form a X ( ) K ( ) p 1 K K p 1 K knot in 2, which we denote by . Then we have 2− . Conversely, suppose 2− is a ( ) 1 ( ) knot in X2. Note that by the previous paragraph, we have K˜ y 1 y1 p− K K1 K2 for 1 ( ) ⇒ 2 ( ) ∪ 26 4. The linking number and the Legendre symbol

1 some knots K1 and K2 in X2. Since p− K , K1 K2 for any two disjoint knots K1 and K2 in X2, 2 ( ) ∪ we ﬁnd K˜ y 1 , y1, i.e., K˜ y 1 K˜ y 1 y2. 1 ( ) 1 ( ) 1 ( ) Using proposition 4.1.2, we get the following equivalences:

lk2 L, K 0 ρ2 K idX ρ2 K y1 y1 K˜ y 1 y1 ( ) ⇐⇒ ([ ]) 2 ⇐⇒ ([ ])( ) ⇐⇒ 1 ( ) 1 p− K K1 K2, ⇐⇒ 2 ( ) ∪ lk2 L, K 1 ρ2 K τ ρ2 K y1 y2 K˜ y 1 y2 ( ) ⇐⇒ ([ ]) ⇐⇒ ([ ])( ) ⇐⇒ 1 ( ) 1 p− K K, ⇐⇒ 2 ( ) which is what we wanted to prove.

Example 4.1.6. We continue the example of the Hopf link and the Whitehead link. Let’s start with the Hopf link K L. According to corollary 4.1.5, since the mod 2 linking number of the ∪ 1 Hopf link is 1, we should have p− L L for some knot L X2. In ﬁgure 4.5 we can see ( ) ⊆ 3 the double cover X2 of XK consisting of two copies Y0 and Y1 of S cut open along S where we S+ S S+ S identify 0 with 1− and where we identify 1 with 0−.

Y S− Y S− 0 0 L 1 1 L S

y0 y1 S+ S+ 0 1

Fig. 4.5: The double cover X2 of XK with the knot L over L.

Now let K and L denote the components of the Whitehead link. Since lk2 K, L 0, we have 1 ( ) p− L L1 L2 for two knots L1 and L2 in X2. This is shown in ﬁgure 4.6. I ( ) ∪

S S Y0 0− Y1 1− L2 L1 S

S+ y L1 S+ y L2 0 0 1 1

Fig. 4.6: The double cover X2 of XK with the knots L1 and L2 over L.

4.2 The mod 2 linking number for primes

For knots we just saw how the linking number lk K, L gave us information about the decompo- ( ) sition of the knot K in the double cover X2 XL. We will now do the same thing for primes. In order to define the mod 2 linking number→ for primes, we need to define the double cover X2 of X q , then associate p to an element σp G q and finally we have to define a map { } ∈ { } ρ2 : G q Gal X2 X q . { } → ( / { }) 4.2. The mod 2 linking number for primes 27

1 Let p and q be odd primes and define X q : Spec q Spec q as an analogue of { } \{ } [ ] the knot complement. To find a double étale cover of X q we need an extension of that’s { } only ramified over q. Since q is ramified in the ring of integers k of a field extension k if and only if q divides the discriminant of k [Neu99, Chap. III, CorollaryO 2.12], we want a/ field extension of with discriminant a power of q. Such a field extension is given by √q , where ( ∗) q : 1 q 1 2q. We define q this way to ensure q 1 mod 4, since the discriminant of ∗ (− )( − )/ ∗ ∗ ≡ √q is equal to 4q if q 3 mod 4, which would mean that 2 also ramifies in √q . However, ( ) ≡ O ( ) the discriminant of √q∗ is q. Now, for extensions of the form √d with d squarefree ( ) + ( ) + and d 1 mod 4 we have 1 √d 2 , so we have q 1 √q 2 . Set ≡ O √d [( )/ ] O √ ∗ [( ∗)/ ] 1 + ( ) ( ) X2 : Spec q , 1 √q 2 . The double étale cover of X q is therefore the map [ ( )/ ] { } h2 : X2 X q −→ { } induced by the embedding 1 1 , 1 + √q 2 . [ q ] −→ [ q ( )/ ] In the knot case, we viewed a knot K as an element of π1 XL . We now want to perform ( ) the arithmetic analogue, i.e., we want p to induce an element σp G q . We have G q Gal urq , where urq is the maximal Galois extension of unramified∈ { } outside q.{ The} problem( / is that) we have next to no insight into the structure of urq . We are going to solve this problem by embedding urq into a field we know a little more of. Namely, we take an ur ur embedding q p of q into the algebraic closure of p, where , p is induced ⊆ ⊆ → ur by the inclusion , p. This induces a homomorphism φp : Gal p p Gal q . → ( / ) → ( / ) Define ˜ p : p ζn gcd n, p 1 , which is the maximal unramified extension of p. Since ( ur| ( ) ) p is unramified in q , the homomorphism φp factors through Gal ˜ p p [Mor12, Example ab ab ( / ) 2.40]. Note that q ˜ p, where q : ζq is the maximal Abelian extension of ⊆ ( ∞ ) unramified outside of q. Also note that since every g Gal ˜ p p can be extended to an ∈ ( / ) automorphism g Gal p p , the map Gal p p Gal ˜ p p is surjective. Therefore, the diagram ∈ ( / ) ( / ) → ( / ) φp f ur 1 ab Gal p p Gal q Gal q ( / ) ( / ) ( / ) f3 f2 f4

Gal ˜ p p ( / ) , p commutes. Let σ˜ Gal ˜ p p be the Frobenius element that is defined by ζn ζn for all ∈ ( / ) 7→ n coprime with p. Let σ Gal p p be the extension of σ˜ to p, i.e., we have f2 σ σ˜ . ∈ ( / ) urq ( ) A logical choice for the element σp Gal G q is the image φp σ . We call σp ∈ ( / ) { } ab ( ) the Frobenius automorphism over p. Since we have field extensions q ζqn n ( | ∈ ) ⊆ p ζn gcd n, p 1 ˜ p, we find that ( | ( ) ) f1 σp f1 φp σ f4 f2 σ f4 σ˜ ( ) ( ◦ )( ) ( ◦ )( ) ( ) abq p ab is the Frobenius automorphism on which sends ζqn to ζ n for all n. We set σ : f1 σp . q p ( ) Now that we have defined σp, it’s time to define our map ρ2 : G q Gal X2 X q . Note { } → ( / { }) that Gal X2 X q Gal √q∗ . It is known that √q∗ ζq [Wei06, Theorem 4.5.1], ( / { }) ( ( )/ ) (ur ) ⊆ ( ) so we get field extensions √q ζq ζq q which means that the natural map ( ∗) ⊆ ( ) ⊆ ( ∞ ) ⊆

ρ2 : G q Gal √q∗ { } −→ ( ( )/ ) 28 4. The linking number and the Legendre symbol

ab sends σp to the restriction of σ Gal ζq to √q . p ∈ ( ( ∞ )/ ) ( ∗) n Deﬁnition 4.2.1. Let τ be the generator of the Galois group Gal √q . Then ρ2 σp τ for some ( ( ∗) ( ) n 2. We deﬁne lk2 q, p : n and we call this the mod 2 linking number of p and q. ∈ / ( ) Let’s refresh our memory a bit regarding the Legendre symbol.

Deﬁnition 4.2.2. Let n be an odd prime and let k be an integer such that n and k are coprime. The Legendre symbol or quadratic residue symbol k is deﬁned ( n ) ( k 1 if k x2 mod n for some x : ≡ n 1 if k . x2 mod n for all x. − Proposition 4.2.3. The Legendre symbol has the following properties.

(1) (quadratic reciprocity) Let m and n be two distinct odd primes. Then

m 1 n 1 n m − − 1 2 · 2 . m n (− )

(2) Let n be an odd prime and let k and l be integers such that gcd k, n 1 gcd l, n . The Legendre symbol is multiplicative, i.e., we have ( ) ( )

k l kl . n n n

(3) Let n be an odd prime. Then we have ( 1 n 1 1 if n 1 mod 4 − 1 2− ≡ n (− ) 1 if n 3 mod 4. − ≡

From these properties it follows that we have

q p q p ∗ 1 and ∗ 1. p q∗ p q

We are now ready to prove the connection between the mod 2 linking number and the Legendre symbol.

Proposition 4.2.4. We have q ∗ lk2 q,p 1 ( ). p (− )

Proof. Deﬁne q 1 Õ− k S : ζk. q q q k1 4.2. The mod 2 linking number for primes 29

ab p We have Sq √q [Wei06, Remark 4.5.6]. Since σ ζq ζq, we have ∗ p ( ) q 1 − k ab ab Õ kp ρ2 σp √q∗ σp √q √q∗ σp Sq ζq ( )( ) | ( ∗)( ) ( ) q k1 q 1 q 1 q p Õ− k q Õ− kp q ∗ ζkp ∗ ζkp ∗ S p q q q p q q p p k1 k1 q ∗ √q , p ∗ from which we get q∗ 1 1 ρ2 σp τ lk2 q, p 1. p − ⇐⇒ ( ) ⇐⇒ ( ) q It follows that ∗ 1 lk2 q,p . ( p ) (− ) ( )

From proposition 4.2.4, we can deduce that lk2 q, p lk2 p, q . The following corollary describes the connection between the mod 2 linking( number) and( the) decomposition of a prime in X2.

Corollary 4.2.5. Let p and q be odd primes and deﬁne h2 : X2 X q as above. We have: → { } ( p p q p 1 1, 2 if lk2 , 0 h2− p { } ( ) ( ) p if lk2 q, p 1. ( )

2 Proof. We have an isomorphism of quotient rings p ¿p X X X q √q∗ √q∗ ∗ 2 O ( )/ O ( ) [ ]/( − − ) ¿p X X q∗ . Since p is not ramiﬁed in √q , we have either p √q p if p √q [ ]/( − ) O ( ∗) O ( ∗) O ( ∗) is prime or p p p if p splits, where p,p p are primes in . Using √q∗ 1 2 √q∗ 1 2 √q∗ proposition 4.2.4O ,( we) get the followingO ( equivalences:) O ( ) q∗ 2 lk2 q, p 0 1 q∗ x mod p for some x ( ) ⇐⇒ p ⇐⇒ ≡ 2 ¿p X X q∗ is not a domain √q p √q is not a domain ⇐⇒ [ ]/( − ) ⇐⇒ O ( ∗)/ O ( ∗) p is not prime √q∗ ⇐⇒ O ( ) q∗ 2 lk2 q, p 1 1 q∗ . x mod p for all x ( ) ⇐⇒ p − ⇐⇒ 2 ¿p X X q∗ is a domain √q p √q is a domain ⇐⇒ [ ]/( − ) ⇐⇒ O ( ∗)/ O ( ∗) p √q is prime, ⇐⇒ O ( ∗) from which the claim follows. 30 4. The linking number and the Legendre symbol CHAPTER 5

Decomposition of knots and primes

In the preliminaries we defined topological covers M S3 ramified over a knot and finite covers Spec B Spec A ramified over a prime. In this chapter→ we will examine the decomposition of knots and→ primes through coverings.

5.1 Decomposition of knots

Let p : M S3 be a finite Galois covering ramified over a link L S3, where M is a connected → 3 ⊆1 oriented closed 3-manifold. Defining X : S L and Y : M p− L gives us the unramified \ 3 \ ( ) 3 finite cover p Y : Y X with Galois group G : Gal M S of maps g : M S that restrict to | → ( / 3 ) → deck transformations g Y : Y X. Set n : #G. Let K S be a knot that is either a component | → 1 ⊆ of L or disjoint from L. Suppose p− K is an r-component link (if K is disjoint from L, this ( ) 1 holds automatically as a consequence of theorem 2.2.4), so take p K K1 ... Kr and let − ( ) t t SK : K1, ... , Kr be the set of knots lying over K. We have the following lemma. { } 1 Lemma 5.1.1. The elements of G, when restricted to p K , define a transitive G-action on SK. − ( )

Proof. Denote with VK a tubular neighborhood of K and let VKi be the connected component of 1 1 1 p VK containing Ki. Let x ∂VK and let p x y1, ... , yn . Let ρ : π1 X Aut p x − ( ) ∈ − ( ) { } ( ) → ( − ( )) be the monodromy permutation representation. We have π1 X p π1 Y G, so since π1 X 1 ( )/ ∗1( ( )) ( ) acts transitively on p− x , we know that G acts transitively on p− x and therefore on SK as well. ( ) ( )

Deﬁnition 5.1.2. Let G, SK, p : Y X and K1, ... , Kr be as above. →

1) The decomposition group of Ki, denoted with DKi , is the stabilizer of Ki SK under the action of G: ∈ DK : g G g Ki Ki . i { ∈ | ( ) }

2) The inertia group of Ki, denoted with IKi , is the subgroup of DKi consisting of all elements that are the identity on Ki: IK : g DK g K idK . i { ∈ i | | i i } 32 5. Decomposition of knots and primes

3) Let Z X be the subcovering of Y X such that Gal Y Z DK . Then the decomposition → → ( / ) i (covering) space of Ki, denoted with ZKi , is the Fox completion (cf. example 2.2.16) of Z.

4) Let Z X be the subcovering of Y X such that Gal Y Z IK . Then the inertia (covering) → → ( / ) i space of Ki, denoted with TKi , is the Fox completion (cf. example 2.2.16) of Z.

Since G acts transitively on SK, the orbit of any Ki is equal to SK. Therefore, by the orbit- stabilizer theorem we have # G DK #SK, so #DK n r is independent of i. ( / i ) i / 1 Let VK be a tubular neighborhood of K and let VKi be the connected component of p− VKi 1 ( ) that contains Ki. Let x ∂VK and let p x y1, ... , yn , with yi ∂VK for every i. Every ∈ − ( ) { } ∈ i element of g induces a homeomorphism g ∂VK : ∂VKi ∂Vg Ki , so if g DKi we have a deck | i → ( ) ∈ transformation g ∂V of ∂VK over ∂VK. This gives a map | Ki i

φ : DK Gal ∂VK ∂VK , g g ∂V . i −→ ( i / ) 7−→ | Ki Proposition 5.1.3. The map φ is a group isomorphism.

Proof. To show that φ is injective, let g DK such that g ∂V idV . Then g x x. Since ∈ i | Ki Ki ( ) p : Y X is a Galois covering, the action of G on p 1 x is regular. Therefore, g can only be the → − ( ) identity, so the kernel of φ is trivial. To show that φ is surjective, let g Gal ∂VKi ∂VK and ∈ ( / ) suppose g x y. Let g0 G such that g0 x y. Then g0 ∂VKi ∂VKi , so g0 DKi . Since ∂V ( ) ∈ ∂V (∂V) p 1 x ( ) ∈g g Ki is connected, the action of Gal Ki K on − is free. This means that 0 ∂VKi , so φ is surjective. ( / ) ( ) |

If we restrict a map g DKi to Ki, we get a deck transformation of Ki over K. This gives us a group homomorphism ∈ DK Gal Ki K i −→ ( / ) with kernel IKi . We have the following result.

Lemma 5.1.4. The homomorphism DK Gal Ki K , g g K is surjective, i.e., the sequence i → ( / ) 7→ | i 1 IK DK Gal Ki K 1 −→ i −→ i −→ ( / ) −→ is exact.

1 Proof. Let ρ : GL π1 X, x Aut p− x be the monodromy permutation representation. Let ( ) → ( ( )) α be a meridian of K and β a longitude of K. Set e : #IKi . For g G with g Ki Kj, we have 1 ∈ ( ) IKj gIKi g− , so e is independent of i. Note that e agrees with the ramification index e0 of Ki e0 1 over K: by definition of a ramified covering, ρ α is the identity on p− x . Let zi1 , ... , zie be 1 ( ) ( ) { 0 } the orbit of zi p x under ρ α . Since every g G that fixes the orbit of zi is the identity 1 ∈ − ( ) h ( )i ∈ 1 on Ki, we have IK ρ α , so e e . i h ( )i 0 Let yi ∂VKi . Note that every element in Gal ∂VKi ∂VK DKi is induced by an element ∈ ( / ) of π1 ∂VK and that we have π1 ∂VK α , β . Since ρ α yi ∂VKi and ρ β yi ∂VKi , ( ) ( ) h[ ] [ ]i ( )( )f ∈ ( )( ) ∈ we find DKi ρ α, β . Let f be minimal such that ρ β y1 y1. Since ρ α is the (h i) ∈ ( )( ) ( ) identity on Ki, we find that f is the covering degree of Ki over K. We have DKi ρ α, β j k (h i) ρ α β 0 j e 1, 0 k f 1 , which gives us #DKi ef and also #Gal Ki K f. Then we{ ( have) | ≤ ≤ − ≤ ≤ − } ( / ) #im DK Gal Ki K # DK IK ef e f #Gal Ki K , ( i → ( / )) ( i / i ) / ( / ) so the map DK Gal Ki K is surjective. i → ( / ) 5.1. Decomposition of knots 33

From n r #DK ef we get efr n. Keeping in mind that DK Gal M ZK and / 1 1 ( / i ) IK Gal M TK , we ﬁnd i ( / i )

DK 1 ZK M e f 1, r n, i ⇐⇒ i ⇐⇒ 3 DK G ZK S ef n, r 1, i ⇐⇒ i ⇐⇒ IK 1 TK M e 1, fr n, i ⇐⇒ i ⇐⇒ 3 IK G TK S e n, f r 1. i ⇐⇒ i ⇐⇒

Let Ki,T be the image of Ki under M TKi and let Ki,Z be the image of Ki,T under TKi ZKi . We have the following theorem. → →

Theorem 5.1.5. The map M TK is a ramiﬁed covering of degree e such that the ramiﬁcation index → i of Ki over Ki,T is e. The map TKi ZKi is a cyclic covering of degree f such that the covering degree of → 3 Ki,T over Ki,Z is f. The map ZK S is a covering of degree r such that K is completely decomposed i → into an r-component link containing Ki,Z as a component.

Suppose the Galois group G of the covering M S3 is Abelian. Then, since the decom- → position groups DKi are conjugate, they are independent of Ki. Therefore, we can write DK. Likewise, we can write IK for the inertia group and we write ZK and TK for the decomposition and inertia spaces, as well. Theorem 5.1.5 then shows how K is decomposed, covered and then 3 ramiﬁed through the coverings M TK, TK ZK and ZK S : → → →

1 M K1, ... , Kr e e ramiﬁed

IK TK K1,T , ... , Kr,T f f covered

DK ZK K1,Z, ... , Kr,Z r r decomposed G S3 K .

p Z S3 K p 1 K Ãr K In the covering 1 : p , is completely decomposed, i.e., we have 1− i1 i,Z. In → ( ) 1 the covering p2 : TK ZK, every Ki,Z is covered with degree f, i.e., we have p− Ki,Z Ki,T , → 2 ( ) p 1 x K K x K p M T but # 2− #Gal i,T i,Z 2 for every i,Z. Lastly, in the covering 3 : K we 1( ) ( / ) ∈ → have p− Ki,T Ki with ramiﬁcation index e for all i. 3 ( ) Let’s illustrate all this with some examples.

Example 5.1.6. Let’s take a look at the Hopf link. We call the components of the Hopf link K and L and we let S be the disk with boundary K. In other words, S is the Seifert surface of K (see also example 4.1.3). Let m . By cutting S3 along the surface S, as in example 2.2.14, we can construct the ∈ 3 3 m-fold cover M of S ramiﬁed over K. Let Y1, ... , Ym be m copies of S . Note that in each Yi we + obtain surfaces Si and Si−, both homeomorphic to S, by cutting Yi along S. We get the m-fold 3 m + cyclic cover M of S ramiﬁed over K by taking the union Yi and identifying S with S− for ti1 i i+1 34 5. Decomposition of knots and primes

S S Y1 1− Y2 2− Ym Sm− L1 L1 L1 K1 S K1 S S K1 ...

S+ S+ S+ 1 2 m

Fig. 5.1: The covering space M of S3 ramiﬁed over K.

S + every i m. The space M is displayed in figure 5.1, where Yi Yi+1 indicates that Si is ∈ / S ↔ identified with i−+1. 3 3 Let p : M S be the covering map. We have G : Gal Y S σ : Yi Yi+1 m. → ( / ) h 7→ i / Since L is unramified, its ramification index is e L 1 the inertia group IL is trivial and the ( ) i inertia space is TK M. This is clear upon inspection, since σ is clearly only the identity on L when Yi Yi, i.e., when i 0. As for the decomposition group, since L decomposes into 7→ the single knot L1, we have r 1 and every element of G fixes L1. Therefore, we have DL G 3 and ZL S . By contrast, the component K is ramified with ramification index e K n, so 3 ( ) the inertia group is IK G and the inertia field is TK S . Furthermore, we have DK G and 3 ZK S . It is easy to see this simply by observing figure 5.1. I Example 5.1.7. In all constructions we’ve seen so far, we either had IKi 1 or IKi G. In this example we’re going to construct a covering such that 1 ( IKi ( DKi ( G. We’re going to add another unknot to the Hopf link. Let S, T and U be their Seifert surfaces, i.e., discs with boundary K, L and M, respectively. See figure 5.2.

K M L

S T U

Fig. 5.2: The 3-component link consisting of knots K, L and M, with Seifert surfaces S, T and U.

We’re now going to construct a covering space Y ramified over K, L and M by cutting S3 along all three of their Seifert surfaces S, T and U. As in the previous example, we let Y1 and Y S3 S S+ S Y 2 be copies of , cut open along the Seifert surface . We obtain surfaces 1 and 1− in 1 and S+ S Y S+ S S+ S surfaces 2 and 2− in 2. By identifying 1 with 2− and identifying 2 with 1−, we obtain a 3 double cover Y1 Y2 of S ramified over K. ( ∪ )/∼ We can now repeat this process by taking a copy Y3 Y4 of Y1 Y2 , where Y3 Y1 and ( ∪ )/∼ ( ∪ )/∼ + Y4 Y2. Now we can cut open Y1 Y2 and Y3 Y4 along T to obtain surfaces Ti , Ti− Yi i ( T∪+ )/∼ (T ∪ )/∼ i ⊆ for every 1 4. We identify i mod 4 with i−+2 mod 4 for 1 4. We now obtain a cover ≤ ≤ 3 ≤ ≤ Y1 Y2 Y3 Y4 of S ramified over K and L. ( ∪ ∪ ∪ )/∼0 We can now do this a third time by taking a copy Y5 Y6 Y7 Y8 0, where Yi Yi 4 for +( ∪ ∪ ∪ )/∼ − 5 i 8. Cutting along U will give us surfaces Ui , Ui− Yi for every 1 i 8. This time ≤ ≤ + ⊆ Ð≤ ≤ 3 we identify Ui with Ui−+ for 1 i 8 to obtain a cover Y : 1 i 8 Yi 00 of S mod 8 4 mod 8 ≤ ≤ ( ≤ ≤ )/∼ 5.1. Decomposition of knots 35

ramiﬁed over K, L and M. Figure 5.3 shows what Y1 looks like after it has been cut along S, T and U.

Y1 T K1 1− M1 S U− 1− 1 L1

L2 S+ U+ 1 1 T + 1

Fig. 5.3: The 3-sphere cut along S, T and U.

Lastly, p : Y S3 be the natural covering map. Figure 5.4 shows the space Y with the → S + identifications between the spaces Yi. Here Yi Yj indicates that Si is identified with Sj− and + ←→ Sj is identified with Si− (likewise for T and U). One can check that p : Y S3 is Galois by the way the Seifert surfaces are identified. It is → 3 clear that every element of G : Gal Y S permutes the spaces Yi. Furthermore, because Y is ( / ) connected, for every pair 1 i, j 8 there is a unique g G such that g Yi Yj. Therefore, we ≤ ≤ ∈ ( ) have #G 8 (in fact, G 2 3). ( / ) For 1 i 8, let gi G be such that gi Y1 Yi. We’ll be focussing on K1. Note that since S+ ≤ ≤ S ∈ S+ S ( ) p 1 K Y p 1 K Y K 1 is identified with 2− and 2 with 1−, we have − 1 − 2. Also note that 1 + + ( ) ∩ ( ) ∩ T T Y T T Y p 1 K K K passes through 1 and 1− in 1 and through 2 and 2− in 2. Hence, we have − 1, 2 1 1 ( ) { } with K1 p K Y1 Y2 Y3 Y4 and likewise K2 p K Y5 Y6 Y7 Y8 . Therefore, − ( ) ∩ ( ∪ ∪ ∪ ) − ( ) ∩ ( ∪ ∪ ∪ ) the elements of G that map K1 onto itself are g1, g2, g3 and g4, so DK g1, g2, g3, g4 . We have { } ZK Y1 Y5 M with the natural covering map Y ZK that sends Y1, Y2, Y3 and Y4 to Y1 ∪ ⊆ 0 → and that sends Y5, Y6, Y7 and Y8 to Y5. Note that g2 transposes Y1 and Y2 as well as Y3 and Y4. S+ S S+ S S+ S S+ S Since 1 is identified with 2− and 2 is identified with 1− (likewise for 3 , 4−, 4 and 3−), we can see that g2 is actually the identity on K1. Therefore, we have IK g1, g2 and therefore { } 1 ( IK1 ( DK1 ( G, as was our goal. The inertia space is TK Y1 Y3 Y5 Y7 with the covering ∪ ∪ ∪ map Y0 TK that sends Yi to Y i 2 for every i. The knot K is decomposed, covered and ramified as follows:→ d / e 36 5. Decomposition of knots and primes

Y T Y T 1 K1 1− M1 2 K1 2− M2 S L U− S L U− 1− 1 1 2− 2 2 S

S+ U+ S+ U+ 1 L2 1 2 L1 2 T + T + 1 2

T T U U Y T Y T 3 K1 3− M1 4 K1 4− M2 S L U− S L U− 3− 1 3 4− 2 4 S

S+ U+ S+ U+ 3 L2 3 4 L1 4 T + T + 3 4

Y T Y T 5 K2 5− M1 6 K2 6− M2 S L U S L U 5− 2 5− 6− 1 6− S

S+ U+ S+ U+ 5 L1 5 6 L2 6 T + T + 5 6 U U T T

Y T Y T 7 K2 7− M1 8 K2 8− M2 S L U S L U 7− 2 7− 8− 1 8− S

S+ U+ S+ U+ 7 L1 7 8 L2 8 T + T + 7 8

Fig. 5.4: The covering space Y of S3 ramiﬁed over K, L and M.

8 2 2 2 Ø 3 Y Yi TK Y1 Y3 Y5 Y7 ZK Y1 Y5 S , ⊇ ∪ ∪ ∪ ⊇ ∪ ⊇ i1 4 1 Ø K1 p− K Yi K1,T K1 Y1 Y3 K1,Z K1 Y1 K, ( ) ∩ ⊇ ∩ ( ∪ ) ⊇ ∩ ⊇ i1 8 1 Ø K2 p− K Yi K2,T K2 Y5 Y7 K2,Z K1 Y5 K. ( ) ∩ ⊇ ∩ ( ∪ ) ⊇ ∩ ⊇ i5 5.2. Decomposition of primes 37

2 Here indicates that the covering degree is 2. As per theorem 5.1.5, we see that the preimage ⊆ of K in ZK is K1,Z K2,Z, the preimage of Ki,Z in TK is Ki,T with Gal Ki,T Ki,Z 2 and the ∪ ( / ) / preimage of Ki,T in Y is Ki with ramiﬁcation index 2. I

5.2 Decomposition of primes

Let k be a finite Galois extension such that the extension k is ramified over a set of / ⊆ O primes S Spec . Then let h : Spec k Spec be the ramified covering induced by the ⊆ O → 1 inclusion , k. If we take X : Spec S and Y : Spec k h− S , we get a finite étale covering p : →Y O X. Let G : Gal Y X : \Gal k and set nO :\ #G.( Let) p Spec and let 1 → ( / ) ( / ) ∈ Sp : h p p1, ... ,pr be the set of primes lying over p. We have the following lemma. − ( ) { } 1 Lemma 5.2.1. The elements of G, when restricted to h p , define a transitive G-action on Sp. − ( )

Proof. Suppose there is i , j such that for every σ G we have σ pi , pj. Since pj is not ∈ ( ) contained in σ pi for any σ G, there exists x pj such that x < σ pi for any σ G (this is a result of the prime( ) avoidance lemma∈ [Eis04, Lemma∈ 3.3]). Using the( ﬁeld) norm of ∈x, deﬁned as Î Nk x : σ G σ x , we notice that idk x x, so Nk x pj. We also have σ x < pi for any / ( ) ∈ ( ) ( ) / ( ) ∈ ( ) σ G, so Nk x < pi. However, we have Nk , so ∈ / ( ) / ∈

Nk x pj p pi , / ( ) ∈ ∩ ∩ which gives a contradiction.

We now deﬁne the arithmetic analogues to deﬁnition 5.1.2.

Deﬁnition 5.2.2. Let G, Sp, h : Y X and p1, ... ,pr be as above. →

1) The decomposition group of pi, denoted with Dp , is the stabilizer of pi Sp under the action of G: i ∈

Dp : g G g pi pi . i { ∈ | ( ) }

2) Any map g Dp induces an isomorphism g : ¿p ¿p that is the identity on ¿p, where ∈ i i → i ¿p : k pi. This gives a map Dp Gal ¿p ¿p , g g. The inertia group of pi, denoted i O / i → ( i / ) 7→ with Ipi , is the kernel of this map:

Ip : g Dp g id¿ . i { ∈ i | pi }

3) Let Zp be the subfield of k such that Gal k Zp Dp . We call Zp the decomposition field of pi. i ( / i ) i i 4) Let Tp be the subfield of k such that Gal k Tp Ip . We call Tp the inertia field of pi. i ( / i ) i i

Since G acts transitively on Sp, the orbit of any pi is equal to Sp. Therefore, we have an equivalence G Dpi G pi Sp, so #Dpi n r is independent of i. Also, for g G with / · 1 / ∈ g pi pj, we have Ip gIp g , so #Ip is independent of i. We set e : #Ip . ( ) j i − i i Recall that Ipi is deﬁned as the kernel of the map Dpi Gal ¿pi ¿p that sends an element → ( / ) g Dpi to g, deﬁned by g x mod pi g x mod pi. Analogous to lemma 5.1.4 for the knot case, we∈ have the following result.( ) ( ) 38 5. Decomposition of knots and primes

Lemma 5.2.3. The homomorphism Dp Gal ¿p ¿p , g g is surjective, i.e., the sequence i → ( i / ) 7→

1 Ip Dp Gal ¿p ¿p 1 −→ i −→ i −→ ( i / ) −→ is exact.

Proof. The ring of integers k is a number ring, so all non-zero primes in k are maximal. So O O ¿p k pi is a ﬁnite ﬁeld. Therefore, Gal ¿p ¿p is generated by the Frobenius automorphism i O / ( i / ) σ. Therefore, all we have to prove is that there is some g Dp such that g σ. ∈ i

Since ¿pi is a finite field, it is a simple extension of the field ¿p. Let θ k be such that ∈ O ¿pi ¿p θ , where θ θ mod pi. We know that all pi are distinct maximal ideals, so they are ( ) ≡ Ñr pairwise coprime. Since i1 pi p , we can apply the Chinese remainder theorem to get an isomorphism ( )

k p k p1 ... k pr ¿p ... ¿p . O /( ) O / × × O / 1 × × r Now let α k be such that α θ mod pi and α 0 mod pj for j , i, i.e., α pj for all j , i. Now consider∈ O the characteristic≡ polynomial ≡ ∈ Ö f X : X g α ( ) g G( − ( )) ∈ of α. Note that G acts transitively on Sp. Then, for any element g G Dp , there exists ∈ \ i some j , i such that g pj pi. Since α pj, this gives g α pi for all g < Dp . If we take ( ) ∈ ( ) ∈ i f X f X mod pi, then we get ( ) ≡ ( )

#G Dp Ö ¿p f X X \ i h X , h X : X g α . i 3 ( ) · ( ) ( ) ( − ( )) g Dp ∈ i

#G Dp Let α α mod pi and note that we have α θ. This gives 0 f α f θ θ \ i h θ and ≡ ( ) ( ) · ( ) since θ , 0, we find h θ 0. ( ) fθ Î X g θ θ ¿ Let ¿ g Gal ¿p ¿p be the minimal polynomial of over p. Note that p ∈ ( i / )( − ( )) θ θ f X X , so f X ¿p X , so h X ¿p X . Then this gives f h X . Since f σ θ 0, ( ) ∈ [ ] ( ) ∈ [ ] ( ) ∈ [ ] ¿p | ( ) ¿p ( ( )) we have h θ 0. Therefore, there exists g Dp such that σ σ θ g α g θ . Since ( ) ∈ i ( ( )) ( ) ( ) ¿p ¿p θ , this means that for such g Dp and any x ¿p , we have σ x g x , so g σ i ( ) ∈ i ∈ i ( ) ( ) and the homomorphism Dp Gal ¿p ¿p is surjective. i → ( i / ) Remark 5.2.4. Recall that in the case of knots, we had IKi g DKi g Ki idKi . One might { ∈ | | } expect the equality Ipi g Dpi g pi idpi to hold in the case of primes, as well. However, take k i with prime p{ ∈1+i |over| 2 } and g G the map i i. Since 1+i i 1 i , ( ) ( ) ( ) ⊆ ∈ 7→ − ·( − ) we find g p p, so g Dp. We now find g < g Dp g p idp . However, the induced map ( ) ∈ { ∈ | | } g : ¿ 1+i ¿ 1+i is the identity on ¿ 1+i , so g Ip. ( ) → ( ) ( ) ∈ This is because the analogy between knots and primes does not refer to primes as ideals of rings, but rather as elements of a spectrum. In this case, pi denotes the embedding Spec ¿p , i → Spec k induced by the quotient map k ¿p . This is why the proper analogue of Gal Ki K O O → i ( / ) is Gal ¿pi ¿p . I ( / ) 5.2. Decomposition of primes 39

Set f to be the degree of ¿pi over ¿p. As in the case of knots, we can relate e, r, f and n. For every 1 i r, let gi G be such that pi gi p1 and let ei be the ramification index of pi over ≤ ≤ e1 e∈r ( ) p, i.e., p k p pr . Applying gi gives O 1 ··· e1 e2 er e1 er e1 er p gi p2 ... gi pr gi p pr gi p k p k p pr . i · ( ) · · ( ) ( 1 ··· ) ( O ) O 1 ··· Being the ring of integers of k, we know that k has unique prime decomposition, so e1 ei. O e By choosing different values of i, we get e1 ... er, so we have p k p1 pr 1 . Also, since O ( ··· ) every gi induces an isomorphism ¿p ¿p , we find that f ¿p : ¿p is independent of i. 1 → i [ i ] Since k has an integral basis ω1, ... , ωn k, we find k ω1 ... ωn as abelian O ∈ O O ⊕ ⊕ n groups. Therefore, k p ω1¿p ... ωn¿p as abelian groups and we find # k p p . By multiplicativity ofO the/( ) ideal norm,⊕ we⊕ have O /( )

e1 ei ei e f # k p k : p k : p p 1 . O /( 1 ) [O ] [O ] e1 Since all pi are coprime, the powers pi are also coprime, so we can apply the Chinese remainder theorem to obtain

n e1 e1 e1fr p # k p # k p ... k pr p O /( ) (O /( 1 ) × × O /( )) (see also [Mar77, Theorem 22]). This gives us n e1fr. By lemma 5.2.3 we know Gal ¿p ¿p ( i / ) Dp Ip , so f n r e, which gives us e n fr e1. We now conclude the following: i / i ( / )/ /( ) Dp 1 Zp k e f 1, r n, i ⇐⇒ i ⇐⇒ Dp G Zp ef n, r 1, i ⇐⇒ i ⇐⇒ Ip 1 Tp k e 1, fr n, i ⇐⇒ i ⇐⇒ Ip G Tp e n, f r 1. i ⇐⇒ i ⇐⇒ Let pi,T : pi Tp and let pi,Z pi Zp . We have the following theorem. ∩ O i ∩ O i Theorem 5.2.5. The extension k Tp is a ramiﬁed extension of degree e such that the ramiﬁcation index / i of pi over pi,T is e. The extension Tp Zp is a cyclic extension of degree f such that the covering degree i / i of pi,T over pi,Z is f and pi,Z is inert in Tp . The extension Zp is an extension of degree r such that p i i / is completely decomposed into r ideals containing pi,Z as a prime factor.

Suppose the Galois group G of the field extension k is Abelian. Then, since the decom- / position groups Dpi are conjugate, they are independent of pi lying over p. Therefore, we can write Dp. Likewise, we can write Ip for the inertia group and we write Zp and Tp for the decomposition and inertia fields, as well. Theorem 5.2.5 then shows how p is decomposed, covered and then ramified through the field extensions k Tp, Tp Zp and Zp : / / /

1 k p1, ... ,pr e e ramiﬁed

Ip Tp p1,T , ... ,pr,T f f covered

Dp Zp p1,Z, ... ,pr,Z r r decomposed G p . 40 5. Decomposition of knots and primes

Îr In the extension Zp , p is completely decomposed, i.e., we have p Z pi,Z. In the / O p i 1 extension Tp Zp, every pi,Z is covered in the ﬁnite étale sense. Recall the pullback diagram in / section 3.3. Here the tensor product of the fraction ﬁeld κ pi,Z Z pi,Z with T is given by ( ) O p / O p f Ö κ pi,Z Z Tp Zp pi,Z Z Tp Tp pi,Z Tp Zp pi,Z. ( ) ⊗O p O O / ⊗O p O O / O O / j1 Therefore, we have a commutative diagram

Ãf Spec Z pi,Z Spec T i 1 O p / O p η

Spec κ pi,Z Spec Z ( ) O p where η is induced by the embedding Z T and where Spec κ pi,Z Spec Z sends O p → O p ( ) → O p the only prime ideal of κ pi,Z to pi,Z. We can see that pi,Z is covered f times through η. Lastly, ( ) e in the extension k Tp we have pi,T k p for every i, so all pi,T ramify. / O i Let’s look at an example.

Example 5.2.6. Let k : ζn be a cyclotomic extension of . We have k ζn and ( ) O [ ] G : Gal k n . A prime number p is ramified in k if and only if p n. Let p be a ( / ) ( / )× | prime of and p a prime in k lying over p. If p - n, then e 1, so the inertia group Ip of a prime p over p is trivial and the inertia field is Tp k. Since Ip is trivial, we know that the decomposition group Dp is isomorphic to Gal ¿p ζn ¿p , which is generated by the Frobenius automorphism ( ( )/ ) p σ : ¿p ζn ¿p ζn , x x . ( ) −→ ( ) 7−→ Therefore, f : #Dp #Gal ¿p ζn ¿p # σ is the order of p in n and we have ( ( )/ ) h i ( / )× Dp σ , where σ is defined h i p σ : ζn ζn , x x . ( ) −→ ( ) 7−→ In particular, Dp is trivial if and only if p 1 mod n, in which case we have r n. In other ≡ words, p splits completely in ζn if and only if p 1 mod n. ( ) ≡ As an example, take p 7 and n 6. The primes over 7 are 7, ζ6 3 and 7, ζ6 5 (see also [Neu99, Chap. 1, Proposition 8.3] and [Lan94, Chap. 1,( Proposition) ( − 25]),) so r ( 2 −n,) since Gal ζ6 6 1 mod 6, 5 mod 6 . ( ( )/ ) ( / )× { } m m Now suppose p n and let m be such that p n but p - n. Set G : Gal ζ24 . The | m | ( ( )/ ) 1 m ramification index of p in ζn is e p 1 p − and our inertia field is Tpi ζn p . Note that we have ( ) ( − ) ( / )

m m 1 ζn : ζn pm ζpm : p × p 1 p − e, [ ( ) ( / )] [ ( ) ] ( / ) ( − ) as claimed in theorem 5.2.5. As an example, let p 3 and n 24. We have

2 2 2 2 3 3, ζ + ζ24 + 2 3, ζ + 2ζ24 + 2 ( ) ( 24 ) ( 24 ) in ζ24 , so r 2. We have m 1 and e 2, so the inertia ﬁeld of p is Tpi ζ8 . Note ( ) ζ24 2 3 2 ( ) that the minimal polynomial of ζ24 over ζ8 is f x + ζ x ζ , so ζ24 : ζ8 2. ζ8 8 8 ( ) ( ) − [ ( ) ( )] 5.2. Decomposition of primes 41

Since ζ24 : φ 24 8, where φ is the Euler totient function, we have f 8 er 2, so [ ( ) ] ( ) /( ) according to theorem 5.2.5 we have 1 ( Ip1 ( Dp1 ( G. Explicitly, we have

2 2 2 k ζ24 Tp ζ8 Zp i√2 , ( ) ⊇ ( ) ⊇ ( ) ⊇ 2 2 p1 3, ζ + ζ24 + 2 p1,T ζ + ζ 1 p1,Z 1 + i√2 p 3 , ( 24 ) ⊇ ( 8 − ) ⊇ ( ) ⊇ ( ) 2 7 6 p2 3, ζ + 2ζ24 + 2 p2,T ζ + ζ 1 p2,Z 1 i√2 p 3 , ( 24 ) ⊇ ( 8 − ) ⊇ ( − ) ⊇ ( ) 2 and we have p Z p1,Zp2,Z, pi,Z T pi,T and pi,T k p . O p O p O i For more details on the various results above regarding cyclotomic ﬁelds, see [Mar77, Chap. 2]. I 42 5. Decomposition of knots and primes CHAPTER 6

Homology groups and ideal class groups

We are now going to make the switch from fundamental groups to homology. In this section we’re going to take a quick look at a homologically motivated analogy between knots and primes.

6.1 Homology groups and the Hurewicz theorem

Recall the construction of the first homology group of a space X: we use part of the singular chain complex ∂3 ∂2 ∂1 ... C2 X C1 X C0 X −→ ( ) −→ ( ) −→ ( ) and define the group of 1-cycles Z1 X : ker ∂1 and the group of 1-boundaries B1 X : im ∂2. ( ) ( ) Then the first homology group of X is H1 X : Z1 X B1 X . The important thing to realize here is that knots can be seen as loops, and( ) therefore( )/ as 1-cycles,( ) i.e., for a knot γ : S1 S3 → with endpoints γ 0 γ 1 x we have ∂1 K x x 0, so γ ker ∂1 and we can speak of ( ) ( ) ( ) − ∈ γ H1 X . ∈ ( ) Let p : M S3 be a finite abelian covering covering ramified over a link L. Define X : S3 L → 1 3 \ and Y : M p− L , so G : Gal M S : Gal Y X . As a consequence of the Hurewicz theorem, we have\ the( ) following theorem.( / ) ( / ) Theorem 6.1.1. We have an isomorphism

H1 X p H1 Y G. ( )/ ∗( ( ))

Proof. We start by noting that we have π1 X p π1 Y G by construction. We can now apply the Hurewicz theorem, along with fact that( )/ for∗( a( group)) H with normal subgroup N we have Hab Nab H N ab. We get / ( / ) ab ab ab ab H1 X p H1 Y π1 X p π1 Y π1 X p π1 Y ( )/ ∗( ( )) ( ) / ∗( ( ) ) ( ) / ∗( ( )) ab ab π1 X p π1 Y G G, ( ( )/ ∗( ( ))) which is what we needed to prove. 44 6. Homology groups and ideal class groups

6.2 Ideal class groups and Artin reciprocity

Let’s consider the case of primes. Let k be a finite Abelian extension of . Just as knots can be considered 1-cycles, primes in k can be considered invertible ideals. O Definition 6.2.1. A fractional k-ideal I is an k-submodule of k such that xI k for some x k∗. A fractional ideal I k is calledO an invertible idealO if there exists a fractional⊆ ideal O J k such∈ that ⊆ ⊆ IJ k. A fractional ideal I k is called integral if I k. O ⊆ ⊆ O In particular, an integral ideal I is invertible if and only if IJ is a non-zero principal ideal for some integral ideal J k. It is a basic number theoretic result that all prime ideals in a ring ⊆ O of integers k are invertible, so the primes in k can indeed be considered invertible fractional O O ideals. For two fractional k-ideals I and J, we define the ideal quotient I : J as follows: O I : J : x k xJ I . { ∈ | ⊆ } The set I k of invertible k-ideals becomes a group under the operations of ideal multiplication and ideal( ) quotients. LetO P k be the subgroup of I k consisting of all principal fractional ( ) ( ) k-ideals. The analogue of the first homology group is now the ideal class group H k of k: O ( ) H k : I k P k . ( ) ( )/ ( ) Suppose k is ramified over a set of primes S p1, ... , pr of . Let f : Spec k Spec and 1 { } O → Sk : f S . Let X : Spec S and Y : Spec k Sk. Define G : Gal Y X : Gal k . − ( ) \ O \ ( / ) ( / ) It is known that for a number field K, any fractional ideal a I K has a unique prime ideal a Î pnp I É∈ (O ) I X decomposition p Spec K , i.e., we have K p K . As the group of ∈ O (O ) ∈O ( ) invertible fractional 1 p1, ... , 1 pr -ideals, then, we define [ / / ] Ê I X : , ( ) 0,p X ∈ and likewise, Ê I Y : . ( ) 0,p Y ∈ Let p X be a non-zero prime and let p Y be a prime lying over p, i.e., p p . Recall ∈ ∈ ∩ from section 5.2 that the decomposition group of p is defined Dp : g G g p p . As we { ∈ | ( ) } saw in section 5.2, for another prime q lying over p, the decomposition groups Dp and Dq are conjugate. Since k is an Abelian extension of , this means that Dp Dq, so the decomposition group is uniquely determined by p and we write Dp. Also, since p is unramified we know Dp Gal ¿p ¿p , so Dp is cyclic and we denote with σp a generator of Dp. For a fractional n Î( / ) np Î p ideal a 0,p X p we can generalize this by defining σa : 0,p X σp . We can now define the following map∈ on I X : ∈ ( ) φ : I X G, a σa. ( ) −→ 7−→ The kernel of φ can be given in terms of principal fractional ideals. Specifically, we define the following subgroups of I X and I Y : ( ) ( ) P X : a P a 1 mod q for all q S ( ) {( ) ∈ ( ) | ≡ ∈ } and

P Y : a P k a 1 mod q for all q Sk and σ a > 0 for all embeddings σ : k , ( ) {( ) ∈ ( ) | ≡ ∈ ( ) → } 6.2. Ideal class groups and Artin reciprocity 45

(note that the requirement that σ a > 0 for all embeddings σ : , puts no further restrictions on P , since every principal( ) fractional ideal in P has a positive→ generator). The ( ) ( ) map φ is surjective and the kernel is given by ker φ Nk I Y P X , where / ( ( )) ( )

Ö np Ö np Nk : I Y I X , p # k p / ( ) −→ ( ) 7−→ ( (O / )) 0,p Y 0,p Y ∈ ∈ is the ideal norm map [Mor12, Ch. 6, Sect. 2]. Deﬁne H X : I X P X and H Y : I Y P Y . We get the following result. ( ) ( )/ ( ) ( ) ( )/ ( ) Theorem 6.2.2 (Artin reciprocity law). The map

σk : H X Nk H Y G, a σa / ( )/ / ( ( )) −→ [ ] 7−→ induced by φ is an isomorphism.

This result can be generalized. For example, if K is a number field and L is the maximal unramified Abelian extension of K (known as the Hilbert class field of kn), the map

σL K : H K Gal L K , a σa, / ( ) −→ ( / ) 7−→ commonly referred to as the Artin map, is actually an isomorphism [Cox13, Theorem 5.23]. We have now established ideal class groups and Artin reciprocity as the arithmetic analogues of homology and the Hurewicz theorem, respectively. We will apply the analogy between homology and ideal class groups further in the next section. 46 6. Homology groups and ideal class groups CHAPTER 7

The Alexander and Iwasawa polynomials

In this chapter we’re going to take a look at the analogy between the Alexander polynomial for knots and the Iwasawa polynomial for primes. We’ll start by laying the groundwork to later compute the Alexander polynomial.

7.1 Diﬀerential modules

In previous chapters, we’ve looked at the fundamental group as a source of information about the structure of knots. The Alexander polynomial is a knot invariant that stems from the desire to understand the homology of knots. Specifically, we will be looking at the first homology group of the infinite cyclic cover H1 X and give a presentation of it as a module over the group ab ( ∞) ring GK . In the next section we will give the definition of the Alexander polynomial, but in order to[ effectively] calculate it, we will need some results regarding group rings and differential modules.

Definition 7.1.1. Let G be a group and R be a ring. The group ring or group algebra R G is the set Í [ ] of formal sums g G rgg with finite support, i.e., ∈ n Õ o R G : rgg rg 0 for almost all g G . [ ] g G | ∈ ∈ Í Í Í Í The ring operations are defined by g G rgg + g G sgg g G rg + sg g and g G rgg Í Í Í ∈ ∈ ∈ ( Í ) ( Í ∈ )· g G sgg g G h G rgshgh . The action of R on R G is defined by r g G rgg g G rrgg. ( ∈ ) ∈ ( ∈ ) [ ] ( ∈ ) ∈ ab ab Example 7.1.2. Let K be a knot and GK the knot group. Since GK , the group ring GK 1 [ ] is isomorphic to the group ring of Laurent polynomials t, t− . As we shall see soon, the 1[ ] Alexander polynomial is defined to be an element of t, t− . I [ ] In order to construct some very useful exact sequences later, we now define the augmentation map and its kernel, the augmentation ideal. 48 7. The Alexander and Iwasawa polynomials

Deﬁnition 7.1.3. Let G be a group and R be a ring. Then the R-algebra homomorphism R G : R G Í Í [ ] [ ] → R, g G rgg g G rg is called the augmentation map. The ideal IR G : ker R G of R G is called∈ the augmentation7→ ∈ ideal. [ ] [ ] [ ]

The augmentation ideal will be instrumental in connecting the homology of the inﬁnite cyclic cover with the Alexander module later on. The next proposition reveals more about the nature of this ideal.

Proposition 7.1.4. Let G be a group and R be a ring. The augmentation ideal IR G is generated by g 1 g G . [ ] { − | ∈ } Í Í Proof. The inclusion g 1 IR G is clear. Let g G rgg IR G . Then g G rg 0, so Í ( − ) ⊆ [ ] ∈ ∈ [ ] ∈ r1 rg. Then we have − g,1 Õ Õ Õ Õ Õ rgg rgg + r1 rgg rg rg g 1 , − ( − ) g G g,1 g,1 g,1 g,1 ∈ so the other inclusion holds, as well.

Any group homomorphism ψ : G H can be extended to a ring homomorphism ψ : Í Í → R G R H , g G rgg g G rgψ g . We continue with the definition of a ψ-differential. [ ] → [ ] ∈ 7→ ∈ ( ) Definition 7.1.5. Let ψ : G H be a group homomorphism and let M be an H-module. Then a map → d : G M is a ψ-differential if, for all g1, g2 G, it satisfies the relation → ∈ d g1g2 d g1 + ψ g1 d g2 . ( ) ( ) ( ) ( ) We put this definition into practice immediately with the ψ-differential module. É Definition 7.1.6. Let ψ : G H be a group homomorphism. Let g G H dg be the free H - → ∈ [ ] [ ] module on the generating set dg g G . Then we define the ψ-differential module Aψ as follows: { | ∈ } Ê Aψ : H dg d g1g2 dg1 ψ g1 dg2 g1, g2 G H . [ ] g G [ ] /h{ ( ) − − ( ) | ∈ }i ∈ Here d g1g2 dg1 ψ g1 dg2 g1, g2 G H is the H -module generated by the elements of h{ ( ) − − ( ) | ∈ }i [ ] [ ] the form d g1g2 dg1 ψ g1 dg2. ( ) − − ( ) By definition, we have d g1g2 dg1 + ψ g1 dg2 in Aψ, so the map G Aψ, g dg is by definition a ψ-differential. Also( useful) to note,( is) that we have d1 0. → 7→ This unwieldy construction occurs naturally as the result of a universal property concerning ψ-differentials.

Proposition 7.1.7. Let ψ : G H be a group homomorphism and let Aψ be the ψ-differential module → with canonical ψ-differential d : G Aψ. For any H -module A with ψ-differential ∂ : G A, → [ ] → there exists a unique φ : Aψ A such that φ d ∂, i.e., the following diagram commutes: → ◦ G ∂ d

Aψ A φ .

This property determines Aψ uniquely up to isormorphism. 7.1. Diﬀerential modules 49

Proof. Since the dg generate Aψ as a H -module, we only have to define φ for elements dg. We have φ dg ∂ g , so φ is uniquely[ determined.] Also, since ( ) ( ) φ d g1g2 ∂ g1g2 ∂ g1 + ψ g1 ∂ g2 φ dg1 + ψ g1 dg2 ( ( )) ( ) ( ) ( ) ( ) ( ( ) ) this map is well-defined. Suppose that for any H -module B with ψ-differential δ : G B, there exists a unique H -homomorphism ξ : A[ ] B such that ξ ∂ δ. Then the unique→ morphism ξ : A A [ ] → ◦ → such that ξ ∂ ∂ is idA. Since there also exists a ξ : A Aψ such that ξ ∂ d, we have ◦ 0 → 0 ◦ φ ξ ∂ φ d ∂, so φ ξ idA. Likewise, we find ξ φ idA , so the universal property ◦ 0 ◦ ◦ ◦ 0 0 ◦ ψ indeed determines Aψ uniquely up to isomorphism.

The Alexander polynomial is obtained by finding a presentation matrix of a differential module and looking at its minors. Therefore, our goal now is to find a free presentation of a differential module. For this we need a few auxiliary results.

Lemma 7.1.8. Let G be a group. We have an isomorphism AidG I G induced by dg g 1. [ ] 7→ − Proof. We’re going to use the universal property of the ψ-differential module. The map δ : G → I G , g g 1 is an idG-differential, since [ ] 7→ − δ g1g2 g1g2 1 g1 1 + g1 g2 1 δ g1 + idG g1 δ g2 ( ) − − ( − ) ( ) ( ) ( ) for all g1, g2 G. For a G -module A with ψ-differential ∂ : G A, a map φ : I G A with φ δ ∂ has∈ to satisfy φ[ g] 1 φ δ g ∂ g , which uniquely→ determines φ[ .] Therefore,→ ◦ ( − ) ( ◦ )( ) ( ) I G satisfies the universal property of ψ-diffential modules and we have AidG I G . [ ] [ ] Lemma 7.1.9. Let ψ : G H be a surjective group homomorphism and ψ˜ : G H the induced ring homomorphism. Let N→: ker ψ. We have an isomorphism [ ] → [ ]

G I N G H [ ]/ [ ] [ ] [ ] of G -modules, where we view H as a G -modules through ψ˜ . [ ] [ ] [ ]

Proof. We are ﬁrst going to show ker ψ˜ I N G . The augmentation ideal I N is the kernel [ ] [ ] Í [ ] of the augmentation map N : N . Therefore, for g N ngg I N , we have [ ] [ ] → ∈ ∈ [ ] Õ Õ Õ Õ ψ˜ ngg ngψ g ng 1 N ngg 1 0 1, [ ] g N g N( ( )) g N · g N · · ∈ ∈ ∈ ∈ Í so I N G ker ψ. Now let α g G rgg ker ψ. We have [ ] [ ] ⊆ ∈ ∈ Õ Õ Õ 0 ψ˜ α rgψ g rg h, ( ) ( ) g G h ψ G ψ g h ∈ ∈ ( ) ( ) Í so ψ g h rg 0 for all h ψ G . Let h ψ G and gh G such that ψ gh h. Since ( ) ∈ (1 ) ∈ ( ) ∈ ( ) N G ψ G , we know Ngh φ h . Then \ ( ) − ( ) Õ Õ Õ Õ Õ rgg rgg rggh rng ngh rng gh − h − h ψ g h ψ g h ψ g h n N n N ( ) ( ) ( ) ∈ ∈ Õ Õ rngh n rngh gh I N G . [ ] n N − n N ∈ [ ] ∈ ∈ 50 7. The Alexander and Iwasawa polynomials

Therefore, we have Õ Õ Õ α rgg rgg I N G , ∈ [ ] [ ] g G h ψ G ψ g h ∈ ∈ ( ) ( ) which gives us ker ψ˜ I N G as was claimed. Since ψ˜ is surjective, this gives us the isomorphism ⊆ [ ] [ ] G I N G H [ ]/ [ ] [ ] [ ] of G -modules. [ ] Proposition 7.1.10. Let ψ : G H be a surjective group homomorphism. Let Aψ be the ψ-diﬀerential module. Let N : ker ψ. Then there→ is an isomorphism

Aψ I G I N I G [ ]/ [ ] [ ] of H -modules deﬁned by dg g 1. [ ] 7→ −

Proof. We will first show Aψ H G AidG using the universal property for ψ-differential [ ] ⊗ [ ] modules. Note that we can view H as a G -module through ψ˜ : G H , the map [ ] [ ] [ ] → [ ] induced by ψ. By extension of scalars, H G AidG is then a H -module. Define the map [ ] ⊗ [ ] [ ]

δ : G H G AidG , g 1 dg. −→ [ ] ⊗ [ ] 7−→ ⊗ This map is a ψ-diﬀerential, since we have

δ g1g2 1 d g1g2 1 dg1 + 1 g1dg2 1 dg1 + ψ g1 dg2 δ g1 + ψ g1 δ g2 . ( ) ⊗ ( ) ⊗ ⊗ ⊗ ( ) ⊗ ( ) ( ) ( ) Now let A be a H -module with ψ-diﬀerential ∂ : G A. We want to deﬁne a H -module [ ] → [ ] homomorphism φ : H G AidG A such that φ δ ∂. This map is uniquely determined, since we have [ ]⊗ [ ] → ◦ φ 1 dg φ δ g ∂ g . ( ⊗ ) ( ◦ )( ) ( )

Therefore, H G AidG satisfies the universal property of ψ-differential modules. We have [ ] ⊗ [ ] Aψ H G AidG and this H -isomorphism is defined by dg 1 dg. Using lemma 7.1.8 and lemma[ ]⊗ 7.1.9[ ]we now find the[ isomorphism] 7→ ⊗

Aψ H G AidG G I N G G I G I G I N I G [ ] ⊗ [ ] ( [ ]/ [ ] [ ]) ⊗ [ ] [ ] [ ]/ [ ] [ ] of G -modules deﬁned by dg g 1. [ ] 7→ − 1 Let β H and α1, α2 ψ− β . Since ψ α1 α2 0 we have α1 α2 ker ψ I N I G . ∈ [ ] ∈ ( ) ( − ) − ∈ [ ] [ ] Therefore, we have an action of H on I G I N I G , where the action of β H is 1 [ ] [ ]/ [ ] [ ] ∈ [ ] multiplication by any α ψ− β . This turns I G I N I G into a H -module. Therefore, ∈ ( ) [ ]/ [ ] [ ] [ ] we have an isomorphism Aψ I G I N I G of H -modules. [ ]/ [ ] [ ] [ ] Before we introduce the main theorem that will allow us to compute the Alexander polynomial, we need to deﬁne the Fox free derivative.

Deﬁnition 7.1.11. Let F be the free group on generators x1, ... , xm with identity element e. The Fox free derivative with respect to x is a map ∂ : F F , where 1 i m, which is determined i ∂xi by the following properties: [ ] → [ ] ≤ ≤

∂ α + β ∂α ∂β (1) ( ) + for any α, β F , ∂xi ∂xi ∂xi ∈ [ ] 7.1. Diﬀerential modules 51

∂xj (2) δij, ∂xi ∂ uv ∂u ∂v (3) ( ) + u for any u, v F. ∂xi ∂xi ∂xi ∈

These properties uniquely determine ∂ [CF63, Chap. VII, 2.9]. Following this deﬁnition, ∂xi we can check that for inverses we have the following rule:

1 ∂ u− 1 ∂u ( ) u− for any u F. ∂xi − ∂xi ∈ The following computational result is going to be important in a later proof.

Lemma 7.1.12. Let F be the free group on the generators x1, ... , xr and consider the Fox free derivative ∂ ∂xj : F F . Let α F. We have / [ ] → [ ] ∈ r Õ ∂α xj 1 α 1. ∂xj ( − ) − j1

Proof. First note that we can recursively work out

n n 1 n 2 ∂x ∂x ∂x − ∂x ∂x ∂x − i i + i i + i + i + + + n 1 xi xi xi ... 1 xi ... xi − δij, ∂xj ∂xj ∂xj ∂xj ∂xj ∂xj ( )

n ∂xi n where δij is the Kronecker delta. Multiplying by xi 1 gives xi 1 x 1 δij. We use ( − ) ∂xj ( − ) ( i − ) this fact to get

r ∂ xn r Õ i Õ + + + n 1 n ( ) xj 1 1 xi ... xi − δij xj 1 xi 1. ∂xj ( − ) ( ) ( − ) − j1 j1

ei e Now let α x 1 x in F. This gives us i1 ··· in ∈ ei e r r ∂ x 1 x in Õ ∂α Õ i1 in xj 1 ( ··· ) xj 1 ∂xj ( − ) ∂xj ( − ) j1 j1 ei ei e r ∂ x 1 ∂ x 2 ∂ x in ! Õ i ei i ei ei i ( 1 ) + 1 ( 2 ) + + 1 n 1 ( n ) xi ... xi xi − xj 1 ∂xj 1 ∂xj 1 ··· n 1 ∂xj ( − ) j1 − e e r ∂ x i1 r ∂ x i2 Õ i ei Õ i ( 1 ) + 1 ( 2 ) + xj 1 xi xj 1 ... ∂xj ( − ) 1 ∂xj ( − ) j1 j1 r ∂ xein ei ei Õ i + 1 n 1 ( n ) xi xi − xj 1 1 ··· n 1 ∂xj ( − ) − j1 e e e e e i1 i1 i2 i1 in 1 ein x 1 + x x 1 + ... + x x − x 1 i1 i1 i2 i1 in 1 in ( − ) ( − ) ··· − ( − ) ei e x 1 x in 1 α 1, i1 ··· in − − which concludes the proof. 52 7. The Alexander and Iwasawa polynomials

Let G be a ﬁnitely presented group with presentation G x1, ... , xr R1, ... , Rs . Let F h | i be the free group on generators x1, ... , xr with the canonical quotient map π : F G. Let ψ : G H be a surjective group homomorphism. Consider the H -module homomorphism→ → [ ] s ! s r Õ ∂Ri d2 : H H , βi 1 i s βi ψ π , [ ] −→ [ ] ( ) ≤ ≤ 7−→ ( ◦ ) ∂xj i1 1 j r ≤ ≤ i.e., d2 is given by multiplication of β1, ... , βs with the matrix ( ) ∂R1 ∂R1 © ψ π ψ π ª ( ◦ ) ∂x1 ···( ◦ ) ∂xr ® . . ® . .. . ® . . . . ® ∂R ∂R ® ψ π s ψ π s ® ( ◦ ) ∂x ···( ◦ ) ∂xr « 1 ¬ We now give the main theorem of this section.

Theorem 7.1.13. We have an isomorphism of H -modules [ ]

Aψ coker d2.

r Proof. We are going to deﬁne the H -module homomorphisms η : H Aψ and ξ : É [ ] ˜ [ ] → g G H dg coker d2 that induce maps η˜ : coker d2 Aψ and ξ : Aψ coker d2. Our goal∈ is to[ get] the→ following commutative diagram: → →

d H s 2 H r [ ] [ ] η

η˜ Aψ coker d2 ξ˜

ξ É g G H dg ∈ [ ] . É Deﬁne the H -module homomorphism ξ : g G H dg coker d2 by [ ] ∈ [ ] → ! ∂f ξ dg : ψ π , ( ) ( ◦ ) ∂xj 1 j r ≤ ≤ 1 1 where f π− g . To show that this is independent of the choice of f, let f π− g and k ker π. We have∈ ( ) ∈ ( ) ∈ ! ! ! ∂ fk ∂f ∂k ξ fk ψ π ( ) ψ π + ψ π f , ( ) ( ◦ ) ∂xj ( ◦ ) ∂xj ( ◦ ) ∂xj 1 j r 1 j r 1 j r ≤ ≤ ≤ ≤ ≤ ≤ 7.1. Diﬀerential modules 53

so it suﬃces to show ψ π f∂k ∂xj 1 j r im d2. Note that G is the quotient group of F by (( ◦ )( / )) ≤ ≤ ∈ 1 the normal closure of R1, ... , Rs , i.e., the normal subgroup generated by elements fR˜ if˜− F, where f˜ F and 1 i { s. Therefore,} k can be written as ∈ ∈ ≤ ≤ 1 1 k fi Ri f− fi Ri f− . ( 1 1 i1 )···( n n in ) For f˜ and 1 i s we have ≤ ≤ 1 1 ∂ fR˜ if˜− ∂f˜ ∂Ri ∂f˜− π ( ) π + f˜ + fR˜ i ∂xj ∂xj ∂xj ∂xj ˜ ˜ ∂f ∂Ri 1 ∂f π + f˜ fR˜ if˜− ∂xj ∂xj − ∂xj ∂Ri fπ˜ , xj

1 1 since ∂f ∂xj f ∂f ∂xj and π Ri 1. We now ﬁnd − / − / ( ) 1 1 ∂k ∂ fi1 Ri1 fi− fin Rin fi− π π (( 1 )···( n )) ∂xj ∂xj 1 1 ∂ fi Ri f− ∂ fi Ri f− ( 1 1 i1 ) + 1 ( 2 2 i2 ) + π fi1 Ri1 fi− ... ∂xj 1 ∂xj 1 ∂ fi Ri f− 1 1 n n in + fi Ri f− fi Ri f− ( ) 1 1 i1 n 1 n 1 in 1 ( )···( − − − ) ∂xj ∂ f R f 1 1 i1 i1 i− ∂ fin Rin fi− π ( 1 ) + ... + π ( n ) ∂xj ∂xj ∂R ∂R i1 + + in fi1 π ... fin π , ∂xj ∂xj which we can rewrite as s ∂k ∂R1 ∂Rn Õ ∂Ri π α1π + ... + αnπ α1π ∂xj ∂xj ∂xj ∂xj i1 for some α1, ... , αn G. Denote βi ψ αi and h ψ π f . We now ﬁnd ∈ ( ) ( ◦ )( ) ! s ! ∂k Õ ∂Ri ψ π f hβi ψ π ξ hβi 1 i s im d2. ( ◦ ) ∂xj ( ◦ ) ∂xj (( ) ≤ ≤ ) ∈ 1 j r i1 1 j r ≤ ≤ ≤ ≤ We conclude that the image of ξ is independent of choice of f.

To show that ξ induces a map ξ˜ : Aψ coker d2, we have to show that for all g1, g2 G → ∈ we have ξ d g1g2 dg1 ψ g1 dg2 0. Let g1, g2 G and f1, f2 F such that π f1 g1 and ( ( ) − − ( ) ) ∈ ∈ ( ) π f2 g2. Now we clearly have ( ) ! ∂ f1f2 ∂f1 ∂f2 ξ d g1g2 dg1 ψ g1 dg2 ψ π ( ) ψ π ψ g1 ψ π 0, ( ( ) − − ( ) ) ( ◦ ) ∂xj − ( ◦ ) ∂xj − ( )( ◦ ) ∂xj 1 j r ≤ ≤ 54 7. The Alexander and Iwasawa polynomials since ∂ f1f2 ∂f1 ∂f2 ∂f1 ∂f2 ψ π ( ) ψ π + f1 ψ π + ψ g1 ψ π , ( ◦ ) ∂xj ( ◦ ) ∂xj ∂xj ( ◦ ) ∂xj ( )( ◦ ) ∂xj so ξ indeed induces a map ξ˜ : Aψ coker d2. → Now we define r r Õ η : H Aψ, βj 1 j r βjdπ xj [ ] −→ ( ) ≤ ≤ 7−→ ( ) j1 and prove that it induces η˜ : coker d2 Aψ. To this end, we have to show η d2 0. Let s → ◦ αi 1 i s H . Let µ : I G I N I G Aψ be the isomorphism defined by g 1 dg (as in) ≤ proposition≤ ∈ [ ] 7.1.10. Using[ lemma]/ [ ] 7.1.12[ ] → we get − 7→ r s Õ Õ ∂Ri η d2 αi 1 i s αi ψ π dπ xj ( ◦ )(( ) ≤ ≤ ) ( ◦ ) ∂xj ( ) j1 i1 s r Õ Õ ∂Ri αi ψ π µ π xj 1 ( ◦ ) ∂xj ( ( ) − ) i1 j1 s r ! Õ Õ ∂Ri αi µ π π xj 1 ∂xj ( ( ) − ) i1 j1 s r !! Õ Õ ∂Ri αiµ π xj 1 ∂xj ( − ) i1 j1 s Õ αiµ π Ri 1 ( ( − )) i1 s Õ αiµ 0 ( ) i1 0, so η d2 0, so η induces a map η˜ : coker d2 Aψ. To conclude the proof we need to show ◦ → η˜ ξ˜ idA and ξ˜ η˜ id d . It suffices to show this for all generators dg Aψ and ◦ ψ ◦ coker 2 ∈ δkj 1 j r coker d2, where g G and 1 k r. Let g G and 1 k r, with f F such that π( f )≤g≤. Then∈ we have ∈ ≤ ≤ ∈ ≤ ≤ ∈ ( ) 7.2. The Alexander polynomial 55

! ! ∂f η˜ ξ˜ dg η˜ ψ π ( ◦ )( ) ( ◦ ) ∂xj 1 j r ≤ ≤ r Õ ∂f ψ π dπ xj ( ◦ ) ∂xj ( ) j1 r Õ ∂f ψ π µ π xj 1 ( ◦ ) ∂xj ( ( ) − ) j1 r !! Õ ∂f µ π xj 1 ∂xj ( − ) j1 µ π f 1 ( ( − )) µ g 1 ( − ) dg and r ! Õ ξ˜ η˜ δkj 1 j r ξ˜ δkjdπ xj ( ◦ )(( ) ≤ ≤ ) ( ) j1

ξ˜ dπ xk ( ( )) ! ∂x ψ π k ( ◦ ) ∂xj 1 j r ≤ ≤ δkj 1 j r, ( ) ≤ ≤ which concludes the proof.

Theorem 7.1.13 has the following immediate consequence.

Corollary 7.1.14. The ψ-diﬀerential module Aψ has a free presentation over H : [ ] Q s ψ r H H Aψ 0. [ ] −→ [ ] −→ −→ Its presentation matrix Qψ is given by ∂R1 ∂R1 © ψ π ψ π ª ! ( ◦ ) ∂x1 ···( ◦ ) ∂xr ® ∂Ri . . . ® Qψ ψ π . . . ® . ∂x . . . ( ◦ ) j 1 i s ® 1≤j≤r ∂Rs ∂Rs ® ≤ ≤ ψ π ψ π ® ( ◦ ) ∂x ···( ◦ ) ∂xr « 1 ¬

7.2 The Alexander polynomial

3 We’ve already seen in lemma 2.2.12 that the first homology group H1 S K of a knot complement is always isomorphic to . However, for an infinite cyclic cover the first( \ homology) group is more 56 7. The Alexander and Iwasawa polynomials

3 interesting. Let K be a knot in S and let XK be the knot exterior. Let p : X XK be an inﬁnite ∞ → cyclic cover of the knot complement XK, so we have π1 X p π1 X GK, GK (recall that p is injective by theorem 2.2.9). In order to make the( jump∞) from∗( the( ∞ fundamental)) [ group] to homology,∗ we will now introduce the Crowell exact sequence.

Theorem 7.2.1. Let ψ 1 N G H 1 −→ −→ −→ −→ be a short exact sequence of groups. Then we have an exact sequence of H -modules [ ] θ1 θ2 H ab [ ] 0 N Aψ H 0, −→ −→ −→ [ ] −→ −→ where θ1 is the homomorphism induced by n dn and θ2 is the homomorphism induced by dg ψ g 1. This sequence is called the Crowell exact7→ sequence. 7→ ( ) −

Proof. The action of H on Aψ and H is clear. The action of H on is deﬁned by Í [ Í] [Í ] [ ] h H nhh m : H h H nhh m h H nhm, where nh, m . As for the action of ( ∈ ab) [ ]( ∈ ) ∈ 1 ab∈ H on N , it is induced by the action h n : ghng− of H on N , where gh G such that [ ] ( ) h ∈ ψ gh h. This action is independent of the choice of gh: suppose g, g0 G with ψ g ψ g0 . Then( )ψ g g 1 1, so g g 1 N, so also gn g 1 gng 1 g g 1 ∈N. Then we( ) have ( ) ( ( 0)− ) ( 0)− ∈ ( 0)− ( − )( ( 0)− ) ∈ 1 1 1 1 1 1 1 1 1 ab gng− g0n− g0 − gn g0 − g0g− gn g0 − − g0g− − 1 N , ( ) ( ( ) )( )( ( ) ) ( ) ∈ so gng 1 g n g 1. Therefore Nab is well-deﬁned as a H -module. − 0 ( 0)− [ ] Let m, n N. Since ∈ 1 1 1 dm− d m− m ψ m− dm d1 dm dm, ( ) − ( ) − − we have

1 1 1 1 1 1 1 1 1 d m− n− mn dm− + ψ m− dn− + ψ m− n− dm + ψ m− n− m dn ( ) ( ) ( ) ( ) dm dn + dm + dn 0, − − so θ1 is well-deﬁned. Now let g1, g2 G. We have ∈

d g1g2 dg1 ψ g1 dg2 ψ g1g2 1 ψ g1 1 ψ g1 ψ g2 1 0, ( ) − − ( ) 7−→ ( ( ) − ) − ( ( ) − ) − ( )( ( ) − ) so θ2 is well-defined. ab To show that θ1 is injective, we are going to construct a left inverse η : Aψ N such that → η θ1 idNab . Let r : H G be a map such that ψ r idH with r 1 1. Note that r is not necessarily◦ a homomorphism.→ We define a group homomorphism◦ τ( :)É H dg Nab g G [ ] → that will induce the group homomorphism η. Since É H dg is free as∈ a group on the set g G [ ] hdg g G, h H of generators, we will define τ on the∈ elements hdg. Define { | ∈ ∈ } 1 τ hdg : r h gr hψ g − . ( ) ( ) ( ( )) Note that this is well-defined, since

1 1 ψ r h gr hψ g − hψ g hψ g − 1. ( ( ) ( ( )) ) ( )( ( )) 7.2. The Alexander polynomial 57

Since we have Now let g1, g2 G. We have ∈

τ hd g1g2 hdg1 hψ g1 dg2 ( ( ) − − ( ) ) 1 1 1 1 1 r h g1g2r hψ g1g2 − r h g1r hψ g1 − − r hψ g1 g2r hψ g1 ψ g2 − − 0, ( ) ( ( )) ·( ( ) ( ( )) ) ·( ( ( )) ( ( ) ( )) ) ab so τ indeed induces a map η : Aψ N . Since we have → 1 η θ1 n η 1dn r 1 nr 1ψ n − n, ( ◦ )( ) ( ) ( ) ( ( )) we ﬁnd that η θ1 id ab , so θ1 is injective. ◦ N The sequence is exact at Aψ by the series of equivalences

dg ker θ2 ψ g 1 0 ψ g 1 dg im θ1. ∈ ⇔ ( ) − ⇔ ( ) ⇔ ∈

Exactness at H can be seen by noting that ker H : I H is generated as a H -module [ ] [ ] [ ] [ ] by elements g 1, and since g 1 θ2 dg im θ2 for all g G, we have ker H im θ2. − − ( ) ∈ ∈ [ ] Lastly, since H is obviously surjective, we can conclude that the sequence is exact. [ ] The Crowell exact sequence helps us bridge the gap between the knot module and the Alexander module. 3 ab Deﬁnition 7.2.2. Let K be a knot in S . Let GK be the knot group and GK its abelianization, with the ab canonical projection map ψ : GK GK . We call AK : Aψ the Alexander module. By the Crowell → ab exact sequence, we can consider H1 X as a GK -module. We then call H1 X the knot module. ( ∞) [ ] ( ∞) ab Since H1 X GK, GK by the Hurewicz theorem, we can apply the Crowell exact sequence to the( ∞ short) [ exact sequence]

ψ ab 1 GK, GK GK G 1. −→ [ ] −→ −→ K −→ This gives us ab 0 H1 X AK GK 0. −→ ( ∞) −→ −→ [ ] −→ −→ This sequence actually occurs naturally when we look at the long exact sequence of homology 1 groups for the space pair X , p− x0 , where x0 XK. This sequence ends as follows ( ∞ ( )) ∈ 1 1 ... H1 p− x0 H1 X H1 X , p− x0 −→ ( ( )) −→ ( ∞) −→ ( ∞ ( )) 1 1 H0 p− x0 H0 X H0 X , p− x0 0. −→ ( ( )) −→ ( ∞) −→ ( ∞ ( )) −→ 1 1 1 Since p− x0 is discrete, H1 p− x0 0. Furthermore, since X is path connected and p− x0 ( ) ( ( 1 )) ∞ ( ) is non-empty, we have H0 X , p− x0 0 [Rot88, Theorem 5.12], so the sequence reduces to ( ∞ ( )) 1 1 0 H1 X H1 X , p− x0 H0 p− x0 H0 X 0. −→ ( ∞) −→ ( ∞ ( )) −→ ( ( )) −→ ( ∞) −→ It is a theorem that there exists a diagram

ab ab 0 GK, GK AK G 0 [ ] [ K ] ∼ ∼ ∼ ∼

1 1 0 H1 X H1 X , p− x0 H0 p− x0 H0 X 0 ( ∞) ( ∞ ( )) ( ( )) ( ∞) 58 7. The Alexander and Iwasawa polynomials where every square commutes [Cro71, p. 233]. In other words, when we apply the Crowell exact sequence to the knot group, we get the long exact sequence of homology groups. ab ab Note that G t and let α be a meridian of K such that ψ : GK G sends α to K h i → K t. From now on, we will denote with Λ : t, t 1 Gab the ring of Laurent polynomials. [ − ] [ K ] Since IΛ t 1 Λ as Λ-modules through t 1 1, we can write the Crowell exact sequence in the form( − ) − 7→ 0 H1 X AK Λ 0. −→ ( ∞) −→ −→ −→ Since Λ is a free Λ-module, this sequence splits [Eis04, Appendix 3, Proposition A3.1], so we have AK H1 X Λ as Λ-modules. We can now reformulate theorem 7.1.14. ( ∞) ⊕ Theorem 7.2.3. Let GK x1, ... , xr R1, ... , Rs be a ﬁnite presentation of the knot group. Deﬁne h | iab F : x1, ... , xr with π : F GK and ψ : GK G the canonical quotient maps. Then the Alexander h i → → K module AK has a free resolution over Λ:

s Q r Λ Λ AK 0. −→ −→ −→ The presentation matrix Q is given by

! ∂Ri Q : ψ π ( ◦ ) ∂xj 1 i s 1≤j≤r ≤ ≤ and is called the Alexander matrix of K. It depends on the choice of presentation of GK.

We are now ready to define Fitting ideals and more Alexander terminology. Definition 7.2.4. Let R be a commutative ring and M a finitely generated R-module with presentation

Q Rs Rr M 0 −→ −→ −→ and s r presentation matrix Q. For d 0 we deﬁne the d-th Fitting ideal Fd M (or the d-th elementary× ideal) of M as follows: ≥ ( )

(1) If 0 < s d r, then Fd M is the ideal generated by the r d-minors of Q. − ≤ ( ) − (2) If s d > r, then Fd M 0. − ( ) (3) If s d 0, then Fd M R. − ≤ ( ) A special case of the Fitting ideal, is the Alexander ideal. Definition 7.2.5. Let K be a knot and let Q be the s r Alexander matrix of K with s r. The × ≤ Alexander ideal of AK is the 0-th Fitting ideal F0 AK Λ generated by the s-minors of Q. We define ( ) ⊆ the Alexander polynomial ∆K t Λ of K to be a generator of F AK . ( ) ∈ ( ) The Alexander ideal exists for any knot, since the Wirtinger presentation gives us an Alexan- der matrix with s n 1 n r for some n. It is known that the Fitting ideal is independent of the choice of presentation− ≤ [Lan02, Chap. XIX, Lemma 2.3]. Also, as we shall see soon, the Fitting ideal of AK is principal, so ∆K t is defined up to multiplication by an element of Λ tn n . ( ) × {± | ∈ } 7.2. The Alexander polynomial 59

Fig. 7.1: The trefoil knot

Example 7.2.6. We’re going to compute the Alexander polynomial of the trefoil knot K, displayed in ﬁgure 7.1. The knot group has a Wirtinger presentation

GK x, y xyx yxy . h | i ab Let F x, y , with π : F GK and ψ : GK G t the canonical quotient maps. We h i → → K h i denote with π and ψ also the induced maps GK and GK Λ, respectively. In the Wirtinger presentation above, x and y are meridians→ of[ K ]and we[ have] →ψ π x t ψ π y . Using ( ◦ )( ) ( ◦ )( ) ∂ xyx yxy ∂x ∂y ∂x ∂y ∂x ∂y ( − ) + x + xy y yx 1 y + xy ∂x ∂x ∂x ∂x − ∂x − ∂x − ∂x − (and, likewise, ∂ xyx yxy ∂y 1 + x yx), we find the following Alexander matrix: ( − )/ − − ! ∂ xyx yxy ∂ xyx yxy Q ψ π ( − ) ψ π ( − ) ( ◦ ) ∂x ( ◦ ) ∂y ψ π 1 y + xy ψ π 1 + x yx (( ◦ )( − )( ◦ )(− − )) 1 t + t2 1 + t t2 . ( − − − ) 2 2 2 We find F0 AK 1 t + t , 1 + t t 1 t + t , so the Alexander polynomial of K is ( ) 2 ( − − − ) ( − ) ∆K t 1 t + t . ( ) − Let’s choose a different presentation of GK:

2 3 GK u, v u v . h | i Since the Alexander polynomial is a knot invariant, we should end up with 1 t + t2 again, up − to multiplication by Λ×. Previously, while using the Wirtinger presentation, we knew that x and y corresponded to meridians of K. In that case, we knew ψ π x t ψ π y . Since we now use a different presentation, this is no longer necessarily( ◦ the)( case.) There( ◦ is no)( ) obvious way in general to find out what ψ π u and ψ π v are, but with the knowledge of the Wirtinger presentation, all we need to( do◦ is)( find) an( isomorphism◦ )( )

u, v u2 v3 x, y xyx yxy . h | i −→ h | i One such isomorphism is given by u xy and v yxy, which gives ψ π u t3 and ψ π v t2 a 7→b 7→K ( ◦ )( ) G . In general, for an , -torus knot 0 with knot group presentation K0 (g ◦g )(ga) gb gdg c( ) K ad bc 1, 2 1 2 , the element 2 1− describes the meridian of 0, where 1 [BZH13, Propositionh | 3.38(b)].i We then have −

ad bc a d bc bd bc d c b g1 g − g g− g g− g g− 1 ( 1 ) 1 2 1 ( 2 1 ) 60 7. The Alexander and Iwasawa polynomials

g gdg c a ψ π g tb ψ π g ta and likewise, 2 2 1− , so 1 and 2 , which agrees with our case (recall that the trefoil( knot) is a 2,( 3◦-torus)( ) knot). The( presentation◦ )( ) matrix for this presentation is therefore ( ) ! ∂ u2 v3 ∂ u2 v3 Q ψ π ( − ) ψ π ( − ) ( ◦ ) ∂u ( ◦ ) ∂v ψ π 1 + u ψ π 1 v v2 (( ◦ )( )( ◦ )(− − − )) 1 + t3 1 t2 t4 . ( − − − ) The 1-minors of Q are 1 + t3 and 1 t2 t4. Using the Euclidean algorithm, we find − − − gcd 1 + t2 + t4, 1 + t3 gcd 1 + t3, 1 t + t2 gcd 1 t + t2, 1 t + t2 1 t + t2, ( ) ( − ) ( − − ) − + 2 + 2 so F0 AK 1 t t and ∆K t 1 t t , as expected. I ( ) ( − ) ( ) − Example 7.2.7. We’re going to compute the Alexander polynomial of the figure-eight knot K, displayed in figure 7.2.

Fig. 7.2: The ﬁgure-eight knot

The knot group of K has the following Wirtinger presentation:

1 1 1 1 1 1 GK w, x, y, z wy− x− y xzy− z− yw− z− w 1 h | i 1 1 1 1 1 1 (i.e., R1 wy− x− y, R2 xzy− z− and R3 yw− z− w). For brevity’s sake, we deﬁne f : ψ π : F Gab . This gives us the following Alexander matrix: ◦ [ ] → [ K ] ∂R1 ∂R1 ∂R1 ∂R1 © ψ π ψ π ψ π ψ π ª ( ◦ ) ∂w ( ◦ ) ∂x ( ◦ ) ∂y ( ◦ ) ∂z ® ∂R ∂R ∂R ∂R ® 2 2 2 2 ® Q ψ π ψ π ψ π ψ π ® ( ◦ ) ∂w ( ◦ ) ∂x ( ◦ ) ∂y ( ◦ ) ∂z ® ∂R ∂R ∂R ∂R ® ψ π 3 ψ π 3 ψ π 3 ψ π 3 ® ( ◦ ) ∂w ( ◦ ) ∂x ( ◦ ) ∂y ( ◦ ) ∂z « ¬ f 1 f wy 1x 1 f wy 1 + wy 1x 1 f 0 ( ) (− − − ) (− − − − ) ( ) © f 0 f 1 f xzy 1 f x + xzy 1z 1 ª ( ) ( ) (− − ) ( − − )® f yw 1 + yw 1z 1 f 0 f 1 f yw 1z 1 « (− − − − ) ( ) ( ) (− − − ) ¬ 1 t 1 1 + t 1 0 − − − − © 0 1 t 1 + tª . − − ® 1 + t 1 0 1 t 1 «− − − − ¬ 2 1 2 1 The 3-minors of Q are all t− 3t− + 1 , so F0 AK t− 3t− + 1 and the Alexander ±(2 − 1 + ) ( ) ( − ) polynomial of K is ∆K t t− 3t− 1. I ( ) − 7.3. The Alexander polynomial as a characteristic polynomial 61

7.3 The Alexander polynomial as a characteristic polynomial

In the last section, we used the structure of AK to compute the Alexander polynomial ∆K t . ( ) Using the Λ-module identity AK H1 X Λ, we shall now see how the Alexander polynomial ( ∞)⊕ is actually determined by H1 X and how it is related to a -endomorphism on H1 X . ( ∞) ( ∞) ⊗ 3 Let K be a knot in S . Let GK x1, ... , xm R1 ... Rm 1 1 be a presentation of h | −ab i the knot group of K. Let π : x1, ... , xm GK and ψ : GK GK be the canonical quotient h 1 i → → maps. We identify Λ : t, t− Gal X XK , where X is the infinite cyclic cover of the [ ] [ ( ∞/ )] ∞ knot complement XK. Take τ to be a generator of the Galois group of X over XK such that τ ∞ corresponds to t, i.e., Gal X XK τ . Let ( ∞/ ) h i Q m 1 K m Λ − Λ AK 0 −→ −→ −→ be a free presentation of AK with m 1 m presentation matrix QK. By applying elementary − × column operations if necessary, we can assume QK Q1 0 , where Q1 is a m 1 m 1 matrix. ( | ) − × − Since AK H1 X Λ as Λ-modules, we get a free presentation ( ∞) ⊕ Q m 1 1 m 1 Λ − Λ − H1 X 0 −→ −→ ( ∞) −→ of H1 X with presentation matrix Q1. Since QK Q1 0 , the m 1-minors of QK are det Q1 ( ∞) ( | ) − ( ) and 0, so the Alexander ideal (the 0-th Fitting ideal) is F0 H1 X det Q1 with Alexander ( ( ∞)) ( ( )) polynomial ∆K t det Q1 . Define Λ : Λ and note that H1 X is finitely generated ( ) ( ) 1 ⊗ ( ∞)⊗ as a Λ-module. Since Λ t, t is a principal ideal domain, the structure theorem for [ − ] finitely generated modules over a principal ideal domain tells us that there are fi Λ, with 1 i s for some s, such that ∈ ≤ ≤ s Ê H1 X Λ fi . ( ∞) ⊗ /( ) i1 És Note that τ acts on Λ (and thus also on Λ fi ) through multiplication by t. i 1 /( ) Theorem 7.3.1. Let ∆K t Λ Λ be the Alexander polynomial of K. We have ( ) ∈ ⊂ ∆K t f1 fs det t id τ H1 X mod Λ× , ( ) ··· ( · − | ( ∞) ⊗ ) where τ : H1 X H1 X is a -linear map. ( ∞) ⊗ → ( ∞) ⊗ Proof. Tensoring with we find a free presentation

s s Q1 s Ê Λ Λ Λ fi 0 −→ −→ /( ) −→ i1 for H1 X , with Fitting ideal F0 H1 X det Q1 . Since the Fitting ideal is ( ∞) ⊗ ( ( ∞) ⊗ ) ( É( s )) independent of the choice of presentation, the ﬁtting ideal of i1 Λ fi is f1 fs [Lan02, Q s s /( ) ( ··· ) f Chap. XIX, Corollary 2.9]. This can be seen if we let 01 be the matrix with the elements i along the diagonal. We get a presentation ×

s Q s 01 s Ê Λ Λ Λ fi 0 −→ −→ /( ) −→ i1 62 7. The Alexander and Iwasawa polynomials

and F0 H1 X det Q0 f1 fs. Considering the Alexander polynomial in the ring ( ( ∞) ⊗ ) ( 1) ··· Λ, we now ﬁnd

∆K t det Q0 f1 fs. ( ) ( 1) ···

Now for the second equality we have to prove. By tensoring H1 X with , we have acquired És ( ∞) a vector space over . Since τ acts on i1 Λ fi through multiplication by t, we know that τ s /( ) É Λ f i τ τ Λ f Λ f is a linear map on i1 i . For every , deﬁne i : Λ fi : i i . Choose /( ) | /( ) /( ) → /( ) ai ai + + + ni 1 + ni λt Λ× such that gi : fi λt is of the form ai0 ai1t ... ai,ni 1t − t , where ∈ · − ai0 , 0 and ni : deg gi . We then have Λ fi Λ gi and Λ gi is a ni-dimensional ( ) ni 1 /( ) /( ) /( ) vector space with basis 1, t, ... , t − . The linear map τi that sends an element p Λ gi to pt has the following matrix( representation:) ∈ /( )

0 0 1 ai,ni 1 « ··· − − ¬

The characteristic polynomial of τi is

t 0 0 0 ······ © 1 t 0 ai0 ª − ······ ® .. ® 0 1 . 0 ai1 ® ni+1 det t id τi det − ® 1 tgi fi mod Λ× . ( · − ) ...... ® (− ) ≡ ...... ® ® 0 0 1 t ai,ni 2 ® · · · − + − 0 0 0 1 t ai,ni 1 « ··· − − ¬ És Since τ is a linear endomorphism on i1 Λ fi , its matrix representation is the following block matrix: /( ) τ1 0 0 © ··· ª 0 τ2 0 ® τ . . ··· . ® . . . .. . ® . . . . ® 0 0 τs « ··· ¬ És Therefore, the characteristic polynomial of τ as a -linear map on i1 Λ fi H1 X is /( ) ( ∞) ⊗

t id τ1 0 0 © · − ··· ª 0 t id τ2 0 ® det t id τ det . · .− ··· . ® ( · − ) . . .. . ® . . . . ® 0 0 t id τs s « ··· · − ¬ Ö det t id τi f1 fs mod Λ× , ( · − ) ≡ ··· i1 which concludes the proof. 7.4. Complete diﬀerential modules 63

7.4 Complete diﬀerential modules

In order to define the analogue of the Alexander polynomial, we need analogues of differential modules that we can apply to étale fundamental groups. These groups are profinite, so we need some profinite analogues. From here on, let R be a compact complete local ring with maximal ideal m, i.e., R Rˆ : lim R mi. We endow R with the Krull topology or the m-adic topology, i.e., i / the topology that has←−− as its basis the sets r + mn, where mn is the kernel of the quotient map R R mn. This implies that for every quotient ring R mi is discrete and has an open cover r→+ mi/ r R . Since R mi is compact, every R mi is in/ fact finite. Therefore, R is a profinite ring.{ | ∈ } / /

Let G be a profinite group and let Nj j J be the set of open normal subgroups of G. Since { | ∈ } the Nj are open and normal, the quotient groups G Nj are finite [RZ10, Theorem 2.1.3]. If we / i0,j0 i i i N N φ( ) R m 0 G N let 0 and j0 j, we can define the natural ring homomorphism i,j : j0 ≥ ⊆ ( ) / [ / ] → i i i0,j0 R m G N R m G N φ( ) j . Then j , i,j forms a projective system of finite rings. / [ / ] { / [ / ] ( ) } Definition 7.4.1. Let G be a profinite group and let Nj j J be the set of open normal subgroups i { | ∈ } of G. The projective limit lim R m G Nj is called the complete group ring or complete group i,j / [ / ] algebra of G over R and is←−− denoted by R G . [[ ]] In other words, R G is the completion of R G [RZ10, Lemma 5.3.5] and R G is a profinite ring. Since we have [[ ]] [ ] [[ ]] i i i R G : lim R m G Nj lim lim R m G Nj lim lim R m G Nj [[ ]] / [ / ] ( / [ / ]) / [ / ] ←−−i,j ←−−j ←−−i ←−−j ←−−i lim R G Nj , [ / ] ←−−j Í + we can write elements of R G in the form gj G Nj rgj gj Nj . [[ ]] ∈ / ( ) j J ∈ Example 7.4.2. The arithmetic analogue of the infinite cyclic covering is called the cyclotomic ab p-extension and has Galois group G isomorphic to p. Analogous to the group ring G for [ K ] knots, the complete group ring p G will be important in defining the Iwasawa polynomial [[ ]] later. I

Let G and H be proﬁnite groups with f : G H a continuous homomorphism. Let Nj j J → { | ∈ } be the set of open normal subgroups of G and Mk k K the set of open normal subgroups of 1 { | ∈ } H. Note that f Mk Nj j J for all k. Then we get an induced continuous homomorphism − ( ) ∈ { | ∈ } Õ + Õ + f : R G R H , rgj gj Nj rgk f gk Mk . [[ ]] −→ [[ ]] ( ) j J 7−→ ( ( ) ) gj G Nj gk G f 1 Mk ∈ / ∈ ∈ / − ( ) When H 1 we get the augmentation map.

Deﬁnition 7.4.3. Let G be a proﬁnite group. Let Nj j J be the set of open normal subgroups of G { | ∈ } and set N0 : G. Then the R-algebra homomorphism Õ + R G : R G R, rgj gj Nj rg0 [[ ]] [[ ]] −→ ( ) j J 7−→ gj G Nj ∈ / ∈ 64 7. The Alexander and Iwasawa polynomials

(here g0 G N0 1 is the unique element of the trivial group) is called the augmentation map. The ∈ / ideal IR G : ker R G of R G is called the augmentation ideal. [[ ]] [[ ]] [[ ]] The group ring R G is embedded in R G through [ ] [[ ]] Õ Õ R G R G , rgg rg g + Nj [ ] −→ [[ ]] g G 7−→ g G ( ) j J ∈ ∈ ∈ (in fact, R G is dense in R G [RZ10, Lemma 5.3.5(c)]) and this embedding is implicitly invoked [ ] [[ Í]] when we talk of an element g G rgg R G . ∈ ∈ [[ ]] The following proposition is a complete analogue of proposition 7.1.4. For a proof, see [RZ10, Lemma 6.3.2(c)].

Proposition 7.4.4. Let G be a proﬁnite group. The augmentation ideal IR G is generated by g 1 g G R G . [[ ]] { − | ∈ } ⊆ [[ ]] We are now ready to introduce the complete analogue of the ψ-diﬀerential module.

Definition 7.4.5. Let ψ : G H be a continuous homomorphism of profinite groups. Let l be a prime É → number. Let g G l H dg be the free l H -module on the generating set dg g G . Then we define the complete∈ ψ-differential[[ ]] module as[[ follows:]] { | ∈ } Ê Aψ : l H dg d g1g2 dg1 ψ g1 dg2 g1, g2 G l H . [[ ]] g G [[ ]] /h{ ( ) − − ( ) | ∈ }i ∈

Here d g1g2 dg1 ψ g1 dg2 g1, g2 G l H is the l H -module generated by the elements h{ ( ) − − ( ) | ∈ }i [[ ]] [[ ]] of the form d g1g2 dg1 ψ g1 dg2. ( ) − − ( )

As in definition 7.1.6, G Aψ, g dg is by definition a ψ-differential, and as in definition 7.1.6, this construction→ arises from7→ the following universal property of complete ψ- differentials.

Proposition 7.4.6. Let ψ : G H be a continuous group homomorphism of profinite groups and let → Aψ be the complete ψ-differential module with canonical ψ-differential d : G Aψ. For any l H - → [[ ]] module A with ψ-differential ∂ : G A, there exists a unique φ : Aψ A such that φ d ∂, i.e., the following diagram commutes: → → ◦ G ∂ d

Aψ A φ .

This property determines Aψ uniquely up to isomorphism.

We can apply this to prove the following analogue to lemma 7.1.8.

Lemma 7.4.7. Let G be a proﬁnite group. We have an isomorphism AidG Il G induced by dg g 1. [[ ]] 7→ − 7.4. Complete diﬀerential modules 65

Proof. Suppose G lim Gi and define δi : Gi Il Gi , g g 1. As in the proof of i → [ ] 7→ − lemma 7.1.8, every δi←−−is an idGi -differential. Taking the projective limit, we obtain an idG- differential δ : G Il G . For a l G -module A with ψ-differential ∂ : G A, a map → [[ ]] [[ ]] → φ : Il G A with φ δ ∂ has to satisfy φ g 1 φ δ g ∂ g , which uniquely [[ ]] → ◦ ( − ) ( ◦ )( ) ( ) determines φ. Therefore, Il G satisfies the universal property of complete ψ-differential [[ ]] modules and we have AidG Il G . [[ ]] Lemma 7.4.8. Let ψ : G H be a continuous homomorphism between profinite groups and ψ˜ : → l G l H the induced ring homomorphism. Let N : ker ψ. We have an isomorphism [[ ]] → [[ ]]

l G Il N l G l H [[ ]]/ [[ ]] [[ ]] [[ ]] of l G -modules, where we view l H as a l G -module through ψ˜ . [[ ]] [[ ]] [[ ]] ˜ Proof. As in the proof of lemma 7.1.9, we are ﬁrst going to show ker ψ Il N l G . Suppose [[ ]] [[ ]] G lim Gi and H lim Hj. For every i, j we have a homomorphism ψij : Gi Hj deﬁned i j → ←−− ←−− ψ by the composite Gi G H Hi. Set Nij : ker ψij and let ψ˜ ij : l Gi l Hj be the → → → ˜ [ ] → [ ] map induced by ψij. Lemma 7.1.9 tells us that ker ψij Il Nij l Gi . Taking the projective ˜ ˜ [ ] [ ] limit, we get ker ψ Il N l G . Since ψ is surjective, this gives us the isomorphism [[ ]] [[ ]]

l G Il N l G l H [[ ]]/ [[ ]] [[ ]] [[ ]] of l G -modules. [[ ]] The following proposition can be proven entirely analogously to proposition 7.1.10. Proposition 7.4.9. Let ψ : G H be a surjective continuous homomorphism between proﬁnite groups. → Let Aψ be the complete ψ-diﬀerential module. Let N : ker ψ. Then there is an isomorphism

Aψ Il G Il N Il G [[ ]]/ [[ ]] [[ ]] of l H -modules deﬁned by dg g 1. [[ ]] 7→ − We will now introduce the appropriate analogue to the Fox free derivative.

Deﬁnition 7.4.10. Let Fˆ l be the free pro-l group on the generators x1, ... , xr. The pro-l Fox derivative ( ) is a map ∂ ∂xi : l Fˆ l l Fˆ l , where 1 i m, which is determined by the following properties: / [[ ( )]] → [[ ( )]] ≤ ≤

∂ α + β ∂α ∂β (1) ( ) + for any α, β l Fˆ l , ∂xi ∂xi ∂xi ∈ [[ ( )]]

∂xj (2) δij, ∂xi ∂ uv ∂u ∂v (3) ( ) + u for any u, v Fˆ l . ∂xi ∂xi ∂xi ∈ ( )

Since l Fˆ l is dense in l Fˆ l and since the proﬁnite ring l Fˆ l is Hausdorﬀ, the fact that these properties[ ( )] uniquely determine[[ ( )]] the Fox free derivative ∂ :[[( F)]]ˆ l Fˆ l implies ∂xi l l ∂ [ ( )] → [ ( )] that they also uniquely determine the pro-l Fox free derivative : l Fˆ l l Fˆ l . ∂xi [[ ( )]] → [[ ( )]] 66 7. The Alexander and Iwasawa polynomials

Let G be a ﬁnitely presented pro-l group with presentation G x1, ... , xr R1 ... Rs . h | i Let Fˆ l be the free pro-l group on generators x1, ... , xr with the canonical quotient map π : Fˆ l ( ) G. Let ψ : G H be a surjective continuous homomorphism of proﬁnite groups. ( ) → → Consider the l H -module homomorphism [[ ]] s ! s r Õ ∂Ri d2 : l H l H , βi 1 i s βi ψ π . [[ ]] −→ [[ ]] ( ) ≤ ≤ 7−→ ( ◦ ) ∂xj i1 1 j r ≤ ≤ Analogous to theorem 7.1.13 and its corollary, we have the following theorems.

Theorem 7.4.11. We have an isomorphism of l H -modules [[ ]]

Aψ coker d2.

Corollary 7.4.12. The complete ψ-diﬀerential Aψ has a free presentation over l H : [[ ]] Q s ψ r l H l H Aψ 0. [[ ]] −→ [[ ]] −→ −→

Its presentation matrix Qψ is given by ∂R1 ∂Rs © ψ π ψ π ª ! ( ◦ ) ∂x1 ···( ◦ ) ∂x1 ® ∂Ri . . . ® Qψ ψ π . . . ® . ∂x . . . ( ◦ ) j 1 i s ® 1≤j≤r ∂R1 ∂Rs ® ≤ ≤ ψ π ψ π ® ( ◦ ) ∂xr ···( ◦ ) ∂xr « ¬

7.5 The Iwasawa polynomial

Now that we’ve laid the groundwork for the presentation of complete ψ-differential modules, let us start the construction of the Iwasawa polynomial. Let p be a prime number and define + : ζp : ζpn n . Then Gal p× ¿p 1 pp , so there is a quotient ∞ ( ∞ ) ( | ∈ ) ( ∞/ ) × ( ) k of such that Gal k p. We call k the cyclotomic p-extension of . Let kn be the ∞ ∞ ( ∞/ ) n ∞ cyclic subextension of k of degree p and let Hn be the Sylow p-subgroup of its ideal class ∞/ group, i.e., Hn : H kn p . Suppose the order of the class group of kn has prime factorization e1 er( )( ) #H kn p pr . Assume without loss of generality p p1. Since H kn is Abelian we know ( ) 1 ··· ( ) that H kn H kn p1 ... H kn pr [Rot10, Proposition 4.42]. Therefore, Hn is isomorphic ( ) ( )( ) × × ( )( ) to the maximal p-quotient H kn p : H kn H kn p2 ... H kn pr of H kn . Lastly, let ( ) ( )/( ( )( ) × × ( )( )) ( ) L0n be the maximal unramified Abelian extension of kn (the Hilbert class field of kn) and let Ln be the maximal unramified Abelian p-extension of kn. It’s a well-known fact the Artin map defined in section 6.2 induces an isomorphism

φ0 : H kn Gal L0 kn , a σa, n ( ) −→ ( n/ ) 7−→ where σa is deﬁned as in section 6.2[Cox13, Theorem 5.23]. On the Sylow p-subgroup of H kn ( ) this is an isomorphism on the maximal unramiﬁed Abelian p-extension of kn, i.e., for each n, we have Hn : H kn p H kn p Gal L kn p Gal Ln kn through ( )( ) ( ) ( 0n/ ) ( / )

φn : Hn Gal Ln kn , a σa, −→ ( / ) 7−→ 7.5. The Iwasawa polynomial 67

where H kn p and Gal L kn p denote the maximal p-quotients of H kn and Gal L kn , ( ) ( 0n/ ) ( ) ( 0n/ ) respectively. We want to define an arithmetic analogue H of H1 X . This has to be a profinite ∞ ( ∞) group and a natural choice is the inverse limit of the projective system Hn n 0, Nn m n m , ≥ / ≥ (( k ) q: k ( p ) ) where Nn m : Hn Hm is the norm map uniquely determined by q p[O n / O m / ] on prime / → 7→ Ð ideals q kn , where p : q km . Hence we define H : lim Hn. Define L : n 0 Ln. ∈ O ∩ O ∞ n ∞ ≥ The diagram ←−−

φn Hn Gal Ln kn ( / ) Nn m / φm Hm Gal Lm km ( / ) commutes, so we have an isomorphism

φ : H Gal L k , p0,p1,p2, ... lim σpn ∞ ∞ −→ ( ∞/ ∞) ( ) 7−→ −−→n

Since L0n is the maximal unramiﬁed Abelian extension of kn, we know that L0n is a Galois extension of [Was97, p. 399], so Ln is a Galois extension. We have #Gal Ln k Ln : / ( / ) [ Ln : kn kn : , so Ln is actually a p-extension. Therefore, L is a pro-p Galois extension.] [ We]·[ will now] state the/ complete analogue of theorem 7.2.1. For∞/ a proof see [Mor12, Theorem 9.17].

Theorem 7.5.1. Let l be a prime number and let

ψ 1 N G H 1 −→ −→ −→ −→ be a short exact sequence of proﬁnite groups. Then we have an exact sequence of l H -modules [[ ]] θ θ H ab 1 2 l[[ ]] 0 N l Aψ l H l 0, −→ ( ) −→ −→ [[ ]] −→ −→ ab ab where N l is the maximal pro-l quotient of the abelianization N of N. The homomorphism θ1 is ( ) induced by n dn and θ2 is the homomorphism induced by dg ψ g 1. This sequence is called the complete7→ Crowell exact sequence. 7→ ( ) −

We now introduce some more terminology.

Definition 7.5.2. Let k be a finite algebraic number field and let S be a finite set of prime ideals of k. et´ ab O Let G : GS k π Spec k S and let H : GS k be its abelianization with the canonical ( ) 1 ( O \ ) ( ( )) projection map ψ : G H. Define N : ker ψ. We call Aψ the complete ψ-Alexander module of S. → ab When H l for some prime number l, we call the l H -module N l the Iwasawa module. [[ ]] ( ) We now apply the complete Crowell exact sequence to the short exact sequence

ψ 1 N Gal L Gal k 1 −→ −→ ( ∞/ ) −→ ( ∞/ ) −→ to get ab 0 N p Aψ p Gal k p 0. −→ ( ) −→ −→ [[ ( ∞/ )]] −→ −→ 68 7. The Alexander and Iwasawa polynomials

Note that N Gal L k H is an Abelian pro-p group, so we have Nab p N p N ( ∞/ ∞) ∞ ( ) ( ) Gal L k H . From now on, we will denote Λˆ : p T and we call Λˆ the Iwasawa algebra. ( ∞/ ∞) ∞ ab [[ ]] By theorem 7.5.1, we have Aψ N p Λˆ as Λˆ -modules. ( ) ⊕ Much like the Alexander polonomial was an element of Λ : t, t 1 defined up to multi- [ − ] plication by Λ×, the Iwasawa polynomial will be defined as an element of Λˆ up to multiplication by Λˆ ×. The Iwasawa polynomial will be defined analogously to the characterization of the Alexander polynomial in section 7.3. For this, we will need a few algebraic lemmas concerning Weierstrass polynomials.

Deﬁnition 7.5.3. Let l be a prime number. A polynomial g T l T is called a Weierstrass polynomial if it is of the form ( ) ∈ [ ]

λ λ 1 g T T + c1T − + ... + cλ, ci 0 mod p. ( ) Lemma 7.5.4. We have the following facts about Weierstrass polynomials.

(1) Let g be a Weierstrass polynomial of degree λ 1. Then any element f Λˆ can be written uniquely in the form ≥ ∈ f qg + r, q Λˆ , r l T , deg r λ 1. ∈ ∈ [ ] ( ) ≤ − (2)( p-adic Weierstrass Preparation Theorem) Any f T Λˆ with f T , 0 can be written uniquely in the form ( ) ∈ ( ) f T pµg T u T , ( ) ( ) ( ) where µ 0, g T is a Weierstrass polynomial and u T Λˆ ×. The integers µ and λ deg g are called∈ the ≥µ-invariant( ) and the λ-invariant of f, respectively.( ) ∈ ( )

For a proof of lemma 7.5.4(1), see [Was97, Proposition 7.2]. For a proof of lemma 7.5.4(2), see [Was97, Theorem 7.3].

Definition 7.5.5. We call two Λˆ -modules N and N0 pseudo-isomorphic, notation N N0, if there is a Λˆ -homomorphism φ : N N such that ker φ and coker φ are finite. ∼ → 0 Lemma 7.5.6. Let N be a compact Λˆ module. − (1) (Nakayama’s lemma) The module N is a finitely generated Λˆ -module if and only if N p, T N is finite. /( ) (2) (Structure theorem for Iwasawa modules) Suppose N is a finitely generated Λˆ -module. Then we have

s t Ê Ê e N Λˆ r Λˆ pmi Λˆ f j , ∼ ⊕ /( ) ⊕ ( j ) i1 j1

where r is a non-negative integer, mi, ej and fj is an irreducible Weierstrass polynomial. The ∈ numbers r, mi, ej and the prime ideals fj are uniquely determined by N. ( ) For a proof of lemma 7.5.6(1), see [Was97, Lemma 13.16]. For a proof of lemma 7.5.6(2), see [NSW08, Theorem 5.3.8]. Now we can ﬁnally deﬁne the Iwasawa polynomial. 7.6. The Iwasawa polynomial as a characteristic polynomial 69

Deﬁnition 7.5.7. Let N be a ﬁnitely generated, torsion Λˆ -module. We have

s t Ê Ê N Λˆ pmi Λˆ fei . ∼ /( ) ⊕ /( i ) i1 i1

e Îs mi Ît i The ideal generated by f : i1 p i1 fi is called the characteristic ideal of N. The polynomial f is deﬁned up to multiplication by Λˆ × and is called the Iwasawa polynomial.

Before we can apply this to our Λˆ -module H , though, we need to know the following. For a proof, see [Mor12, proposition 11.10]. ∞

Proposition 7.5.8. The Λˆ -module H is finitely generated and torsion. ∞ It is known that if a Λˆ -module N has no non-trivial finite Λˆ -submodule, the characteristic ideal of N equals the 0-th fitting ideal F0 N [Mor12, Example 9.18]. Hence its Iwasawa polynomial ( ) then equals ∆0 N . In particular, the Iwasawa module for the short exact sequence 1 N Gal L Gal( ) k 1 is known to satisfy this condition since is totally real→ [Mor12→, Example( ∞/ 9.18].) → Furthermore,( ∞/ ) → since the extension L is a subextension of urp (where urp is the maximal Galois extension of unramified∞ outside/ p), by the Galois correspondence/ for finite étale coverings we know that Gal L is a quotient of the prime group G p . This brings the analogy with knots full circle. If we( could∞/ ) find a presentation of the group Gal{ } L , then we could apply corollary 7.4.12 to compute the Iwasawa polynomial explicitly. Alas,( ∞/ much) of the structure of G p and its quotients is still unknown. { }

7.6 The Iwasawa polynomial as a characteristic polynomial

Let k be a finite algebraic number field and let p be a prime. Let k be the cyclotomic p- ∞ extension of k and define Λˆ : p T . Let γ be a topological generator of the Galois group of [[ ]] k over k. We have γ Gal k k p. It is known that γ 1 + T induces an isomorphism ∞ h i ( ∞/ ) 7→ˆ ˆ p Gal k k p T [NSW08, Proposition 5.3.5]. Define Λp : Λ p p and note that [[ ( ∞/ )]] [[ ]] ⊗ Λˆ p T . By lemma 7.5.6 we have p [[ ]] s t Ê Ê ˆ mi ˆ ei H Λ p Λ fi . ∞ ∼ /( ) ⊕ /( ) i1 i1

e Îs mi Ît i ˆ Let f T i1 p i1 fi Λ be the Iwasawa polynomial of H . Analogous to theorem 7.3.1( ) , we now have the following∈ result. ∞

Theorem 7.6.1. Let f T Λˆ be the Iwasawa polynomial of H . We have ( ) ∈ ∞ ˆ f T det T id γ 1 H p p mod Λp ×. ( ) ( · − ( − ) | ∞ ⊗ ) ( )

mi Proof. Note that p Λˆ × , so tensoring with p gives us ∈ p

t Ê ˆ ei H p p Λp fi . ∞ ⊗ ∼ /( ) i1 70 7. The Alexander and Iwasawa polynomials

ˆ Since Λp is a principal ideal domain, the structure theorem for ﬁnitely generated modules over a principal ideal domain tells us that we have

r Ê ˆ H p p Λp gi , ∞ ⊗ /( ) i1

e0i where the gi are primary ideals. In a PID this means that gi g0i for some irreducible ( ) µi ( ) (( ) ) g0i. By lemma 7.5.4(2) we can write g0i p hi ui, where hi is a Weierstrass polynomial · · e e0 0i and u Λˆ × . Therefore, we have gi g0 i h for every 1 i r. Note that h0 is ∈ p ( ) (( i) ) ( i ) ≤ ≤ i irreducible as well. Therefore, we have

r Ê e ˆ 0i H p p Λ hi . ∞ ⊗ ∼ /( ) i1 Since the fi and ei are uniquely determined by H p p, we ﬁnd t r and for every 1 i r ∞ ⊗ ≤ ≤ we have hi fi and e0i ei, so

t Ê ˆ ei H p p Λp fi . ∞ ⊗ /( ) i1

Since we have an isomorphism p Gal k k p T induced by γ 1 + T, we know t [[ ( ∞/ )]] [[ ]] 7→ É ˆ ei ˆ ei that γ 1 acts on i1 Λp fi through multiplication by T. Since we have Λp fi − ei ei /( ) /( ) p T fi p T fi [NSW08, Corollary 5.3.3], the rest of the proof is analogous to the proof[[ ]]/( of theorem) 7.3.1[ ]/(. ) Bibliography

[Ada05] C.C. Adams. The Knot Book. An Elementary Introduction to the Mathematical Theory of Knots. Reprint of the 1994 original. Providence: American Mathematical Society, 2005. [BZH13] G. Burde, H. Zieschang, and M. Heusener. Knots. 3rd ed. Vol. 5. De Gruyter Studies in Mathematics. Berlin: De Gruyter, 2013. [CF63] R.H. Crowell and R.H. Fox. Introduction to Knot Theory. 1st ed. Vol. 57. Graduate Texts in Mathematics. New York: Springer, 1963. [Cox13] D.A. Cox. Primes of the form x2 +ny2. Fermat, class ﬁeld theory, and complex multiplication. 2nd ed. Pure and Applied Mathematics. Hoboken: Wiley, 2013. [Cro71] R.H. Crowell. “The Derived Module of a Homomorphism”. In: Advances in Math- ematics 6.2 (1971), pp. 210–238. url: http://www.sciencedirect.com/science/ article/pii/0001870871900168 (visited on 08/04/2017). [Die10] T. tom Dieck. Algebraic Topology. Vol. 7. Textbooks in Mathematics. Corrected 2nd printing. Zürich: European Mathematical Society, 2010. [Eis04] D. Eisenbud. Commutative Algebra. with a View Toward Algebraic Geometry. 1st ed. Vol. 150. Graduate Texts in Mathematics. New York: Springer, 2004. [Ful95] W. Fulton. Algebraic Topology. A First Course. 1st ed. Vol. 153. Graduate Texts in Mathematics. New York: Springer, 1995. [Kos80] C. Kosniowski. A First Course in Algebraic Topology. 1st ed. Cambridge: Cambridge University Press, 1980. [Lan02] S. Lang. Algebra. 3rd ed. Vol. 211. Graduate Textsin Mathematics. New York: Springer, 2002. [Lan94] S. Lang. Algebraic Number Theory. 2nd ed. Vol. 110. Graduate Texts in Mathematics. New York: Springer, 1994. [Mar77] D.A. Marcus. Number Fields. 1st ed. Universitext. New York: Springer, 1977. [Mor12] M. Morishita. Knots and Primes. An Introduction to Arithmetic Topology. 1st ed. Univer- sitext. London: Springer, 2012. [Mum99] D. Mumford. The Red Book of Varieties and Schemes. Includes the Michigan Lectures (1974) on Curves and their Jacobians. Vol. 1358. Lecture Notes in Mathematics. 2nd, expanded edition. Berlin: Springer, 1999. [Neu99] J. Neukirch. Algebraic Number Theory. 1st ed. Vol. 322. Grundlehren der mathematischen Wissenschaften. Berlin: Springer, 1999. 72 BIBLIOGRAPHY

[NSW08] J. Neukirch, A. Schmidt, and K. Wingberg. Cohomology of Number Fields. 2nd ed. Vol. 323. Grundlehren der mathematischen Wissenschaften. Berlin: Springer, 2008. [Rol03] D. Rolfsen. Knots and Links. Reprint of the 1990 edition. Providence: American Math- ematical Society, 2003. [Rot10] J.J. Rotman. Advanced Modern Algebra. 2nd ed. Vol. 114. Graduate Studies in Mathe- matics. Providence: American Mathematical Society, 2010. [Rot88] J.J. Rotman. An Introduction to Algebraic Topology. 1st ed. Vol. 119. Graduate Texts in Mathematics. New York: Springer, 1988. [RZ10] L. Ribes and P.Zalesskii. Proﬁnite Groups. 2nd ed. Vol. 40. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Berlin: Springer, 2010. [Ser79] J.-P. Serre. Local Fields. 1st ed. Vol. 67. Graduate Texts in Mathematics. New York: Springer, 1979. [Was97] L.C. Washington. Introduction to Cyclotomic Fields. 2nd ed. Vol. 83. Graduate Texts in Mathematics. New York: Springer, 1997. [Wei06] S.H. Weintraub. Galois Theory. 1st ed. Universitext. New York: Springer, 2006.