Stark's Conjectures

Stark’s Conjectures

Pieter Mostert

Thesis presented for the degree of Master of Science in the Department of Mathematics and Applied Mathematics University of Cape Town

February 2008 Acknowledgements

Firstly, I would like to thank my supervisor, Ken Hughes, for much advice given and many books lent. To all the other teachers and lecturers who’ve contributed to my mathematical education over the years, I thank you. To my fellow masters students, thanks for keeping me company during the past two years. May the roads you travel be less bumpy than mine. To everyone at IkamvaYouth, thank-you for the opportunity to do some- thing practical with my mathematical knowledge. You will always be in- spiring. Finally, a big thank-you to my family − Ma, Da, Ingrid, Esther, and Karen − for all your support and love over the years.

1 Synopsis

We give a slightly more general version of the Rubin-Stark conjecture, but show that in most cases it follows from the standard version. After covering the necessary background, we state the principal Stark conjecture and show that although the conjecture depends on a choice of a set of places and a certain isomorphism of Q[G]-modules, it is independent of these choices. The conjecture is shown to satisfy certain ‘functoriality’ properties, and we give proofs of the conjecture in some simple cases. The main body of this dissertation concerns a slightly more general version of the Rubin-Stark conjecture. A number of Galois modules connected with the conjecture are defined in chapter 4, and some results on exterior powers and Fitting ideals are stated. In chapter 5 the Rubin-Stark conjecture is stated and we show how its truth is unaffected by lowering the top field, changing a set S of places ap- propriately, and enlarging moduli. We end by giving proofs of the conjecture in several cases. A number of proofs, which would otherwise have interrupted the flow of the exposition, have been relegated to the appendix, resulting in this dissertation suffering from a bad case of appendicitis. I know the meaning of plagiarism and declare that all of the work in this document, save for that which is properly acknowledged, is my own.

2 Contents

Acknowledgements 1

Synopsis 2

1 Introduction 5 1.1 Background, notation and conventions ...... 6 1.1.1 Algebraic Number Theory ...... 6 1.1.2 R[G]-modules and group representations ...... 8 1.1.3 Some conventions ...... 11

2 Preliminaries to Stark’s conjecture 12 2.1 Artin L-functions ...... 12 2.1.1 The Augmented Artin L-Function and the Functional Equation . 14 2.2 Brauer’s Theorem ...... 16 2.3 The Stark regulator ...... 16 2.3.1 Behaviour under addition, induction and inﬂation of characters . 22

3 Stark’s conjecture 25 3.1 Statement of Stark’s principal conjecture ...... 25 3.2 Independence of f ...... 26 3.3 Independence of S ...... 27 3.4 Reduction to special cases ...... 28 3.5 Proofs of Stark’s conjecture in special cases ...... 29 3.5.1 The trivial character ...... 29 3.5.2 The symmetric group on 3 letters ...... 29

4 Preliminaries to Rubin’s conjecture 31 4.1 R[G]-modules ...... 31 4.1.1 R[G]-lattices ...... 31

3 CONTENTS 4

4.1.2 Exterior powers ...... 32 4.1.3 Rubin’s modiﬁed exterior power ...... 33 4.2 Basic deﬁnitions ...... 34

4.2.1 The Z[G]-module structure of QK,m ...... 36 4.2.2 Fitting ideals ...... 38 4.3 The Stickelberger function ...... 39

5 The Rubin-Stark conjecture 41 5.1 Statement of Rubin’s conjecture ...... 41 5.1.1 An equivalent formulation ...... 43 5.2 Relation to Stark’s principal conjecture ...... 45 5.3 Changing S ...... 47 5.4 Changing m ...... 49 5.5 Subextensions ...... 51 5.6 Proofs of the Rubin-Stark conjecture in special cases ...... 52 5.6.1 More than r places of S split completely ...... 52 5.6.2 The case r =0 ...... 53 5.6.3 Quadratic extensions ...... 55

A Appendices 59 A.1 G-modules...... 59 A.1.1 Hom groups ...... 59 A.1.2 Cohomology of G-modules ...... 60 A.2 Fitting ideals ...... 62 A.3 An approximation theorem ...... 64 A.4 Unit and Picard groups of commutative rings ...... 65 A.5 Algebraic number theory calculations ...... 67

A.5.1 A generator of Fitt (Q n ) ...... 68 Z[G] K,v A.6 Some exact sequences ...... 69 A.6.1 The exact sequence 4.2.2 ...... 69 A.6.2 The exact sequence 5.6.3 ...... 71 A.7 Some results required for the proof of the Rubin-Stark conjecture for quadratic extensions...... 73 Chapter 1

Introduction

Many conjectures and results concern the interplay between algebraic objects (groups of units, class-groups, etc.) and analytic objects (special values of L-functions, etc.) The classic case of this is Dirichlet’s class number formula ([BS66] theorem 2, p 313), which states that for any number ﬁeld F , the leading coeﬃcient of the Taylor expansion of the

Dedekind zeta-function ζF at zero is given by h R F F , wF where hF , RF and wF are the class number, regulator and number of roots of unity of F respectively. It is this relation that Stark tried to generalise to L-functions corresponding to arbitrary characters. While we deal exclusively with number fields, Stark’s conjectures have function- field analogues, and in fact the first-order Rubin-Stark conjecture has been proven for function-fields (see chapitre V of [Tat84]). There are also p-adic analogues (see [Sol02] for example), which we do not examine. The first Stark-type conjectures appeared in a series of papers written by Harold Stark (as one might expect) between 1971 and 1980 ([Sta71], [Sta75],[Sta76], [Sta80]). The conjectures were reformulated by John Tate and given a more accessible treatment in the book [Tat84], based on a course given by Tate at the University of Orsay. No doubt this book also served to further general interest in the conjectures. In [Sta80], Harold Stark gave a refined conjecture for abelian conjectures concerning the first derivatives of Artin L-functions at zero (conjecture 1, p 198), which was subse- quently reformulated and slightly strengthened by Tate ([Tat84] conjecture St(K/k, S), p 89). This was generalised by Karl Rubin in [Rub96], with the purpose of applying so-called ‘Stark units’ to Kolyvagin systems (see [Rub92]).

5 CHAPTER 1. INTRODUCTION 6

The most recent major advance is perhaps the work of David Burns ([Bur07], among others), who has shown that the Rubin-Stark conjecture follows from the equivariant Tamagawa number conjecture, and has formulated reﬁned Stark-type conjectures generalising those of Gross in [Gro88] (conjecture 8.8).

1.1 Background, notation and conventions

We will assume results from basic algebraic number theory, representation theory, homological algebra etc. However, we give a brief summary, without proofs, of some results from algebraic number theory which will be used in this thesis, and use this to ﬁx notation and terminology. We also state some facts and make some conventions regarding group representations. Other results can be found in the appendices.

1.1.1 Algebraic Number Theory

Proofs of most of the results in this subsection can be found in [Cas67] or [Nar90], although the terminology may not agree. If F is a ﬁeld, a valuation on F is a map

φ : F → R≥0 satisfying i) φ(xy) = φ(x)φ(y) for all x, y ∈ F , ii) φ(x) = 0 ⇔ x = 0, iii) ∃C ≥ 1 s.t. φ(x + y) ≤ C max{φ(x), φ(y)} for all x, y ∈ F . Every field has at least one valuation, the trivial one which takes on the value 0 at 0 and is 1 everywhere else. Each valuation determines a topology on F which turns it into a topological field, and we define two valuations on F to be equivalent if they determine the same topology.

This gives an equivalence relation on the set of valuations of F , and one can show that φ1 λ and φ2 are in the same equivalence class iff there is a λ > 0 such that φ1(x) = φ2(x) for all x ∈ F . A valuation is said to be non-archimedean if we can take C = 1 in condition iii) above, and archimedean otherwise. Clearly this terminology can be extended to equivalence classes of valuations. From now on we suppose F is a number field (a finite field extension of the ratio- nals). The ring of integers, discriminant and group of roots of unity of a number field

F will be denoted by OF , dF and µF respectively. We deﬁne a place of F to be an equivalence class of non-trivial valuations of F . By Ostrowski’s theorem ([Nar90],p90), CHAPTER 1. INTRODUCTION 7 the archimedean places are in one-to-one correspondence with embeddings of F in C modulo complex conjugation, while the non-archimedean places are in one-to-one correspondence with non-trivial prime ideals of OF . We say that an archimedean place is real if the corresponding embedding maps F into R, and complex if it does not. For every place v we single out a normalised valuation, written x 7→ |x|v and deﬁned as follows: −ordp(x) If v is non-archimedean, corresponding to the prime ideal p, |x|v = (Np) , where ordp(x) := ordp(xOF ) is the exponent of p in the prime ideal decomposition of ∗ xOF , and where Np = #(OF /p) . We will also write ordv = ordp and Nv = Np. δ If v is archimedean, corresponding to an embedding ϕ : F → C, then |x|v = |ϕ(x)| , where δ is 1 or 2 depending on whether v is real or complex respectively.

If v is a non-archimedean place of F , corresponding to the prime ideal p, F(v) = OF /p will denote the residue ﬁeld of v.

The set of archimedean places of F will be written as S∞,F . A modulus of F is an element of the free abelian group on the non-archimedean and real archimedean places of F , where all coefficients are non-negative, and all coefficients corresponding to the real archimedean places are at most 1. Thus we may also think of a modulus as being a pair consisting of an integral OF -ideal and a set of real places of F . We write this free abelian group multiplicatively, and say that the modulus m divides m0 iff m0m−1 is a modulus. In this case we write m|m0.

Deﬁne the support of a fractional OF -ideal I to be

supp(I) = {v a non-archimedean place of F : ordv(I) 6= 0}.

If I is the integral ideal corresponding to a modulus m, define supp(m) = supp(I) and ordv(m) = ordv(I) if v is a non-archimedean. L The signature group SgnF of F is defined to be real v{1, −1}, where {1, −1} is the group of units of Z (the direct sum is taken over all real archimedean places of F ). We also define a signature map

× sgnF : F → SgnF : x → (sign(φv(x)))real v, where φv is the embedding of F in R corresponding to v. If m is a modulus of F , define the modified signature group by M SgnF,m = {1, −1}, real v|m where the direct sum is taken over all real archimedean places of F dividing m. The × modified signature map sgnF,m : F → SgnF,m is defined analogously to sgnF .

∗If S is a set, #S denotes its cardinality. CHAPTER 1. INTRODUCTION 8

Given an extension K/k, any valuation of K restricted to k is clearly a valuation of k. If w is a place of K which contains a valuation whose restriction to k is a element of the place v, then the same is true for all valuations in w, and we say that w divides, or lies above v, and write w|k = v. For every place v of k there exists a finite, non-empty set of places of K which lie above v; we use the notationv ¯ to denote some fixed place of K dividing v. Suppose w divides v. If w (and hence v) is non-archimedean, let p and P be the prime ideals of Ok and OK corresponding to v and w respectively. Then Ok/p is canonically embedded in OK /P, and we define

f(w/v) = dimOk/p (OK /P) , e(w/v) = ordP(pOK ).

On the other hand, if v and w are archimedean, deﬁne ( 2 if v is real and w is complex f(w/v) = 1 , e(w/v) = † 1 otherwise

e(w/v)f(w/v) This ensures that |x|v = |x|w for all x ∈ k. We say that w is ramified in K/k if e(w/v) > 1, and unramified otherwise, and that w splits completely in K/k iff it is unramified and f(w/v) = 1. If the extension K/k is Galois, then e(w/v) and f(w/v) depend only on v, and we talk about v being ramified or splitting completely in K/k if v¯ is ramified or splits completely in K/k respectively.

If S is a set of places of k, then SK denotes the set of places of K dividing those in S. Likewise, if m = Q vnv is a modulus of k such that all archimedean places dividing v n Q Q e(w/v) v m are unramiﬁed in K/k, deﬁne mK = v w|v w , which is then a modulus of K.

1.1.2 R[G]-modules and group representations

Let R and S be commutative unital rings, and suppose A and B are S-modules. If A N is also an R-module, we give A S B an R-module structure by considering A as an R-S-bimodule and B as a left S-module. If G is a group and there is a representation

ρ : G → AutS(A), then A is naturally an S[G]-module. We write σ·a for ρ(σ)(a) when the representation is clear from the context. If there are representations ρ : G → AutR(A) N and ρ : G → AutS(B), we obtain a representation of G associated to A S B by deﬁning σ·(a ⊗ b) = (σ·a) ⊗ (σ·b).

†See [Gra03] for a dissenting view. CHAPTER 1. INTRODUCTION 9

For any abelian group A, we will write RA as short-hand for the R-module R N A. Z We always give R the trivial G-action, so if A is an Z[G]-module, R[G] acts on RA by ! X X rσσ ·(r ⊗ a) = rσr ⊗ σ·a. σ∈G σ∈G N If f : A → B is a homomorphism, we will again write f for the homomorphism 1R f : Z RA → RB. Although this notation is ambiguous, it should be clear from the context which function f refers to. We will usually take R to be a subring of C, and will abbreviate r ⊗ a by ra.

Remark 1.1.1. If G is abelian and M is a left Z[G]-module, we may regard R[G] as an R[G]− [G]-bimodule and form the left R[G]-module R[G] N M. This is isomorphic Z Z[G] N P to the R[G]-module RM = R M, the isomorphism sending ( rσσ) ⊗ m to Z σ∈G P σ∈G rσ ⊗ σ·m.

Let E be a subfield of C. Suppose V is a (left) E[G]-module, of dimension n over E, corresponding to the representation ρ : G → AutE(V ). If α is a field embedding of E in C, we define Cα to be C with a C-E-bimodule structure given by γ ∈ C acting on the left α N by γ·x = γx, and η ∈ E acting on the right by x·η = xα(η). We define V = Cα E V . α N If φ : V → W is a homomorphism of E-vector spaces, we define φ = 1Cα E φ. If f ∈ EndE(V ) corresponds to the matrix Mij with respect to the basis {b1, . . . , bn} of V , then

n n α X X α f (1 ⊗ bi) = 1 ⊗ f(bi) = 1 ⊗ Mijbj = Mij ·(1 ⊗ bj), j=1 j=1

α α α so f corresponds to the matrix Mij with respect to the basis {1 ⊗ b1,..., 1 ⊗ bn} of V . α α If ρ is the representation associated to V , then ρ : G → AutC(V ) is deﬁned by ρα(σ) = ρ(σ)α for σ ∈ G. Therefore if χ : G → E is the character of ρ, χα := α ◦ χ : G → C is the character of ρα. Any Z[G]-module may also be regarded as a Z-module/abelian group. To avoid constant repetition in chapters 4 and 5, we adopt the convention that when referring to a Z[G]-module M, the word ‘torsion’ will mean Z-torsion, and not Z[G]-torsion. We use Mf to denote M modulo torsion, and we write xe for the image of x ∈ M under the natural map M → Mf. If M is torsion-free, we identify M with Mf. If R is a unital commutative ring containing Q, we will identify Mf with the image of M in RM. Since we will frequently be dealing with exterior powers modulo torsion, we use the notation Vr M to denote V^r M. gR R CHAPTER 1. INTRODUCTION 10

If G is a finite group, we will use 1G to denote the trivial one-dimensional character ‡ P of G (and not the identity element of G ). We define NG = σ∈G σ ∈ Z[G]. P P The map augG : Z[G] → Z : σ∈G nσσ 7→ σ∈G nσ is called the augmentation map, and we denote its kernel by IG. Define MG = M/IG ·M.

Results in the remainder of this section may be found in [Ser77]. Let G be a finite group, F a field. A finite-dimensional representation of G over F is a homomorphism

ρ : G → AutF (V ), where V is a finite-dimensional F -vector space (we will refer to finite-dimensional representations as representations, it being understood that all representation we consider are finite-dimensional). If for i = 1, 2, ρi : G → Vi is a representation of G over F , a morphism from ρ1 to ρ2 is an F -linear map Φ : V1 → V2 such that Φ ◦ (ρ1(σ)) = ρ2(σ) ◦ Φ for all σ ∈ G. Each representation χ gives a function χ = Tr ◦ ρ : G → F , called the character of the representation, and any two representations are isomorphic iff they have the same characters. We can form the direct sum of two representations of the same group, and a representation is said to be irreducible if it is not isomorphic to the direct sum of two non-zero representations. For any irreducible character χ of G, the central idempotent associated to χ is defined by 1 X e = χ(σ)σ−1 ∈ [G]. χ #G C σ∈G We now assume that F is algebraically closed. Every representation can then be uniquely written as the direct sum of irreducible representations (up to isomorphism and order). If χ and ψ are two characters of G, define 1 X hχ, ψi = χ(σ)ψ(σ−1). G #G σ∈G If χ and ψ are irreducible, then ( 1 χ = ψ hχ, ψi = G 0 χ 6= ψ

If G → AutF V is a representation, with character φ, the idempotents eχ (χ irreducible), ∼ L can be used to give V a direct sum decomposition V = χ eχ ·V , where dim(eχ ·V ) = G hχ, φiG. This implies that h1G, φiG = dimF V . Suppose H is a subgroup of G, and χ and ψ are characters of H and G respectively. G G Let IndH χ be the character of IndH V , where V is an F [H]-module giving rise to χ, and G let ResH ψ = ψ|H . Then G G IndH χ, ψ G = χ, ResH ψ H .

‡ Note that if G is abelian, 1G is the identity element of the character group Gb. CHAPTER 1. INTRODUCTION 11

1.1.3 Some conventions

If V is a ﬁnite-dimensional vector space and f ∈ Aut(V ), then det(f|W ) is the deter- § f f|W minant of f|W : W → W . We also write WW/ when we mean WW/ . When A and B are modules as in the previous subsection, we sometimes abbreviate N N the R-module A S B by A B when it is clear what S is. Many objects we will deal with in this dissertation have a number of subscripts. At times we will omit the subscripts when it is clear from the context what they are. In some of the later sections we will give advance warning as to which subscripts will be omitted, but this will not always be done. If G is a group and M is a left G-module which is written multiplicatively, we α (αβ) β α sometimes write m in place of α·m (α ∈ Z[G]). Note that this implies m = (m ) . This should note be confused with M G, which is the submodule of all m ∈ M for which σ·m = m for all σ ∈ G. The trivial group will be written as 1, while the identity morphism of an object X

(in some category) will be denoted by 1X . + Finally, N = {0, 1, 2,...} and N = {1, 2, 3,...}.

§ This is purely for typographical reasons. It would be more natural to write det(f|W ), were it not for the fact that W may be given by a relatively complicated expression which would be awkward to read if it were a subscript. Chapter 2

Preliminaries to Stark’s conjecture

2.1 Artin L-functions

Let K/k be a finite normal extension of number fields with Galois group G. We turn the set of places of K into a G-set by defining

−1 |x|σ·w = |σ (x)|w for all x ∈ K, σ ∈ G. The action of G on the non-archimedean places corresponds to the action of G on prime ideals. If w lies above the place v of k, then so does σ·w for every σ ∈ G, and one can show that G acts transitively on set of places of K lying above v ([Tat67] proposition 1.2 (ii), p 163). Let v be a place of k, and put w =v ¯. Deﬁne

Dw = {σ ∈ G : σ·w = w} to be the decomposition group of w. It is equal to the image of Gal(Kw/kv) in G. Suppose now that w is non-archimedean, corresponding to a prime ideal P. One sees that

Dw = {σ ∈ G : σ(P) = P}.

We also denote this by DP. If i ∈ N, the i-th ramiﬁcation group of w is

i+1 Gw,i = σ ∈ G : σ(x) − x ∈ P (∀x ∈ OK ) , and we call Iw := Gw,0 the inertia group of w. Since all elements of Dw ﬁx P, there is an obvious map Dw → Gal(F(w)/F(v)), which can be shown to be onto. Clearly the

12 CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 13 kernel of this map is Iw, so Dw/Iw is isomorphic to Gal(F(w)/F(v)) which is cyclic of order f(w/v) with generator x 7→ x#F(v) (see [Sma] p 13, for example). The element of # (v) Dw/Iw that corresponds to x 7→ x F will be denoted by (w, K/k) and is called the Frobenius automorphism corresponding to w. 0 −1 −1 If w = σ·w is another place extending v, then Dw0 = σDwσ and Gw0,i = σGw,iσ .

Thus if G is abelian, Dw and Gw,i depend only on v, and we sometimes denote these subgroups by Dv and Gv,i, and write (v, K/k) in place of (w, K/k). If the extension

K/k is clear from the context, we abbreviate (w, K/k) by σw (or σv if w|v and K/k is abelian).

Let ρ : G → AutC(V ) be a complex ﬁnite-dimensional representation of G. Since Iw Iw is a normal subgroup of Dw, V is a ρ(Dw)-invariant subspace of V . Thus we obtain a Iw representation G → AutC(V ), whose kernel clearly contains Iw, and so we can deﬁne a representation Iw ρw : Dw/Iw → AutC(V ): σIw 7→ ρ(σ)|V Iw . For any s ∈ C with <(s) > 1, the element −s 1 − ρw(σw)Nv of End(V Iw ) has an inverse f−1 −fs −1 X r −rs (1 − Nv ) ρw(σw) Nv r=0

(where f = f(w/v) is the order of σw), and so its determinant is non-zero. This determinant is independent of the choice of w extending v. Let S be a ﬁnite set of places of k including the archimedean ones. The function deﬁned for {s ∈ C : <(s) > 1} by Y −s−1 s 7→ det 1 − ρv¯(σv¯)Nv v∈ /S (where v runs over the places of k not in S) can be analytically continued to a meromorphic function on C. This only depends on the character χ of ρ, and we call this the Artin L-function of the extension K/k with respect to χ, and denote it by LS(s, χ; K/k).

We will also write LS(s, χ) for LS(s, χ; K/k) when it is not necessary to mention which extension we are dealing with. If S contains only the archimedean places, we will simply write L(s, χ) in place of LS(s, χ). One can show that L(s, χ) is deﬁned and non-zero at s = 1 if χ is a non-trivial irreducible character, while L(s, 1G) has a simple pole at s = 1. Artin L-functions satisfy certain useful properties which we list below (proofs may be found in [Neu86] theorem 4.2, pp 123-124). CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 14

Additivity:

For any two characters χ1 and χ2 of G,

LS(s, χ1 + χ2) = LS(s, χ1)LS(s, χ2)

Hence every Artin L-function may be written as a product of Artin L-functions corresponding to irreducible representations. G This also shows that LS(s, χ) has a pole of order h1G, χiG = dim V at s = 1.

Induction:

If H is a subgroup of G and F = KH is the ﬁxed ﬁeld of H, then for a character χ of H,

G LSF (s, χ; K/F ) = LS(s, IndH χ; K/k).

Inﬂation:

If F is a Galois extension of k containing K, then there is a canonical surjection

π : Gal(F/k) → Gal(K/k): σ 7→ σ|K ,

H and any character χ of G gives a character InﬂG χ := χ ◦ π of H = Gal(F/k). One can show that

H LS(s, χ; K/k) = LS(s, InﬂG χ; F/k).

Suppose χ is the character of the complex representation ρ : G → AutC(V ). Then ρ 0 0 factors through π : G → G/ ker(ρ) (i.e. ρ = ρ ◦ π, where ρ : G/ ker(ρ) → AutC(V ) is a (faithful) representation). Therefore if χ0 is the character of ρ0,

0 ker(ρ) LS(s, χ ; K /k) = LS(s, χ; K/k).

Thus every Artin L-function is equal to one deﬁned by a faithful character. In particular, every L-function given by a one-dimensional character is equal to an L-function associated to a character of an abelian Galois group.

2.1.1 The Augmented Artin L-Function and the Functional Equation

We now introduce the augmented Artin L-function, following the exposition in [Mar77] and [Tat84]. Deﬁne Υ(s) = π−s/2Γ(s/2), CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 15 a meromorphic function with a pole of order 1 at s = 0 (Here Γ is Euler’s gamma function).

Let ρ : G → AutC(V ) be a complex representation of G with character χ. If v is a real archimedean place of k, let

Dv¯ Dv¯ av = dim V , bv = codimV

(these are independent of the choice ofv ¯). We then deﬁne

v av bv Υχ(s) = Υ(s) Υ(s + 1) .

v χ(1) If v is a complex archimedean place of k, deﬁne Υχ(s) = [Υ(s)Υ(s + 1)] . Finally, we deﬁne Υ (s) = Q Υv (s). χ v∈S∞ χ v Note that at s = 0, Υχ(s) has a pole of order av if v is real, and of order χ(1) = Dv¯ ∗ dim V = dim V if v is complex . Thus Υχ(s) has a pole of order

X X X Dv¯ X Dv¯ X Dv¯ av + χ(1) = dim V + dim V = dim V .

v real v complex v real v complex v∈S∞

Also note that Υχ(s) is deﬁned and non-zero at s = 1. Let v be a non-archimedean place of k. Deﬁne

∞ 1 X n(χ, v) = #G codimV Gv,i¯ . #I v,i¯ v¯ i=0

One can show that n(χ, v) is always an integer (see [Ser79], p100), and since Gv,i¯ = 0 for almost all v and i, n(χ, v) is almost always zero. We then define f(χ), the Artin conductor of χ, by Y n(χ,v) f(χ) = pv , v-∞ χ(1) where pv is the prime ideal corresponding to v. Finally, define B(χ) = |dk| N(f(χ)). The augmented Artin L-function Λ is then defined by

s/2 Λ(s, χ) = B(χ) Υχ(s)L(s, χ), and satisﬁes the functional equation

Λ(s, χ) = W (χ)Λ(1 − s, χ¯), where W (χ) ∈ C is a constant with absolute value 1, the Artin root number (see [Mar77] p 14).

∗ If v is complex, Dv¯ is trivial. CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 16

We may use this to determine the order of the zero of L(s, χ) at s = 0; namely

ords=0L(s, χ) = ords=1L(s, χ¯) + ords=1Υχ¯(s) − ords=0Υχ(s) X = − dim V G + dim V Dw , (2.1.1)

v∈S∞ G since ords=1L(s, χ¯) = − dim V .

2.2 Brauer’s Theorem

A character of a ﬁnite group G is said to be monomial if it is induced by a one- dimensional character of some subgroup of G. The following useful theorem is due to Brauer ([Bra47] theorem 1, p 503):

Theorem 2.2.1. Every character of a ﬁnite group G can be written as a Z-linear combination of monomial characters. P Hence for any character χ of G = Gal(K/k), we can write χ = j njΨj, where n ∈ and Ψ = IndG (ψ ) for a one-dimensional character ψ of a subgroup H of G. j Z j Hj j j j H If we let kj = K j , then by the addition and induction properties of Artin L-functions, X LS(s, χ, K/k) = L(s, njΨj; K/k) j

Y G nj = LS(s, IndHj (ψj); K/k) j Y = L (s, ψ ; K/k )nj . Skj j j j

ker ψ Replacing K/kj by K j /kj expresses LS(s, χ, K/k) is a product of integral powers of L-functions corresponding to abelian extensions. Thus we may often reduce questions about general L-functions to questions about L-functions corresponding to abelian extensions (with one-dimensional characters).

2.3 The Stark regulator

This section follows [Tat84], chapitre I. Let K/k be a finite normal extension of number fields, and let S be a finite set of places of k, including all the archimedean ones.

Let YK,S be the free abelian group on SK . This has a natural left G-action, induced by the action of G on SK . If we give Z the trivial G-action, the augmentation map X X augK : YK,S → Z : nww 7→ nw w∈SK w∈SK CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 17 is a surjective G-module homomorphism with kernel    X X  XK,S = nww ∈ YK,S : nw = 0 .

w∈SK w∈SK 

To simplify the notation, we will omit the subscripts K and S at times. Tensoring the short exact sequence 0 → X → Y → Z → 0 with C over Z gives a short exact sequence of C[G]-modules

0 → CX → CY → C → 0. ∼ L By semi-simplicity, CY = CX C (as C[G]-modules), and so if χX and χY are the characters of CX and CY respectively, we have χY = χX + 1G. ∼ L Note that Y = v∈S Yv, where Yv is the free abelian group generated by the places G of K above v, and thus has character IndDv¯ 1Dv¯ . Definition 2.3.1. If χ is the character of a finite-dimensional complex representation of G, define

rK/k,S(χ) = ords=0LS(s, χ; K/k). We will usually omit the subscript K/k.

Proposition 2.3.1.

X Dv¯ G N G rS(χ) = dim V − dim V = hχ, χX iG = dim(V CXK,S) v∈S Proof. * + X G hχ, χX iG = χ, IndDv¯ 1Dv¯ − 1G v∈S G X = χ, IndG 1 − hχ, 1 i Dv¯ Dv¯ G G G v∈S X G G = Res χ, 1D − dim V Dv¯ v¯ Dv¯ v∈S X = dim V Dv¯ − dim V G, v∈S so the second equality holds. The third equality follows from

dim(V N X)G = hχχ , 1 i = hχ, χ i = hχ, χ i , CC X G G X G X G CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 18

where the last equality comes from the fact that χX takes on integer values. It remains to show that

X Dv¯ G rS(χ) = dim V − dim V . (2.3.1) v∈S Using Brauer’s theorem we may assume that χ is one-dimensional. Note that in this case ( 0 ρ (σ ) 6= 1 I −s Iv¯ v¯ v¯ V v¯ ords=0 det 1 − ρv¯(σv¯)Nv |V = 1 ρv¯(σv¯) = 1V Iv¯ = dim(V Iv¯ )Dv¯/Iv¯ = dim(V Dv¯ ).

By equation 2.1.1, 2.3.1 is true if S = S∞,k. Suppose the equality holds for a particular set S; if v is a place of k not in S, then

−s Iv¯ ords=0LS∪{v}(s, χ) = ords=0LS(s, χ) + ords=0 det 1 − ρv¯(σv¯)Nv |V

Dv¯ = ords=0LS(s, χ) + dim V , and so by induction 2.3.1 holds for all S ⊇ S∞,k.

Remark 2.3.1. If V is one-dimensional, then dim V Dw is 1 or 0 depending on whether G χ is trivial on Dw or not, respectively, while dim V is 1 if χ = 1G, and zero otherwise. This shows that ( # {v ∈ S : χ|Dv¯ = 1Dv¯ } if χ 6= 1G rS(χ) = #S − 1 if χ = 1G

Suppose F is a number field, and S a finite set of places of F containing S∞,F . Let × UF,S = {x ∈ F : |x|w = 1 (∀w∈ / S)} be the group of S-units of F . Define the R-linear map X † λF,S : RUF,S → RXF,S : ru 7→ r log |u|w w . w∈S

The proof of Dirichlet’s units theorem can be modiﬁed to show that λF,S is an isomorphism (see [Nar90] theorem 3.5, pp 101-103). The images of Z-bases for UgF,S and XF,S in RUF,S and RXF,S give R-bases for the respective vector spaces, and we deﬁne the S-regulator RF,S to be the determinant of λF,S with respect to these bases.

† P Q This is well-deﬁned, since w∈S log |u|w = log w∈S |u|w = log 1 = 0 by the product formula (see [Nar90] p 93, for example). CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 19

Dedekind’s zeta function ζF is deﬁned by

X −s ζF (s) = (Na) (2.3.2) a for <(s) > 1 (the sum running over integral ideal a), and defined on the rest of C r {1} by analytic continuation. More generally, one defines the S-modified Dedekind zeta function ζF,S by restricting the sum in 2.3.2 to those integral ideals prime to the non- archimedean places in S. Alternatively, one may define it as taking the sum over integral

OF,S-ideals, where

OF,S = {x ∈ F : ordv(x) ≥ 0 for all places v of F not in S} is the ring of S-integers of F . The S-modiﬁed Dedekind zeta function has a product expansion Y −s −1 ζF,S(s) = (1 − Np ) , p6∈S which is simply LS(s, 11; F/F ). By the Dirichlet class-number formula ([BS66]),

1−#S∞ hF RF lim s ζF (s) = − . s→0 wF

Let AF,S denote the S-class group of F , that is, the quotient of the group of fractional

OF,S-ideals by the subgroup of principal fractional OF,S-ideals. We claim that Dirichlet’s class-number formula can be generalised as follows:

1−#S hF,SRF,S lim s ζF,S(s) = − . (2.3.3) s→0 wF where hF,S = #AF,S. If S = S∞,F , this is just the usual formula. Suppose 2.3.3 holds for a set S ⊇ S∞,F , and let v be a place of F not in S, corresponding to the prime ideal d p. Let d be the order p in AF,S, and let u be a generator of p . Then hF,S∪{v} = hF,S/d, and RF,S∪{v} = RF,S |log |u|v| = RF,Sd log Nv. Since adding v to S increases the left- −1 −s hand side of equation 2.3.3 by lims→0 s (1 − Nv ) = log Nv, the result follows by induction. We consider again a normal extension of number ﬁelds K/k with Galois group G.

If S ⊇ S∞,k is a ﬁnite set of places of k, we write UK,S = UK,SK , λK,S = λK,SK ,

OK,S = OK,SK , etc. Let jK/k,S : Yk,S → YK,S be the Z[G]-homomorphism deﬁned by X j(v) = NG ·v¯ = #Dv¯ σ·v¯

σDv¯∈G/Dv¯ CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 20

for v ∈ S. Since augK ◦ j = #Gaugk, j restricts to a map Xk → XK , which we again denote by j. If u ∈ Uk,S, then X X X λK (˜u) = log |u|ww = #Dv¯ log |u|vσ·v¯

w∈SK v∈S σDv¯∈G/Dv¯ X = log |u|vj(v) = j(λk(˜u)). v∈S In other words,

λK (2.3.4) RUK / RXK O O

jK/k

λk RUk / RXk commutes. Likewise, if u ∈ UK,S, then

λK (2.3.5) RUK / RXK

NK/k |k

λk RUk / RXk is commutative.

Since λK,S gives an R[G]-isomorphism between RU and RX, these two modules give rise to the same character. Tensoring a Q[G]-module with R over Q leaves the character unchanged, so QU and QX have the same character and are thus isomorphic ‡ as Q[G]-modules . Hence we can find a Q[G]-isomorphism f : QX → QU, which after tensoring with C over Q gives a C[G]-isomorphism f : CX → CU. Definition 2.3.2. Let V be a C[G]-module with character χ, and let f : QX → QU be a Q[G]-isomorphism. The Stark regulator is defined to be

N N G R(χ, f) = det 1V (λ ◦ f)| (V CX) .

‡although there does not seem to be a canonical way of choosing such an isomorphism. CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 21

Like the regulator of a number ﬁeld, the Stark regulator is a combination of logarithms of normalised valuations of units.

Remark 2.3.2. We deﬁne V ∗ = Hom (V, [G]), and give it the obvious [G]-module C[G] C C ∗ structure. Let (λ ◦ f)V be the image of λ ◦ f under the functor HomG(V , ), i.e.

(λ ◦ f) : Hom (V ∗, X) → Hom (V ∗, X): h 7→ λ ◦ f ◦ h. V C[G] C C[G] C

(V N X)G is naturally isomorphic to Hom (V ∗, X) (see appendix A.1.1), and C C[G] C N N G under this isomorphism, the restriction of 1V (λ ◦ f) to (V CX) corresponds to (λ ◦ f)V . Hence

R(χ, f) = det((λ ◦ f)V ). This is the deﬁnition used in the book by Tate ([Tat84] p 26).

Remark 2.3.3. Although the Q[G]-modules QX and QU are isomorphic, this is not in general true of the [G] modules X and Ue. For example, consider the case K = (ζ), Z √ Q 2 4 k = Q(ζ ), S = S∞, where ζ = i 3. The Galois group Gal(K/k) has order 2, and is generated by τ : ζ 7→ −ζ. The places in SK correspond to the embeddings

w1 : ζ 7→ ±ζ, w+ : ζ 7→ iζ, w− : ζ 7→ −iζ, hence {w1 − w+, w1 − w−} is a Z-basis for X. Using PARI-GP, we ﬁnd that the image 2 3 2 of {ζ − 2, ζ + ζ + ζ + 2} in Ue gives a Z-basis for Ue. With respect to these integral bases, τ corresponds to the matrices ! ! 0 1 1 0 and 1 0 0 −1 respectively. One can easily show that these two matrices are not conjugate by an element of GL2(Z), hence no Z[G]-isomorphism f : X → Ue exists. Remark 2.3.4. There is an alternative way of describing R(χ, f) when χ is one- dimensional. Let (λ ◦ f)χ be λ ◦ f with domain and codomain restricted to eχ¯ ·CX. There is an isomorphism of C-vector spaces

N G § ξχ :(V CX) → eχ¯CX : γ ⊗ x 7→ γx , CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 22

and since the diagram

1 N(λ◦f) N G N G (V CX) / (V CX)

ξχ ξχ

(λ◦f)χ eχ¯ ·CX / eχ¯ ·CX commutes, R(χ, f) = det(λ ◦ f)χ.

2.3.1 Behaviour under addition, induction and inﬂation of characters

Let us consider the general case again. We will show that RS(χ, f) behaves similarly to

LS(s, χ) under addition, induction and inﬂation of characters.

Additivity

For any two characters χ1 and χ2 of G,

RS(χ1 + χ2, f) = RS(χ1, f)RS(χ2, f).

The proof is straightforward, and therefore omitted.

Induction

Let L/k be a Galois subextension of K/k, let H = Gal(K/L), and let V be a C[H]- module. There is a natural isomorphism

Hom (V ∗, X )=∼ Hom (IndG V ∗, X ) C[H] C K,SL C[G] H C K,S

(see A.1.2), so the diagram

(λ◦f)V Hom (V ∗, X ) / Hom (V ∗, X ) C[H] C K,SL C[H] C K,SL

=∼ =∼

(λ◦f) IndG (V ) H Hom (IndG V ∗, X )/ Hom (IndG V ∗, X ) C[G] H C K,S C[G] H C K,S commutes. The deﬁnition of R(χ, f) given by Tate (remark 2.3.2), and the fact that G ∗ G ∗ G IndH V = (IndH V ) , then shows that RSL (χ, f; K/L) = RS(IndH χ, f; K/k).

§ We identify the underlying C vector space of V with C. CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 23

Inﬂation

Suppose E is a Galois extension of k containing K, and let H = Gal(E/k). Suppose we have a commutative diagram of the following form:

fE (2.3.6) QXE / QUE O O

jE/K

fK QXK / QUK

¶ where fE and fK are isomorphisms . If we regard QXK and QXK as H-modules via the restriction-to-K map H → G, then jE/k is an H-homomorphism since the inclusion N N G QUK → QUE is. Therefore, for any C[H]-module V , 1V jE/K maps (V CXK ) = N H N H (V CXK ) into (V CXE) . Combining 2.3.4 and 2.3.6 gives the following commutative diagram

1 N(λ ◦f ) N H V E E N H (2.3.7) (V CXE) / (V CXE) O O

N N 1V jE/K 1 jE/K

1 N(λ ◦f) N G K N G (V CXK ) / (V CXK )

Since the L-functions satisfy the induction property,

N H H N G dim(V CXE) = rS(InﬂG χ; E/k) = rS(χ; K/k) = dim(V CXK ) .

Therefore, since the vertical maps in diagram 2.3.7 are injective, they are isomorphisms, and so H RS(χ, fK ; K/k) = RS(InﬂG χ, fE, E/k).

Some ﬁnal deﬁnitions before we are able to state Stark’s conjecture: Let

−rS (χ) cS(χ; K/k) = lim s LS(s, χ) s→0 be the leading coeﬃcient in the Taylor expansion of LS(s, χ) about s = 0.

¶ Given an isomorphism fK : QXK → QUK , we can always ﬁnd an isomorphism fE : QXE → QUE to make the diagram commute, since the category of Q[G]-modules is semi- simple. CHAPTER 2. PRELIMINARIES TO STARK’S CONJECTURE 24

Choose an isomorphism of Q[G]-modules f : QXK,S → QUK,S and deﬁne

RS(χ, f; K/k) AS(χ, f; K/k) = . cS(χ; K/k) Since R(χ, f) and c(χ) satisfy the same additivity, induction and inﬂation properties, so does A(χ, f). Chapter 3

Stark’s conjecture

As in the previous section, most of the results in this chapter are taken from [Tat84] chapitre I.

3.1 Statement of Stark’s principal conjecture

As before, let K/k be a Galois extension of number ﬁelds with Galois group G, let S be a ﬁnite set of places of k containing the archimedean ones, and choose an isomorphism of Q[G]-modules f : QXK,S → QUK,S. Stark’s principal conjecture, as reformulated by Tate, is Conjecture 3.1.1. For any complex character χ of G, and any α ∈ Aut(C), A(χ, f)α = A(χα, f).

This may not appear to be a generalisation of Dirichlet’s class number formula (and in fact it isn’t), but we will show that it generalises a weaker form of Dirichlet’s formula. Deﬁne Q(χ) = Q(χ(G)), and let ρ be a representation of G with character χ. If σ ∈ G, the eigenvalues of ρ(σ) are g-th roots of unity (g = #G), so Q(χ) ⊆ Q(ζ), where ζ is a primitive g-th root of unity. Hence every embedding of Q(χ) in C is the restriction of a a νa : ζ 7→ ζ for some a coprime to g. Since νa(χ(σ)) = χ(σ ) ∈ Q(χ) for every σ ∈ G, we see that Q(χ)/Q is normal. Stark’s conjecture implies that for any α ∈ Gal(C/Q(χ)), A(χ, f)α = A(χα, f) = A(χ, f), and hence A(χ, f) ∈ Q(χ). One sees thus that Stark’s conjecture is equivalent to: ( A(χ, f) ∈ Q(χ), α α A(χ, f) = A(χ , f)(∀α ∈ Gal(Q(χ)/Q)).

25 CHAPTER 3. STARK’S CONJECTURE 26

−1 This implies that c(χ) = A(χ, f) R(χ, f) is the product of an element of Q(χ) and a combination of logarithms of normalised valuations of SK -units, generalising the class number formula.

3.2 Independence of f

As it stands, it appears that the conjecture depends on the choice of the isomorphism f : QX → QU. However, we shall show that if the conjecture holds for one choice of f, it holds for all others. 0 Suppose f, f : QX → QU are Q[G]-isomorphisms. It suﬃces to show that given any α ∈ Gal(C/Q), A(χ, f 0)α A(χα, f 0) = , A(χ, f)α A(χα, f)

A(χ,f 0) α α so if θ(χ) = A(χ,f) , we wish to show that θ(χ) = θ(χ ). Since

0 0 N 0 N G A(χ, f ) R(χ, f ) det 1V (λ ◦ f )| (V CX) θ(χ) = = = C C A(χ, f) R(χ, f) det (1 N (λ ◦ f)| (V N X)G) V C CC = det 1 N (f −1 ◦ f 0) (V N X)G = det 1 N (f −1 ◦ f 0) (V N X)G , V C CC V Q QQ it remains to prove that for any Q[G]-isomorphism g : QX → QX,

N N G α N αN G det 1V g (V QX) = det 1V α g (V QX) . (3.2.1) N (We have omitted the subscript Q from ). If h : W → W is an endomorphism of a ﬁnite-dimensional complex vector space W , then det (h|W )α = det (hα|W α), hence the left-hand side of equation 3.2.1 is equal to N α N Gα N α N α G det (1V g) (V QX) = det (1V g) ((V QX) ) .

Since α leaves elements of Q ﬁxed, there is a well-deﬁned map

α O O α V QX → (V QX) :(γ ⊗ v) ⊗ x 7→ γ ⊗ (v ⊗ x), which is clearly an isomorphism. Since the diagram

1 α N g α N V α N V QXXV/ V QX

=∼ =∼ (1 N g)α N α V N α (V QX) / (V QX) commutes, 3.2.1 holds. CHAPTER 3. STARK’S CONJECTURE 27

3.3 Independence of S

In this section, we prove that for a given extension, the truth of the conjecture is independent of the set S (so it suffices to prove it for S = S∞). By Brauer’s theorem and the additivity, induction and inflation properties of AS(χ, f), it is enough to prove this for faithful one-dimensional characters. Suppose χ(1) = 1. To prove the independence of S, it is enough to show that if S is a finite set of places of k containing the archimedean ones, and v is a place of k not in S, then the conjecture is true for S if and only it is true for S0 = S ∪ {v}. 0 0 Let U = US, X = XS, U = US0 , X = XS0 . In general, we will use a prime to 0 0 indicate that a quantity is defined using S , so for example c (χ) = cS0 (χ). Note that by proposition 2.3.1, r0(χ) = r(χ) + dim V Dv¯ . Suppose f : QX → QU is a Q[G]-isomorphism. By semi-simplicity, we may view 0 0 QX and QU as direct summands of QX and QU respectively, so f can be extended to 0 0 0 a Q[G]-isomorphism f : QX → QU . It suffices to show that if

Ω(χ) = A(χ, f)/A0(χ, f 0),

α α then Ω(χ) = Ω(χ ) for all α ∈ Gal(C/Q). Let w =v ¯. We consider two cases:

Case 1: Dw is non-trivial

Since the representation is faithful and V is one-dimensional, V Dw = {0}, and so r0(χ) = 0 r(χ). We may view CX as a submodule of CX , and thus there is a canonical embedding N G N 0 G of (V CX) in (V CX ) . Since these spaces have the same dimension over C, this 0 0 embedding is an isomorphism; since λ ◦ f with domain and codomain restricted to CX is λ ◦ f, it is clear that R(χ, f) = R0(χ, f 0). Therefore

0 Iw Ω(χ) = c (χ)/c(χ) = det(1 − σw|V ),

Iw which is either 1 − χ(σw) or 1, depending on whether V is V or {0}. In either case, α α Ω(χ) = Ω(χ ) for all α ∈ Gal(C/Q), and we are done.

Case 2: Dw is trivial

0 In this case, σw = 1, and r (χ) = r(χ) + 1 . Suppose w corresponds to the prime ideal h P; then P K = πOK for some π ∈ OK . Pick any w0 ∈ S, and deﬁne

0 x = w − e1G ·w0 ∈ QX . CHAPTER 3. STARK’S CONJECTURE 28

0 ∼ L 0 ∼ L Note that QU = QU Q[G]·π and QX = QX Q[G]·x, and let j : Q[G]·x → Q[G]·π be the Q[G]-module isomorphism that sends x to π. Since the truth of the Stark conjecture for S0 is independent of the choice of f 0, we may assume that f 0 = f L j. 0 Choose ordered Q-bases for QU and QX and extend these to Q-bases for QU and 0 QX by adding {σ·π : σ ∈ G} and {σ·x : σ ∈ G} respectively (choose some ordering of G). We also view these as bases for the vector spaces obtained by tensoring with C. For each σ ∈ G,

0 X λ (σ·π) = λ(σ·π) + log |σ·π|γ·wγ·w = λ(σ·π) + log |π|wσ·w γ∈G

= λ(σ·π) + log |π|wσ·x + log |π|we1G ·w0 ≡ log |π|wσ·x (mod CX). Therefore, if we let M(λ) and M(f) be the matrices corresponding to λ and f with respect to the chosen bases for CU and CX respectively, the matrices corresponding to λ0 and f 0 with respect to the extended bases are respectively ! ! M(λ) ∗ M(f) 0 M(λ0) = and M(f 0) = , 0 log |π|wIg 0 Ig where Ig is the g × g identity matrix and ∗ represents some unspeciﬁed matrix. Because N 0 G ∼ N G M N G (V CX ) = (V CX) (V C[G]·x) , N G ∼ 0 0 where (V C[G]·x) = Ceχ¯ ·x is one-dimensional, R (χ, f ) = log |π|wR(χ, f). Finally, the fact that

0 −r(χ)−1 −s c (χ) = lim s (1 − Nv )LS(s, χ) (since σw = 1) s→0 −1 −s −r(χ) = lim s (1 − Nv ) lim s LS(s, χ) s→0 s→0 = log Nv c(χ)

−h α α shows that Ω(χ) = log Nv/ log |π|w = log Nv/ log Nw = −1/h, and so Ω(χ) = Ω(χ ) for all α ∈ Gal(C/Q).

3.4 Reduction to special cases

Given a Galois extension of number fields K/k, let F be the normal closure of K over Q, and put H = Gal(F/Q), Γ = Gal(F/k). Then Γ ¯ H Γ ¯ A(χ, f; K/k) = A(InflGχ, f; F/k) = A(IndΓ InflGχ, f; F/Q).

Therefore Stark’s conjecture is true if it holds for every Galois extension of Q. By Brauer’s theorem, we see that Stark’s conjecture is true if it holds for all irreducible characters of abelian Galois groups. CHAPTER 3. STARK’S CONJECTURE 29

3.5 Proofs of Stark’s conjecture in special cases

Stark’s conjecture has been proven in a number of special cases, the most signiﬁcant being in [Sta75], where Stark gives a proof of his conjecture in the case where the character takes on rational values (see also [Tat84] chapitre II and [Das] chapter 9). As this proof is rather long, we omit it, and prove the conjecture in some easier cases.

3.5.1 The trivial character

We will show that Stark’s conjecture is true for the trivial character 1G. Since

G A(1G, f; K/k) = A(Inﬂ1 11, f; K/k) = A(11, f0; k/k),

(where f : QXK → QUK extends f0 : QXk → QUk), we may assume K = k. In this case the L-function concerned is just the Dedekind zeta function of k. By Dirichlet’s class number formula,

hkRk c(1G) = − , wk so if we can show that R(1G, f)/Rk ∈ Q, we will be done. We have deﬁned Rk to be the absolute value of the determinant of λk : Ufk → Xk with respect to Z-bases for Ufk and Xk. If f : QXk → QUk is an isomorphism, then R(1G, f) = ±Rk det(f), where det(f) is calculated with respect to Z-bases for Ufk and Xk, considered as Z-bases for QUk and QXk respectively. Since f is a function between rational vector spaces, det(f) ∈ Q, completing the proof. Remark 3.5.1. From the above result, we may easily prove that the conjecture holds for extensions K/k with [K : k] = 2. Let χ be the non-trivial irreducible representation G of G. Then since 1G + χ = Ind1 11, we see by using the additivity, induction and inﬂation properties of A( , ) that

0 0 A(11, f , K/K) = A(11, f, k/k)A(χ, f , K/k), and so the conjecture is true for χ since it holds for the trivial character.

3.5.2 The symmetric group on 3 letters

Stark’s conjecture holds for quadratic extensions since the non-trivial character can be built up out of trivial characters by applying the inﬂation, induction and additivity properties of L-functions, and clearly this is true in general. A somewhat more complicated example of this is given below (The core idea is taken from [San01] p 7). CHAPTER 3. STARK’S CONJECTURE 30

Suppose K/k is a Galois extension of number fields with Galois group G isomorphic to S3, the symmetric group on three letters. Let τ be an element of G of order 2, let σ be an element of G of order 3, and let F be the subfield of K fixed by hτi. The group G has three irreducible characters; let φ be the non-trivial one-dimensional character, and ψ be the irreducible two-dimensional character (the character table is given below).

{1} {τ, τσ, τσ2} {σ, σ2}

1G 1 1 1 φ 1 −1 1 ψ 2 0 −1

∼ Let ϕ be the non-trivial character of Gal(K/F ) = hτi = Z/2Z. The identities

G G ψ = Indhτi1hτi − 1G and φ = Indhτiϕ − ψ show that Stark’s conjecture holds for all characters of G. Since all characters of a symmetric group take on rational values ([JK81] theorem 1.2.17, p 15), this also follows from the theorem proved by Stark. Let L be the subﬁeld of K ﬁxed by the elements of G of order 3, and let χ and χ be ∼ the non-trivial irreducible characters of H = Gal(K/L) = Z/3Z. Then

G G IndH χ = IndH χ = ψ, so Stark’s conjecture holds for the cubic extension K/L as well∗. However, there does not appear to be a proof of Stark’s conjecture for general cubic extensions.

∗In general, if K/k is a normal extension of number ﬁelds with Galois group G having a normal subgroup H, then Stark’s conjecture holds for K/KH and all characters of H if it holds for K/k and all characters of G. Likewise, Stark’s conjecture holds for KH /k and all characters of G/H if it holds for K/k and all characters of G. Chapter 4

Preliminaries to the Rubin-Stark conjecture

4.1 R[G]-modules

Let G be a ﬁnite abelian group, R a commutative unital ring.

4.1.1 R[G]-lattices

Definition 4.1.1. An R[G]-lattice is an R[G]-module whose underlying R-module is free on a finite number of generators. Let M be an R[G]-module, which we may also view as an R-module. Define M ∗ =

HomR[G](M,R[G]). We have a natural isomorphism of R[G]-modules

∼ HomR[G](M,R[G]) = HomR(M,R)

(see appendix A.1.1 for this isomorphism and the R[G]-module structure on HomR(M,R)). Suppose now that M is finitely generated∗ and that R is a principal ideal domain; by the structure theorem for finitely generated modules over principal ideal domains (see [Bly77] theorem 16.6, p 300), the underlying R-module of M is a direct sum of free and torsion. Thus the underlying R-module of M ∗ is free on a finite number of generators. Hence M ∗ is an R[G]-lattice, and we see that if M is an R[G]-lattice to begin with, M ∗∗ is naturally isomorphic to M. We prove the following homological algebra lemma (based on [Pop02] lemma 5.2.1, pp 15 - 16) for later reference

∗either as an R[G]-module or an R-module; G is ﬁnite so it makes no diﬀerence.

31 CHAPTER 4. PRELIMINARIES TO RUBIN’S CONJECTURE 32

Lemma 4.1.1. Let D be a Dedekind domain and M a ﬁnitely generated D[G]-module. n i) If M is D-torsion-free, then ExtD[G](M,D[G]) = 0 for all n ≥ 1. n ii) ExtD[G](M,D[G]) = 0 for all n ≥ 2.

Proof. Let U : D[G]-Mod → D-Mod be the forgetful functor. Since the functors

U(HomD[G]( ,D[G])) and HomD(U( ),D) (from D[G]-Mod to D-Mod) are naturally isomorphic, the right derived functors of U(HomD[G]( ,D[G])) and HomD(U( ),D) are too. If P is a projective D[G]-module, then U(P ) is a projective D-module†, so U applied to any projective resolution of M gives a projective resolution of U(M). Thus the right n derived functors of HomD(U( ),D) are ExtD(U( ),D) (up to natural isomorphism), and so there exist natural isomorphisms

n ∼ n U(ExtD[G]( ,D[G])) = ExtD(U( ),D).

Therefore it is enough to show that n i) If M is D-torsion-free, then ExtD(U(M),D) = 0 for all n ≥ 1. n ii) ExtD(U(M),D) = 0 for all n ≥ 2. Part i) follows from the fact that D-torsion-free modules are projective D-modules ([Coh03] p 373). To show that ii) holds, let

0 → F1 → F0 → M → 0 (4.1.1) be a short exact sequence of D-modules, where F1 and F0 are D-torsion free (choose

F0 to be free and let F1 be the kernel of F0 → M). The long exact sequence of Ext modules, together with i), shows that ii) holds.

Corollary 4.1.1. If 0 → A → B → C → 0 is an exact sequence of Z[G]-lattices, then 0 → C∗ → B∗ → A∗ → 0 is also exact.

4.1.2 Exterior powers V Let R be a commutative unital ring, M an R-module. We write R M for the exterior algebra of M. It has the structure of a -graded R-algebra V M =∼ L Vr M (we Z R r∈Z R Vr use the convention that R M = {0} if r < 0). If f : N → M is a homomorphism of V V V R-modules, there is a unique R-algebra homomorphism R f : R N → R M which V1 ∼ preserves grading and, when its domain and codomain are restricted to R N = N and † L 0 ∼ ` L 0 ∼ ` If P P = S D[G], then U(P ) U(P ) = S×G D. CHAPTER 4. PRELIMINARIES TO RUBIN’S CONJECTURE 33

V1 ∼ (r) Vr Vr R M = M respectively, coincides with f. We write f : R N → R M for the V restriction of R f. If N and M are both free of rank r with bases {n1, . . . , nr} and {m1, . . . , mr} re- (r) spectively, then f (n1 ∧...∧nr) = det(f)m1 ∧...∧mr, where the determinant is taken with respect to the given bases. One can show that r M M ^r (M N) ∼ ^r−iMN ^i N , (4.1.2) R = R R R i=0 and that if M is an S-module, A an S-algebra, then AN ^r M ∼ ^r (AN M) R R = A R

([Mat86] p 284). This isomorphism applied to the case where R = Z[G] and A = Q[G], together with remark 1.1.1, shows that

^r ∼ ^r Q M = QM. Z[G] Z[G] Vr Vr In general, if N is a submodule of M, R N will not be a submodule of R M (unless V2 V2 r = 0 or 1). For example, if I is an ideal of R, then R R = 0, but R I may not be zero if I is nonprincipal (see [Mat86] pp 283 - 284). However, in certain cases we will have inclusion. For example, if N is a direct summand of M, say M =∼ N L N 0, then

r r ! M M M ^r M ∼ ^r−iNN ^i N 0 ∼ ^r N ^r−iNN ^i N 0 . R = R R R = R R R R i=0 i=1 We will use the following proposition.

Proposition 4.1.1. Suppose N is a submodule of the G-module M. Then for every n ∈ , Vn N may be identiﬁed with a submodule of Vn M. N gZ[G] gZ[G] Proof. Vn N =∼ Vn N is a submodule of Vn M =∼ Vn M since N is Q Z[G] Q[G] Q Q Z[G] Q[G] Q Q a direct summand of M. But Vn N and Vn M are embedded in Vn N and Q gZ[G] gZ[G] Q Z[G] Vn M respectively, so Vn N → Vn M must be injective. Q Z[G] gZ[G] gZ[G]

4.1.3 Rubin’s modiﬁed exterior power

∗ V V For any r ∈ N, there is an R-linear map ι : M → HomR ( R M, R M) deﬁned by

t X j+1 ι(φ)(m1 ∧ ... ∧ mt) = (−1) φ(mj)·m1 ∧ ... ∧ mj−1 ∧ mj+1 ∧ ... ∧ mt j=1 CHAPTER 4. PRELIMINARIES TO RUBIN’S CONJECTURE 34

V on monomials and extended R-linearly to all of R M (we use the convention that a sum from j = 1 to j = 0 is 0). Observe that if f : M → M 0 is an R-module isomorphism, ι(φ ◦ f −1) f (t)(m) = f (t−1)(ι(φ)(m)). (4.1.3)

V ∗ V V We extend ι to a function ι : R M → HomR ( R M, R M) by deﬁning ι(φ1 ∧ ... ∧ ‡ Vr ∗ Vt φr) = ι(φ1)◦...◦ι(φr) . It is clear that for every r, t ∈ N, if φ ∈ R M and m ∈ R M, Vt−r then ι(φ)(m) ∈ R M. V0 One can show that under the identiﬁcation of R M with R,

ι(φ1 ∧ ... ∧ φr)(m1 ∧ ... ∧ mr) = det(φi(mj))

(cite) From now on we only consider the case where R = Z[G] and M is finitely generated. In [Rub96], Rubin defines the following modified exterior power: ∗ ^r M = ι ^r M ∗ . Z[G],0 Z[G] ∗ Let C be the cokernel of the inclusion ι Vr M ∗ → Vr M . By tensoring with Z[G] Z[G] and writing M = L e · M, we see that C = 0. Thus C is finite, and so C∗ = 0. C C χ∈Gb χ C C Therefore by lemma 4.1.1 there is an exact sequence

^ r ^r 1 0 → M → M → Ext [G](C, Z[G]) → 0. g Z[G] Z[G],0 Z Since C is ﬁnite, so is Ext1 (C, [G]). It follows that we may identify Vr M with Z[G] Z Z[G],0

n ^r ^r ∗o m ∈ Q M : ι(φ)(m) ∈ Z[G] ∀φ ∈ M . Z[G] Z[G]

Note that if φ ∈ Vt M ∗, then Z[G] ι(φ) ^r M ⊆ ^r−t M. Z[G],0 Z[G],0

4.2 Basic deﬁnitions

Definition 4.2.1. Let m be a modulus of a number field F . We say that α ∈ F × is × congruent to 1 mod m (in symbols, α ≡ 1 mod m), if ordv(α − 1) ≥ ordv(m) for all v ∈ supp(m), and if sgnF,m(α) is the identity of SgnF,m. Given a set of places S ⊃ S∞,F of a number field F , and a modulus m of F such that supp(m) ∩ S = ∅, we define the following:

‡Of course one needs to check that this is well-deﬁned. CHAPTER 4. PRELIMINARIES TO RUBIN’S CONJECTURE 35

m × • F = {x ∈ F : ordv(x) = 0 (∀v ∈ supp(m))}. Observe that this is a subgroup of F ×. m × × × • F1 = {x ∈ F : x ≡ 1 mod m}. Note that if x ≡ 1 mod m, then ordv(x) = ordv(1) = 0 for all v ∈ supp(m) (otherwise ordv(x − 1) = min{ordv(x), ordv(1)} ≤ m m 0). Thus F1 is a subgroup of F . m m§ • QF,m = F /F1 . We will give a more concrete description of QF,m in due course.

• IF,S,m is the free abelian group on the places of F not in S ∪ supp(m). There is a group homomorphism

m X divF,S,m : F1 → IF,S,m : x 7→ ordv(x)v. v6∈S∪supp(m)

m • AF,S,m = coker(divF,S,m) = IF,S,m/divF,S,m(F1 ) − the ‘S-ray class group modulo m’. One could also describe this as the group of fractional OF,S-ideals prime to the

non-archimedean part of m, modulo the subgroup of principal fractional OF,S-ideals with a generator congruent to 1 mod m. m m • UF,S,m = ker(divF,S,m) = {u ∈ F1 : ordv(u) = 0 (∀v 6∈ S)} = F1 ∩ UF,S. Thus there is an exact sequence

m 0 → UF,S,m → F1 → IF,S,m → AF,S,m → 0 (4.2.1)

m • µF,m is the torsion subgroup of UF,S,m, i.e. µF,m = µF ∩ UF,S,m = µF ∩ F1 . We also deﬁne wF,m = #µF,m. In appendix A.6, we show that there exists a long exact sequence

0 → UF,S,m → UF,S → QF,m → AF,S,m → AF,S → 0. (4.2.2)

From this, we see that

¶ [UF,S : UF,S,m]hF,S,m = #QF,mhF,S . (4.2.3)

Since λF,S,m is λF,S composed with the inclusion UF,S,m/µF,m ,→ UF,S/µF , the deﬁnition of RF,S,m gives

RF,S,m = [UF,S/µF : UF,S,m/µF,m]RF,S [UF,S : UF,S,m] [UF,S : UF,S,m]wF,m = RF,S = RF,S. (4.2.4) [µF : µF,m] wF

§ × This is non-standard notation. The notation (OK /m) is more commonly used. ¶ We will compute QF,m shortly, and show that it is ﬁnite. CHAPTER 4. PRELIMINARIES TO RUBIN’S CONJECTURE 36

Putting equations (2.3.3), (4.2.3) and (4.2.4) together gives

1−#S hF,S,mRF,S,m #QF,m lim s ζF,S(s) = − . (4.2.5) s→0 wF,m

For the rest of this dissertation, K/k will be a ﬁnite abelian extension of number ﬁelds with Galois group G, and all characters of G will be assumed to be one-dimensional.

Suppose S ⊇ S∞,k is a finite set of places of k containing all places which ramify in K/k, and suppose m is a modulus of k such that supp(m) ∩ S = ∅, and such that no real archimedean place dividing m is ramified in K/k. We then define AK,S,m = AK,SK ,mK ,

UK,S,m = UK,SK ,mK , QK,m = QK,mK , etc. (In general, we replace S and m by SK and mK ). All the groups deﬁned above have natural G-module structures, and all the group homomorphisms are G-homomorphisms. Putting F = k and F = K in the exact sequence 4.2.1 gives two sequences which ﬁt together to give the following commutative diagram:

m 0 / UK,S,m / K1 / IK,S,m / AK,S,m / 0 O O O O A iK/k iK/k

m 0 / Uk,S,m / k1 / Ik,S,m / Ak,S,m / 0 P The ﬁrst two vertical maps are the natural inclusions, iK/k sends v ∈ Ik,S,m to w|v w ∈ A IK,S,m, and iK/k is induced by the previous ones. We use the map iK/k : Ik,S,m → IK,S,m to identify Ik,S,m with a submodule of IK,S,m. We also have

m 0 / UK,S,m / K1 / IK,S,m / AK,S,m / 0

N I A NK/k K/k NK/k NK/k

m 0 / Uk,S,m / k1 / Ik,S,m / Ak,S,m / 0

× × I The two ﬁrst two vertical maps are restrictions of NK/k : K → k , NK/k sends P 0 A w ∈ IK,S,m to NG · w = #Dw w0|v w = #Dwi(v) (where v = w|k), and NK/k is induced by the others.

4.2.1 The Z[G]-module structure of QK,m

The following proposition shows that the calculation of QK,m can be reduced to the case where m is the power of a single place. CHAPTER 4. PRELIMINARIES TO RUBIN’S CONJECTURE 37

k ∼ Proposition 4.2.1. If m and n are relatively prime moduli of k, then QK,mn = L QK,m QK,n L Proof. There is an obvious map QK,mn → QK,m QK,n, which is injective since x ≡ × × × 1 mod mK and x ≡ 1 mod nK imply x ≡ 1 mod mK nK . We now show that the map is onto. Let a and b be representatives of elements of

QK,m and QK,n respectively. By theorem A.3.1, we can ﬁnd x ∈ K such that:

• ordw(x − b/a) ≥ ordw(nK ) − ordw(a) for all w ∈ supp(nK ),

• ordw(x − 1) ≥ ordw(mK ) for all w ∈ supp(mK ),

• sgnK,n(x) = sgnK,n(a/b),

• sgnK,m(x) is the identity element of SgnK,m.

For all w ∈ supp(nK ),

ordw(ax/b − 1) = ordw(x − b/a) + ordw(a) − ordw(b) ≥ ordw(nK ),

n since ordw(b) = 0. Also, sgnK,n(ax/b) is the identity of SgnK,n, so y := ax/b ∈ K1 . By m mK nK mK nK construction, x ∈ K1 . Therefore ax = by ∈ K ∩ K = K , and it is clear that the equivalence class of ax = by gets mapped to the equivalence classes of a and b in

QK,m and QK,n respectively.

n + Suppose now that m = v , where v is a place of k and n ∈ N . We consider two cases:

Case 1: v archimedean

m × m In this case K = K . The natural map K → SgnK,m is onto by proposition A.3.1, m ∼ ∼ L and has kernel K1 . Thus QK,m = SgnK,m = w|v Z/2Z as an abelian group, and if L ∼ ∼ we consider the natural action of G on w|v Z/2Z, we see that QK,m = (Z/2Z)[G] = Z[G]/2Z[G] as a G-module.

Case 2: v non-archimedean

m First we show that every element of QK,m has a representative in OK . Given x ∈ K , the approximation theorem A.3.1 implies that we can ﬁnd y ∈ K× such that

• ordw(y − 1) ≥ ordw(m) for all w ∈ supp(mK ),

• ordw(y) ≥ −ordw(x) for all non-archimedean w 6∈ supp(mK ),

kWe say that two moduli are relatively prime if no place divides both. CHAPTER 4. PRELIMINARIES TO RUBIN’S CONJECTURE 38

• sgnK,m(y) is the identity of SgnK,m. m Then y ∈ K1 and xy ∈ OK . Let p be the ideal of Ok corresponding to v. Because every element of QK,m has a m representative in OK , QK,m can be described as the multiplicative monoid OK ∩ K = T P|p(OK r P) modulo the congruence

x ∼ y ⇔ ordw(x/y − 1) ≥ n for all w|v

⇔ ordw(x − y) ≥ n for all w|v (since ordw(y) = 0) n ⇔ x − y ∈ p OK .

n × T ∼ Any representative of an element of (OK /p OK ) must lie in P|p(OK rP), so QK,m = n × (OK /p OK ) .

4.2.2 Fitting ideals

Let R be a commutative unital ring, and let M be a ﬁnitely generated R-module. Choose a free presentation

f F1 / F0 / M / 0 , (4.2.6) where F0 has finite rank, equal to to n. The first Fitting ideal FittR(M) is defined to be the image of f (n) : ^n F → ^n F ∼ R. R 1 R 0 = It is clear that this is an ideal of R, and independent of the choice of isomorphism Vn ∼ R F0 = R, but it takes more work to show that FittR(M) does not depend on the choice of the presentation (see [Nor] theorem 1, p 58). One can define other Fitting ideals, but we will not make use of them and will thus refer to the first Fitting ideal as simply the Fitting ideal. Some properties of Fitting ideals can be found in appendix A.2. Fitting ideals of a number of Galois module occur in connection with the Rubin-Stark conjecture, and it appears that the Fitting ideal of QK,m plays a part. In appendix A.5.1 we show that

( −1 n−1 (1 − σv Nv)Nv if v is non-archimedean δK,vn := 2 if v is archimedean is a generator of Fitt (Q n ). Z[G] K,v Q nv + To deﬁne δK,m in general, write m = v|m av, where av = v , nv ∈ N . We then Q deﬁne δK,m = v|m δK,av , and the direct sum decomposition of QK,m, together with CHAPTER 4. PRELIMINARIES TO RUBIN’S CONJECTURE 39 the fact that Fitt is multiplicative on direct sums (see appendix A.2), shows that Z[G] Fitt (Q ) is generated by δ . Z[G] K,m K,m

Remark 4.2.1. By checking cases, one sees that |augG(δK,m)| = #Qk,m. One could also prove this using the fact that QK,m is cohomologically trivial (proposition A.5.1), and that for a cohomologically trivial G-module A, π Fitt (A) = Fitt (AH ), Z[G] Z[G/H] where H is a normal subgroup of G and π : Z[G] → Z[G/H] is the ring homomorphism G ∼ coming from the quotient map G → G/H (take H = G, and observe that QK,m = Qk,m, as one sees by checking cases or by using lemma A.6.2).

4.3 The Stickelberger function

Let S be a finite set of places of k containing all archimedean places. We define the S-modified Stickelberger function by X ΘS(s; K/k) = LS(s, χ¯)eχ ∈ C[G](s ∈ C r {1}). (4.3.1) χ∈Gb

If S also contains all places which ramify in K/k, there is another way of deﬁning

ΘS(s; K/k):

Proposition 4.3.1. ([Tat84] proposition 1.6, p 86) For s ∈ C with <(s) > 1,

Y −s −1−1 ∗∗ ΘS(s; K/k) = 1 − Nv σv . (4.3.2) v∈ /S

Proof. For every χ ∈ Gb, extended by linearity to C[G],

Y −s −1 −1 χ(ΘS(s)) = LS(s, χ) = det 1 − Nv σv |V v∈ /S ! Y −s −1−1 Y −s −1 −1 = χ 1 − Nv σv = χ (1 − Nv σv ) . v∈ /S v∈ /S

The last equality comes from the fact that χ : C[G] → C is continuous, as one can easily verify.

Let H be a subgroup of G = Gal(K/k), with L the ﬁxed ﬁeld of H. Let π : C[G] → C[G/H] be the C-algebra homomorphism induced by the quotient map G → G/H.

q ∗∗ P P 2 C[G] with the norm σ∈G nσσ = g σ∈G |nσ| is a Banach algebra. CHAPTER 4. PRELIMINARIES TO RUBIN’S CONJECTURE 40

Proposition 4.3.2. ([Tat84] proposition 1.8, p 87) With the notation as above,

π (ΘS(s; K/k)) = ΘS(s; L/k)

Proof. For every χ ∈ G/H[ ,

χ (π (ΘS(s; K/k))) = LS (s, Inﬂχ; K/k) = LS(s, χ; L/k) = χ (ΘS(s; L/k)) ,

0 and so π (ΘS(s; K/k)) = ΘS(s; k /k).

The S-modiﬁed Stickelberger function ΘS(s) extends to a meromorphic function on †† C which is analytic at s = 0 . Deﬁne

rK/k,S = ords=0ΘS(s; K/k) = min{rK/k,S(χ): χ ∈ Gb}, and for r = 0, 1, . . . , rK/k,S, deﬁne r (r) −r 1 d ΘK/k,S = lim s ΘS(s; K/k) = r ΘS(s; K/k) ∈ C[G], s→0 r! ds s=0 (r) (r) ‡‡ and ΘK/k,S,m = δK,mΘK/k,S . We shall usually omit the subscript K/k when there is no confusion as to the extension concerned.

Remark 4.3.1. We could define ΘS,m(s) by defining an analytic C[G]-valued function δm(s) which is equal to δm ∈ C[G] (as we have defined it) at s = 0. When m is the product of distinct non-archimedean places, which is essentially the case considered by Rubin in Q −1 1−s [Rub96], one defines δm(s) = v|m(1 − σv Nv ). This ensures that ΘS,m(−n) ∈ Z[G] for all n ∈ N, thanks to a theorem of Deligne and Ribet (see [DR80]). However, since we are only interested in the value of ΘS(s) at s = 0, we will not be concerned with extending the definition of δm(s) to the case where m is not the product of distinct non-archimedean places.

A ﬁnal deﬁnition: Let λK,S,m : RUK,S,m → RXK,S be the restriction of λK,S, and for any r ∈ N, let (r) ^r ∼ ^r ^r ∼ ^r λ : R UK,S,m = RUK,S,m → RXK,S = R XK,S K,S,m Z[G] R[G] R[G] Z[G] be the isomorphism induced by λK,S,m.

†† g We may identify the underlying topological space of C[G] with C ; the terms meromorphic and analytic are used in this sense. ‡‡ (r) (r) Of course, ΘK/k,S,m = ΘK/k,S = 0 if r < rK/k,S . It should be noted that in most (r) (r) of the relevent literature, ΘK/k,S is usually denoted by ΘK/k,S (0). Since we will not be (r) concerned with the value of ΘK/k,S (s) at any point other than s = 0, we have opted to simplify the notation by omitting the (0). Chapter 5

Rubin’s reﬁnement of Stark’s conjecture in the abelian case

In [Sta80], Stark made some refined conjectures about the first derivatives of Artin L-functions in the case where the extension K/k was abelian and all L-functions associated to irreducible characters of the Galois group had order of vanishing at least 1. In cases where the order of vanishing of all L-functions is strictly greater than 1, the first derivatives are zero, and these conjectures are trivial. In [Rub96], Rubin extended Stark’s refined conjectures for abelian extensions to give not-necessarily-trivial statements about higher-order derivatives of the corresponding L-functions.

5.1 Statement of Rubin’s conjecture

We will in fact state a slightly more general version of Rubin’s integral refinement of Stark’s conjecture, although, as we shall see, this follows from Rubin’s original conjecture in most cases. Consider the following hypotheses: Hypotheses H(K/k, S, m, r): i) S is a finite set of places of k containing all archimedean places and all places ramified in K/k. ii) m is a modulus of k, such that supp(m)∩S = ∅, and such that no real archimedean place dividing m is ramified in K/k.

iii) UK,S,m is torsion-free. iv) At least r places of S split completely in K/k.

41 CHAPTER 5. THE RUBIN-STARK CONJECTURE 42

v) r ≤ #S − 1.

Remark 5.1.1. Condition iii) is easily satisﬁed. For example, if there is an archimedean place dividing m, then K has a real archimedean place, whence µK = {1, −1} and

µF,m = {1}.

Alternatively, if ζ is an non-trivial root of unity in UK,S,m,

N (1 − ζ) = N N (1 − ζ) = N (1 − ζ)[K:Q(ζ)] , K/Q Q(ζ)/Q K/Q(ζ) Q(ζ)/Q and since ( p n is a power of p N (ζ)/ (1 − ζ) = (5.1.1) Q Q 1 otherwise

(see [Was82] for example), one simply needs to choose m = v, a non-archimedean place where the characteristic of F(v) is prime to wK , or choose m = v1v2 where F(v1) and F(v2) have diﬀerent characteristics. Even if the characteristic of F(v) is not prime to ` wK , one could put m = v , where ` is chosen suﬃciently large.

At times we will work with a subset of the five conditions above, and we write H(K/k, S) for i), H(K/k, S, m) for i) − iii), and H(K/k, S, r) for i), iv) and v) of the hypotheses above. (r) Note that by remark 2.3.1, iv) and v) imply r ≤ rK/k,S, so ΘS,m is well-defined. An (apparently) slightly stronger version of the conjecture put forward by Rubin ([Rub96] conjecture B, p 39) is: Conjecture 5.1.1. RS(K/k, S, m, r) If hypotheses H(K/k, S, m, r) are satisfied, then

(r) ^ r (r) ^r Θ · XK,S ⊆ λ UK,S,m (5.1.2) K/k,S,m g Z[G] K,S,m Z[G],0

We will refer to this as the r-th order Rubin-Stark conjecture. Note that because δ generates Fitt (Q ), 5.1.2 is equivalent to K,m Z[G] K,m

(r) ^ r (r) ^r Fitt [G](QK,m)·Θ · XK,S ⊆ λ UK,S,m . Z K/k,S g Z[G] K,S,m Z[G],0

Remark 5.1.2. The conjecture proposed by Rubin in [Rub96] amounts to RS(K/k, S, m, r) with the extra condition that m be a product of distinct non-archimedean places.

Remark 5.1.3. It may happen that, given S and m as above, the maximum r for which hypotheses H(K/k, S, m, r) are satisﬁed is strictly less than rK/k,S, in which case Rubin’s CHAPTER 5. THE RUBIN-STARK CONJECTURE 43 conjecture is trivially true. One may hope to generalise RS(K/k, S, m, r) by conjecturing that if hypotheses H(K/k, S, m) are satisﬁed, then 5.1.2 holds with r = rK/k,S. However, according to Popescu ([Pop]), this is false. Instead, Popescu conjectures that a certain cyclic Z[G]-submodule of (r) ^ r Θ · XK,S, r = rK/k,S K/k,S,m g Z[G] is contained in the right-hand side of 5.1.2∗. This submodule is generated by a certain Vr Vr element wS ∈ XK,S, which is a free generator of eS· XK,S as an eS· [G]- min gZ[G] C Z[G] C module†.

5.1.1 An equivalent formulation

It is often more convenient to work with another version of Rubin’s conjecture, which we derive in this subsection.

If hypotheses H(K/k, S, m, r) are satisﬁed, let ΞS,r be the set of all characters χ ∈ Gb such that rS(χ) = r. Deﬁne X eS,r = eχ,

χ∈ΞS,r (r) (r) and note that eS,rΘS = ΘS . Lemma 5.1.1. ([Rub96] lemma2.6 (ii), pp 41 - 42) Suppose hypotheses H(K/k, S, r) are satisﬁed, and let {v1, . . . , vr} be a set of r places of S which split completely. If w is a place of SK not dividing any vi, then ( 0 if 1G 6∈ ΞS,r eS,r ·w = e1G ·w if 1G ∈ ΞS,r

Proof. If χ ∈ ΞS,r r {e1G }, then by remark 2.3.1, χ|Dw 6= 1Dw . Thus 1 X e ·w = χ¯(σ)σ·w χ g σ∈G 1 X X = χ¯(σµ)σµ·w g µ∈Dw σDw∈G/Dw 1 X X = χ¯(µ) χ¯(σ)σ·w g µ∈Dw σDw∈G/Dw = 0. ∗This is under the assumption that m is the product of distinct non-archimedean places, although this assumption does not appear to be necessary. † Here eS is the sum of the idempotents eχ associated to those characters χ ∈ Gb for

which rS (χ) = rK/k,S . The deﬁnition of wSmin can be found in [Emm06], while the proof that it is a free generator will appear in a forthcoming paper by Popescu. CHAPTER 5. THE RUBIN-STARK CONJECTURE 44

Let W = (w0, w1, . . . , wr) be an (r + 1)-tuple of places of SK , no two of which lie over the same place of k, and such that w1, . . . , wr split completely in K/k. Throughout the rest of this dissertation, whenever the symbol W reappears, it will be defined in this Vr way. Define xK,S,W to be the image of (w1 − w0) ∧ ... ∧ (wr − w0) in XK,S. gZ[G] Lemma 5.1.2. ([Rub96] lemma 2.6 (ii), pp 41 - 42) Under hypotheses H(K/k, S, m, r), (r) Vr Vr (r) the [G]-module Θ · XS ⊆ XS is cyclic, generated by Θ ·xS,W . Z S,m gZ[G] C Z[G] S,m Vr Proof. It will be sufficient to show that eS,r · XS is generated by eS,r ·xS,W . Put gZ[G] vi = wi|k for i = 1, . . . , r, and let T = S r {v1, . . . , vr}. Every x ∈ XS can be written as r X X x = αi ·(wi − w0) + βv ·v,¯ i=1 v∈T P P for some αi, βv ∈ Z[G]. Observe that v∈T augG(βv) = augK v∈T βv ·v¯ = 0. P Suppose 1G ∈ ΞS,r. Then #S = r + 1 and so the sum v∈T βv ·v¯ consists of only one term, say βv ·v¯. By lemma 5.1.1 and the observation above,

eS,r ·(βv ·v¯) = βve1G ·v¯ = augG(βv)e1G ·v¯ = 0. Vr Therefore if x1 ∧ ... ∧ xr is a monomial in XS, Z[G]

eS,r ·(x1 ∧ ... ∧ xr) = (eS,r ·x1) ∧ ... ∧ (eS,r ·xr) = α·(w1 − w0) ∧ ... ∧ (wr − w0) for some α ∈ Z[G]. Vr Now suppose 1G 6∈ ΞS,r. Every monomial x1 ∧ ... ∧ xr ∈ XS can be written in Z[G] the form X α·(w1 − w0) ∧ ... ∧ (wr − w0) + αy ·y, y where each y is a monomial containing at least one element of TK . By lemma 5.1.1, eS,r ·y = 0.

Suppose hypotheses H(K/k, S, m, r) are satisﬁed, and let W be as before. Deﬁne Vr εK,S,W ∈ UK,S by C Z[G] (r) −1 (r) εK,S,W = ΘK/k,S ·(λK,S) (xK,S,W ). Observe that

(r) ^ r (r) ^r Θ · XK,S ⊆ λ UK,S,m K/k,S,m g Z[G] K,S,m Z[G],0 (r) (r) ^r ⇔ Θ ·xK,S,W ∈ λ UK,S,m K/k,S,m S,m Z[G],0 (r) −1 (r) ^r ⇔ δK,m ·εK,S,W = Θ ·(λ ) (xK,S,W ) ∈ UK,S,m, K/k,S,m S,m Z[G],0 CHAPTER 5. THE RUBIN-STARK CONJECTURE 45 and so we have the following formulation of Rubin’s conjecture: Conjecture 5.1.2. RS∗(K/k, S, m, r) If hypotheses H(K/k, S, m, r) are satisﬁed, then

^r δK,m ·εK,S,W ∈ UK,S,m. Z[G],0 Note that this is independent of the choice of W . (r) Vr Lemma 5.1.2 shows that Θ · XS is a cyclic [G]-module, and so it is a cyclic S,m gZ[G] Z eS,rZ[G]-module. It turns out that more is true. The set SK is by definition a Z-basis ◦ for the free abelian group YK,S. This gives us a dual basis Z-basis {w : w ∈ SK } for ∗ ◦ HomZ(YK,S, Z), and we define w to be the image of w under the natural isomorphism Hom (Y , ) → Hom (Y , [G]) (see equation A.1.3). In other words, Z K,S Z Z[G] K,S Z X w∗(w0) = w◦(σ·w0)σ−1. σ∈G ∗ ∗ ∗ ∗ Vr Vr Let W be as above. Define φK,S,W = ι(w ∧...∧w ) ∈ YK,S = YK,S . 1 r Z[G] gZ[G] Vr Then φK,S,W (xK,S,W ) = 1 (we may regard xK,S,W as an element of YK,S by propo- gZ[G] sition 4.1.1). Thus, for any α ∈ Z[G], α · xK,S,W = 0 ⇒ α = 0. This shows that (r) Vr Θ · XS is a free rank 1 eS,r [G]-module. S,m gZ[G] Z Remark 5.1.4. Rubin and other authors define a Z[G]-lattice

n ^r o ΛS,m = u ∈ US,m : eS,r ·u = u , Z[G],0 (r) and a so-called regulator map: ηW := φK,S,W ◦ λK,S : CΛS,m → eS,rC[G], which can be shown to be an isomorphism of C[G]-modules. One can show that εK,S,W,m := (r) δK,m ·εK,S,W is the unique element of CΛS,m that gets mapped to ΘS,m ∈ eS,rC[G] by ηW , and an equivalent formulation of the Rubin-Stark conjecture would be to conjecture that εK,S,W,m ∈ ΛS,m (we identify ΛS,m with its image in CΛS,m).

5.2 Relation to Stark’s principal conjecture

(r) Vr Let RS(K/k, S, m, r) be RS(K/k, S, m, r) with λ UK,S,m replaced by Q K,S,m Z[G],0 (r) Vr λ UK,S,m in equation 5.1.2. Q K,S,m Z[G],0 The following proposition shows how Stark’s principal conjecture is related to Ru- bin’s reﬁnement.

Proposition 5.2.1. ([Rub96] proposition 2.3, p 41) Let K/k be an abelian extension with Galois group G. Under hypotheses H(K/k, S, m, r), conjecture QRS(K/k, S, m, r) is true if and only if Stark’s principal conjecture is true for all χ ∈ ΞS,r. CHAPTER 5. THE RUBIN-STARK CONJECTURE 46

Vr Vr Proof. Firstly, note that since US,m is of ﬁnite index in US,m and US,m is of gZ[G] Z[G],0 ﬁnite index in US,

(r) ^r (r) ^r Qλ US,m = Qλ US . S,m Z[G],0 S Z[G] (r) (r) Since ΘS,m is equal to ΘS multiplied by an invertible element of Q[G], QRS(K/k, S, m, r) is equivalent to (r) (r) ^r Θ ·xS,W ∈ Qλ US . S S Z[G] Let f : QXS → QUS be a Q[G]-isomorphism. Since every χ ∈ Gb is one-dimensional, R(χ, f) = det(λS ◦ f)χ (see remark 2.3.4). Deﬁne

−r aS(χ, f) = lim s LS(s, χ)/RS(χ, f) s→0 ( A (χ, f)−1 if χ ∈ Ξ = S S,r 0 if χ∈ / ΞS,r ( c (χ)/ det(λ ◦ f) if χ ∈ Ξ = S S χ S,r , 0 if χ∈ / ΞS,r and put X X θ = aS(χ, f)eχ¯ = aS(χ, f)eχ¯ ∈ C[G]. χ∈Gb χ∈ΞS,r Any α ∈ Gal(C/Q) acts on C[G] in the obvious way, and one sees immediately that α eχ¯ = eχ¯α = eχ¯α . Note that

Stark’s conjecture is true for all χ ∈ Ξr α α ⇔ aS(χ, f) = aS(χ , f)(∀χ ∈ Ξr)(∀α ∈ Gal(C/Q)) α X α α X α ⇔ θ = aS(χ, f) eχ¯ = aS(χ , f)eχ¯α = θ (∀α ∈ Gal(C/Q)) χ∈ΞS,r χ∈ΞS,r ⇔ θ ∈ Q[G].

If χ ∈ ΞS,r, then r = rS(χ) = dim(eχ¯ ·CXS) and {eχ¯ ·(w1 − w0), . . . , eχ¯ ·(wr − w0)} is a (r) basis for eχ¯ ·CXS. Therefore (λS ◦ f) (eχ¯ ·xS,W ) = det(λS ◦ f)χeχ¯ ·xS,W , and so (r) (r) X (r) λS ◦ f (θ·xS,W ) = θ (λS ◦ f) (eχ¯ ·xS,W ) χ∈Gb     X X =  a(χ, f)eχ¯  det(λS ◦ f)χeχ¯ ·xS,W  χ∈Gb χ∈Gb X −r = lim s LS(s, χ)eχ¯ ·xS,W s→0 χ∈Gb (r) = ΘS ·xS,W . (5.2.1) CHAPTER 5. THE RUBIN-STARK CONJECTURE 47

Suppose QRS(K/k, S, m, r) holds true. Then

(r) (r) (r) (r) ^ r λ ◦ f (θ·xS,W ) = Θ ·xS,W ∈ Qλ US . S S S g Z[G]

(r) Therefore, since λS is injective,

(r) −1 ^ r ^r θ·xS,W ∈ Q(f ) US = Q XS, g Z[G] Z[G]

Vr so θ = φS,W (θ·xS,W ) ∈ φS,W XS ⊆ [G]. Q Z[G] Q (r) (r) (r) Conversely, if θ ∈ Q[G], then by equation 5.2.1, ΘS ·xS,W = λS ◦ f (θxS,W ) ∈ (r) Vr λ US , and so RS(K/k, S, m, r) is true. Q S gZ[G] Q Remark 5.2.1. One seemingly unsatisfactory thing about this proposition is that the truth of Rubin’s conjecture for the set of data (K/k, S, r) only implies the validity of

Stark’s conjecture for characters χ satisfying rS(χ) = r. However, if the Rubin-Stark conjecture is true in general, then Stark’s conjecture is true in general. We have shown that the principal Stark conjecture is true if it is true for all faithful characters, so let χ be a faithful character (of the Galois group of an abelian extension of number ﬁelds).

Since Stark’s conjecture is true for the trivial character, we may assume χ 6= 1G. Thus

rS(χ) = #{v ∈ S : χ|Dv = 1Dv } = #{v ∈ S : v splits completely in K/k}

Now let S be the set of archimedean places, those ramified in K/k, and some ad- ‡ ditional place which does not split completely in K/k , so that #S ≥ rS(χ) + 1. It is easy to find m such that hypotheses H(K/k, S, m, rS(χ)) are satisfied (see remark 5.1.1), and by the previous proposition, Stark’s principal conjecture is true for χ if

QRS(K/k, S, m, rS(χ)) is true.

5.3 Changing S

We wish to look at how increasing the set S affects the truth of the conjecture. It will suffice to consider the effect of adding a single place to S, so suppose v is place of k not in S, and put S0 = S ∪ {v}. In this section, K and m will be fixed, so we omit them from the subscripts.

‡ Our assumption that χ 6= 1G implies that K 6= k, so by the Cebotarevˇ density theorem (see [Nar90] theorem 7.11, p 382, for example) it is possible to ﬁnd a non-archimedean, unramiﬁed place which does not split completely. CHAPTER 5. THE RUBIN-STARK CONJECTURE 48

Proposition 5.3.1. ([Rub96] proposition 3.6, p 49) If hypotheses H(K/k, S, m, r) are satisﬁed, then with the notation as above,

RS(K/k, S, m, r) ⇒ RS(K/k, S0, m, r).

Vr Proof. Suppose RS(K/k, S, m, r) holds. If we identify XS with a submodule of gZ[G] Vr X 0 (which we may do by proposition 4.1.1), then x 0 = xS,W . Consequently gZ[G] S S ,W (r) (r) ΘS0 ·xS0,W = (1 − σv)ΘS ·xS,W (r) ^r (r) ^r ∈ λ US ⊆ λ 0 US0 , S Z[G],0 S Z[G],0 and we are done by lemma 5.1.2.

If v splits completely in K/k, then the order of vanishing of each L-function increases by one, so RS(K/k, S0, m, r) is trivially true. One might hope to show that in this case RS(K/k, S, m, r) implies RS(K/k, S0, m, r + 1), but it appears that stronger hypotheses are necessary for this to be true (see [Rub96], theorem 5.3, (iii)). However, the reverse implication does hold, namely

Proposition 5.3.2. ([Rub96] theorem 5.3 (i)) With the notation as above, if v splits completely, then RS(K/k, S0, m, r + 1) ⇒ RS(K/k, S, m, r).

∗ ∗ Proof. Recall the deﬁnition of w ∈ YS for w ∈ SK given in subsection 5.1.1. Let γ = (log Nv)−1, and deﬁne

∗ Φv¯ = γ v¯ ◦ λS0 : RUS0 → R[G].

For every u ∈ US0 ,   ∗ X X ∗ X ∗ Φv¯(˜u) = γ v¯  log |u|ww = γ log |u|wv¯ (w) = − ordw(u)¯v (w) ∈ Z[G], 0 w∈SK w|v w|v ∗ so we may regard restrict the domain and codomain of Φv¯ to obtain an element of US0 . 0 Deﬁne W = (w0, w1, . . . , wr, v¯), and note that if xS,W = xe, then xS0,W 0 is equal to the Vr+1 ∗ § image of x ∧ (¯v − w0) in XS0 . Therefore ι(¯v )(xS0,W 0 ) = ±xS,W , and so gZ[G]

∗ −1(r+1) −1(r) ∗ ι(¯v ◦ λS0 ) λS0 (xS0,W 0 ) = λS0 ι(¯v )(xS0,W 0 ) (see eqn 4.1.3) −1(r) = ± λS0 (xS,W ) −1(r) = ± λS (xS,W ) . §the sign depends on the parity of r, but is irrelevant for our purposes. CHAPTER 5. THE RUBIN-STARK CONJECTURE 49

(r+1) (r) −1 (r) Thus, since ΘS0 = log NvΘS = γ ΘS , (r) −1(r) εS,W = ΘS · λS (xS,W ) (r+1) ∗ −1(r+1) = ±γΘS0 ·ι(¯v ◦ λS0 ) λS0 (xS0,W 0 ) ∗ (r+1) −1(r+1) = ±ι(γ v¯ ◦ λS0 ) ΘS0 · λS0 (xS0,W 0 )

= ±ι(Φv¯)(εS0,W 0 ). Vr+1 Vr If RS(K/k, S, m, r + 1) holds, εS0,W 0 ∈ US0 and so εS,W ∈ US0 . It is not Z[G],0 Z[G],0 ∗ ∗ hard to show that US0 /US is torsion-free, so corollary 4.1.1 implies that US0 → US is onto. Therefore, given φ ∈ Vr U ∗, we can ﬁnd φ0 ∈ Vr U ∗ such that Z[G] S Z[G] S0 0 ι(φ)(εS,W ) = ι(φ )(εS,W ) ∈ Z[G]. Vr Thus εS,W ∈ US. Z[G],0

5.4 Changing m

Since K and S will be ﬁxed in this section, we omit them from subscripts in most cases, and write Qm = QK,m, U = UK,S, Um = UK,S,m, etc. We will show that it is suﬃcient to prove the Rubin-Stark conjecture for minimal moduli m; in other words,

Proposition 5.4.1. Suppose m|m0, and that hypotheses H(K/k, S, m, r) and H(K/k, S, m0, r) are satisﬁed. Then RS(K/k, S, m, r) ⇒ RS(K/k, S, m0, r)

Proof. (basic ideas taken from [Pop02] proposition 5.3.1, pp 17 - 18). It will be suﬃcient to show this under the assumption m0 = vm, v a place of k. m Let ∆m0 be the kernel of the canonical homomorphism Qm0 → Qm. By looking at the m direct sum decomposition of Qm, one sees that ∆m0 is the kernel of the homomorphism Qvn+1 → Qvn , where n is the largest power of v dividing m (which may be 0). If n = 0, m ∼ × ∼ ¶ then ∆m0 = Qv, which is either Z[G]/2Z[G] or (OK /pOK ) = Z[G]/δvZ[G] , depending on whether v is archimedean or non-archimedean. If n > 0, then v is non-archimedean m and ∆m0 is the kernel of n+1 × n × OK /p OK → (OK /p OK )

(p the prime ideal of Ok corresponding to v), and this can be shown to be isomorphic to Lf i=1 Z[G]/pZ[G], where p is the characteristic of F(v) and f = f(p/pZ) (see equation 1 m ∼ m A.5.3). Observe that in each case, Ext (∆ 0 , [G]) = ∆ 0 . Z[G] m Z m ¶See appendix A.5.1. CHAPTER 5. THE RUBIN-STARK CONJECTURE 50

Let Im and Im0 be the images of U in Qm and Qm0 respectively (under the map in the long exact sequence 4.2.2), so that we have a commutative diagram with exact rows:

0 / Um0 / U / Im0 / 0 = 0 / Um / U / Im / 0 Deﬁne m ∼ m Im0 = Um/Um0 = ker (Im0 → Im) ⊆ ∆m0 . m m ∗ Since ∆m0 is ﬁnite, Im0 = 0. By lemma 4.1.1, we have the exact sequences

∗ ∗ 1 m 0 → U → U 0 → Ext (I 0 , [G]) → 0, m m Z[G] m Z 1 m 1 m Ext (∆ 0 , [G]) → Ext (I 0 , [G]) → 0. Z[G] m Z Z[G] m Z ∗ ∗ 1 m ∼ m Thus U 0 /U may be identiﬁed with a quotient module of Ext (∆ 0 , [G]) = ∆ 0 , m m Z[G] m Z m and so m ∗ ∗ Fitt (∆ 0 ) ⊆ Fitt (U 0 /U ) Z[G] m Z[G] m m (see proposition A.2.1). Lemma A.2.2 applied to the short exact sequence

∗ ∗ ∗ ∗ 0 → Um → Um0 → Um0 /Um → 0 gives

m ^ r ∗ ^ r ∗ Fitt [G](∆m0 )· Um0 ⊆ Um. (5.4.1) Z g Z[G] g Z[G] m m In appendix A.5.1, we show that Fitt (∆ 0 )Fitt = Fitt , so Fitt (∆ 0 ) Z[G] m Z[G](Qm) Z[G](Qm0 ) Z[G] m is generated by δm0 /δm. Therefore 5.4.1 is equivalent to

m ^ r ∗ ^ r ∗ δm0 · Um0 ⊆ Um. g Z[G] g Z[G] Thus for any φ ∈ Vr U ∗ , we have δm ·φ ∈ Vr U ∗ , and so gZ[G] m0 m0 gZ[G] m m ι(φ)(δm0 ·εW ) = ι(δm0 ·φ)(δm ·εW ) ∈ Z[G].

Remark 5.4.1. This shows that the only cases where conjecture RS(K/k, S, m, r) might not follow from Rubin’s original conjecture is where an archimedean place v divides m and m/v, or where vn (n > 1) divides m. Remark 5.1.1 implies that if the original Rubin-Stark conjecture holds, one need only check ﬁnitely many values of m to determine whether RS(K/k, S, m, r) holds for all m. CHAPTER 5. THE RUBIN-STARK CONJECTURE 51

5.5 Subextensions

We will show that the truth of the r-th order Rubin-Stark conjecture for K/k implies its truth for all normal subextensions. Let H be a subgroup of G, and put L = KH . For the rest of this subsection, we omit the subscripts S and m, since these will be

ﬁxed. Thus UK = UK,S,m, UL = UL,S,m, etc. We will need the following lemma for the next proposition and the proof of the Rubin-Stark conjecture for quadratic extensions.

Lemma 5.5.1. ([Hay04] lemma 3.1, p 105)If hypotheses H(K/k, S, m, r) are satisﬁed, then UK /UL is torsion-free.

n Proof. Let u ∈ UK be a representative of a torsion element of UK /UL. Then u ∈ UL for some non-zero integer n. But then for every σ ∈ H,(uσ−1)n = (un)σ−1 = 1. Thus σ−1 σ−1 u is a torsion element of UK , so u = 1, and as this is true of all σ ∈ H, we have u ∈ UL.

Proposition 5.5.1. With K, L and k as above, if hypotheses H(K/k, S, m, r) are sat- isﬁed, then RS(K/k, S, m, r) ⇒ RS(L/k, S, m, r).

Proof. ([Hay04, Tat84]) It is clear that H(K/k, S, m, r)⇒ H(L/k, S, m, r). Let Γ = G/H. By proposition 4.3.2, if π : Z[G] → Z[Γ] is the ring homomorphism coming from the (r) (r) quotient map G → G/H, then π ΘK/k = ΘL/k. Note that we may identify the Vr Vr G-modules UL and UL, if we give the latter a G-module structure using Z[G] Z[Γ] Vr Vr π. The same goes for XL and XL. If W = (w0, w1, . . . , wr), let W |L = Z[G] Z[Γ] (w0|L, w1|L, . . . , wr|L). We will use the ad-hoc notation restrL to denote the map XK → (r) (r) XL : x 7→ x|L. Since ΘK/k ·xK,W = λK (εK,W ), the commutative diagram 2.3.5 shows that Θ(r) ·x = π Θ(r) ·x L/k K,W |L K/k K,W |L (r) (r) (r) (r) (r) (r) = restrL ΘK/k ·xK,W = restrL λK (εK,W ) = λL NK/L(εK,W ) , and therefore ε = N (r) (ε ). L,W |L K/L K,W By lemma 5.5.1,

0 → UL → UK → UK /UL → 0 is an exact sequence of Z[G]-lattices. Corollary 4.1.1 implies that there is a surjective homomorphism Hom (U , [G]) → Hom (U , [G]), Z[G] K Z Z[G] L Z CHAPTER 5. THE RUBIN-STARK CONJECTURE 52 and composing this with the isomorphism

Hom (U , [G]) ∼ Hom (U , [Γ]) Z[G] L Z = Z[Γ] L Z (see equation A.1.4) gives a surjection

N : Hom (U , [G]) → Hom (U , [Γ]) : ϕ 7→ (u 7→ π(ϕ(u) )), K/L Z[G] K Z Z[Γ] L Z 0 where N ·ϕ(u) = ϕ(u). Pick ϕ ∈ Hom (U , [G]), then for every u ∈ U , H 0 Z[G] K Z K NK/L(ϕ) ◦ NK/L (u) = NK/L(ϕ)(NH ·u) = π(ϕ(u)). It follows that (r) (r) ι NK/L(ϕ) NK/L(u) = π(ι(ϕ)(u)) Vr Vr for all ϕ ∈ Hom (UK , [G]) and all u ∈ UK . Z[G] Z[G] Z Z[G] Vr ∼ Vr Let φ ∈ Hom (UL, [Γ]) = Hom (UL, [Γ]) be given. Since N Z[Γ] Z[Γ] Z Z[G] Z[Γ] Z K/L Vr (r) is surjective, there exists ϕ ∈ Hom (UK , [G]) such that N (ϕ) = φ. The Z[G] Z[G] Z K/L result follows from ι(φ)(ε ) = ι N (r) (ϕ) N (r) (ε ) = π(ι(ϕ)(ε )) ∈ π( [G]) = [Γ]. L,W |L K/L K/L K,W K,W Z Z

5.6 Proofs of the Rubin-Stark conjecture in special cases

5.6.1 More than r places of S split completely

We follow [Rub96] proposition 3.1, pp 44 - 45. If more than r places split completely, −r remark 2.3.1 shows that for all χ 6= 1G we have rS(χ) > r, and so lims→0 s LS(s, χ) = 0. −r If r < #S − 1, then lims→0 s LS(s, 1G) = 0 as well and the conjecture is trivial. Thus we may suppose that r = #S − 1, in which case all places in S split completely in K/k. In this subsection, S and m will be fixed, so we omit them from subscripts. Thus (r) (r) Uk = Uk,S,m, RK = RK,S,m, hK = hK,S,m,Θ = ΘS,m, etc. Let S = {v0, v1, . . . , vr}, and for i = 1, . . . , r, define xi = vi − v0 ∈ Xk andx ¯i = v¯i − v¯0 ∈ XK . Let {ui : i = 1, . . . , r} be a Z-basis for Uk; then from the definition of the regulator, (r) λk (u1 ∧ ... ∧ ur) = ±Rkx1 ∧ ... ∧ xr. From the commutative diagram 2.3.4, we see that (r) (r) (r) λK (u1 ∧ ... ∧ ur) = jK/k ◦ λk (u1 ∧ ... ∧ ur) (r) = jK/k (±Rkx1 ∧ ... ∧ xr) = ±Rk(NG ·x¯1) ∧ ... ∧ (NG ·x¯r) r r = ±RkNG ·x¯1 ∧ ... ∧ x¯r = ±Rkg e1G ·xW , (5.6.1) CHAPTER 5. THE RUBIN-STARK CONJECTURE 53 where W = (¯v0,..., v¯r). By remark 4.2.1 and equation 4.2.5,

(r) −r Θ = δK lim s ζk(s)e1 = ±hkRke1 , s→0 G G and this, together with 5.6.1, shows that h ε = ± k u ∧ ... ∧ u . W gr 1 r ˆ If φ ∈ HomZ(UK , Z), let φ be the image of φ under the isomorphism HomZ(UK , Z) → Vr Hom (UK , [G]). To show that εW ∈ UK , it is enough to show that if {φi : Z[G] Z Z[G],0 i = 1, . . . , r} is a set of elements of HomZ(UK , Z), then h k det φˆ (u ) ∈ [G]. gr j i Z

ˆ P −1 Since G acts trivially on ui, φj(ui) = σ∈G φj(σ·ui)σ = φj(ui)NG, and so h h k det φˆ (u ) = k det (φ (u )N ) gr i j gr j i G h h = k N r det (φ (u )) = k N det (φ (u )) . gr G j i g G j i Since S contains all places which ramify, and all places in S split completely, K/k is k unramiﬁed. Therefore g divides hk by class-ﬁeld theory , and we are done. Note that this gives another proof of Stark’s principal conjecture for trivial characters.

5.6.2 The case r = 0

Proposition 5.6.1. If hypotheses H(K/k, S, m) are satisﬁed, then δ ∈ Ann (µ ). K,m Z[G] K Proof. Condition iii) of hypotheses H(K/k, S, m), together with the exact sequence

(4.2.2), implies that µK is embedded in QK,m. Thus

δ ∈ Fitt (Q ) ⊆ Ann (Q ) ⊆ Ann (µ ) K,m Z[G] K,m Z[G] K,m Z[G] K (see appendix A.2 for the ﬁrst inclusion).

We consider the conjecture RS(K/k, S, m, 0). If a place in S splits completely, we are done by the result of the previous subsection; otherwise, no places split completely, in which case k is totally real, K totally complex. The following theorem is a consequence of results in [DR80]:

ksee [Gra03]. CHAPTER 5. THE RUBIN-STARK CONJECTURE 54

Theorem 5.6.1. If k is totally real and K is totally complex,

Ann (µ )·Θ (0) ⊆ [G]. Z[G] K K/k,S Z

Thus, by proposition 5.6.1,

Θ(0) = δ Θ (0) ∈ Ann (µ )·Θ (0) ⊆ [G]. (5.6.2) K/k,S,m K,m K/k,S Z[G] K K/k,S Z

V0 ∼ (0) V0 ∼ Since XK,S = λ UK,S,m = [G], the Rubin-Stark conjecture follows. Z[G] K,S,m Z[G],0 Z This shows that Stark’s principal conjecture is true for characters χ with rS(χ) = 0, a fact which also follows from results of Siegel and Klingen ([Sie70]), and Shintani ([Shi76]). If S satisﬁes hypotheses H(K/k, S), deﬁne

MK/k,S = {m : Hypotheses H(K/k, S, m) are satisﬁed}.

While we do not need it here, we will prove the following proposition for later reference:

Proposition 5.6.2. ([Tat84] lemme 1.1, p 82) If hypotheses H(K/k, S) are satisﬁed, then Ann (µ ) is generated as a [G]-module by {δ : m ∈ M }. Z[G] K Z K,m K/k,S Proof. We will prove that Ann (µ ) is equal to the [G]-ideal A generated by Z[G] K Z

{δK,v : v ∈ MK/k,S, v a non-archimedean place of k}.

By proposition 5.6.1, A ⊆ Ann (µ ). For each σ ∈ G we may ﬁnd a non-archimedean Z[G] K ∗∗ P place vσ ∈ MK/k,S whose Frobenius automorphism is σ . Let σ∈G aσσ be an element of Ann (µ ). Then since Z[G] K X X X X aσσ = (aσNvσ + aσ(σ − Nvσ)) = aσNvσ + aσσδvσ , σ∈G σ∈G σ∈G σ∈G P it is enough to show that σ∈G aσNvσ ∈ A. But proposition 5.6.1 implies that P a Nv ∈ Ann (µ ), so must be a multiple of w = #µ since µ is cyclic as σ∈G σ σ Z[G] K K K K an abelian group. Thus it remains to show that wK ∈ A. Let d be the greatest common divisor of

{δv : v ∈ MK/k,S, v non-arch, σv = 1} = {1 − Nv : v ∈ MK/k,S, v non-arch, σv = 1}, and let ζ be a primitive d-th root of unity. Let τ be an element of Gal(K(ζ)/K) ⊆ Gal(K(ζ)/k)††. Let S0 be the union of S and all places of k which ramify in K(ζ)/k.

∗∗This follows from remark 5.1.1 and the Cebotarevˇ density theorem. ††K(ζ)/k is Galois (in fact abelian), since K(ζ) is the compositum of K and k(ζ). CHAPTER 5. THE RUBIN-STARK CONJECTURE 55

∗∗ Then τ = (v, K(ζ)/k) for some non-archimedean place v ∈ MK(ζ)/k,S0 . Note that by 1−Nvτ −1 proposition 5.6.1, ζ = 1. Since (v, K/k) = τ|K = 1, it follows that d|1 − Nv, and consequently ζτ = ζNv = ζ. Thus τ is trivial on K(ζ), so K(ζ) = K and hence

ζ ∈ K. Therefore d|wK , and we are done.

5.6.3 Quadratic extensions

This proof follows [Rub96] theorem 3.5, pp 47 - 48. We consider extensions with Galois ∼ group G = Z/2Z. In this subsection, S and m will be ﬁxed, so we omit the subscripts S and m, and write UK = UK,S,m, Uk = Uk,S,m, hK = hK,S,m, εK,W = εK,S,W,m, etc.

Lemma 5.6.1. ([Rub96] lemma 3.4, p 46) Suppose that hypotheses H(K/k, S, m) hold, and in addition that G is cyclic and S contains an element v such that Dv = G. Then

a) hk divides hK , 1 b)#H (G, UK ) divides hk, 0 1 c) If G is a p-group (p prime) and Hb (G, UK ) = H (G, UK ) = 0, then hK /hk is prime to p iﬀ hk is prime to p.

0 0 0 Proof. Put S = S r {real v |m}, and note that v ∈ S since any real archimedean 0 place dividing m splits completely in K/k. Let HK (resp. Hk) be the (SK , mK )-ray class ﬁeld of K (resp. the (S0, m)-ray class ﬁeld of k), so that we have isomorphisms ∼ ∼ † 0 Gal(HK /K) = AK and Gal(Hk/k) = Ak . Since all places in S split completely in Hk,

Hk ∩ K = k (otherwise v would split completely and be inert in a proper ﬁeld extension of k). Therefore we have an onto map Gal(HK /K) → Gal(Hk/k), which proves i). The exact sequence

1 G 0 0 → H (G, UK ) → Ak → AK → Hb (G, UK ) (5.6.3)

(derived in appendix A.6) shows that ii) holds. ∼ G Under the assumptions of iii), 5.6.3 shows that Ak = AK , and so we may view Ak as a subgroup of AK . A Suppose p|hk. Since NK/k : AK → Ak is corresponds to the map Gal(HK /K) → Gal(Hk/k), it is surjective. Therefore to show that p|(hK /hk), it suﬃces to show that ˇ p divides # ker(NK/k). If NK/k : Ak → Ak is the restriction of NK/k to Ak, then

† If F is a number ﬁeld, HF,S,m is deﬁned to be maximum subextension of HF,m in which all places in S split completely. Details can be found in the book by Georges Gras ([Gra03]), but note that in his notation H = F S¯ , where m is the integral ideal F,S,m (m0) 0 corresponding to m and S¯ = S r {complex archimedean places of F }. CHAPTER 5. THE RUBIN-STARK CONJECTURE 56

ˇ ˇ NK/k is simply multiplication by g (if we write Ak additively), and so # ker(NK/k) = ˇ ˇ #coker(NK/k) is divisible by p (recall that g is a power of p). Since ker(NK/k) is a subgroup of ker(NK/k), the result follows. G Conversely, if p|(hK /hk), then p|hK = #AK , and since G is a p-group, p|#(AK ) = hk (we use the fact that if a p-group Γ acts on a ﬁnite set B, then #B ≡ #(BΓ) mod p − see [Ser79] p138 for example).

We wish to show that RS(K/k, S, m, r) holds if [K/k] = 2. Without loss of generality, ∼ we may assume r = rK/k,S. Let χ be the non-trivial character of Gal(K/k) = Z/2Z. If r(1G) < r(χ), then r + 1 places split completely, so the conjecture is true by the result of subsection 5.6.1. Thus we may suppose r(1G) ≥ r(χ), and that exactly r places split.

Note that this implies Dv = G for some v ∈ S. Deﬁne d = #(S) − r − 1, so that

#(SK ) = #S + r = d + 2r + 1. ∼ L By lemma 5.5.1, UK = Uk UK /Uk (as abelian groups), so we may choose a Z-basis {ui : i = 1, . . . , d + 2r} for UK such that {ui : i = 1, . . . , d + r} is a Z-basis for Uk. By 1 lemma A.7.1, we may assume that if H (G, UK ) 6= 0, then NK/k(ud+r+1) = 1. Write S = {vi : i = 1, . . . , d + r + 1}, and choose the numbering so that the places vi, for i = d + 1, . . . , d + r, split completely. Write SK = {wi : i = 1, . . . , d + 2r + 1}, and choose the numbering so that wi is the unique place of K dividing vi for i = 1, . . . , d, wi and wi+r are the two places of K dividing vi for i = d + 1, . . . , d + r, and wd+2r+1 is the unique place of K dividing vd+r+1. With respect to the ordered Z-bases {ui : i = 1, . . . , d + 2r} and {wi − wd+2r+1 : i = 1, . . . , d + 2r}, λK is represented by the matrix

(log |uj|wi )1≤i,j≤d+2r, which we can write in the form !T ABB CDE where A is (d + r) × d, B is (d + r) × r, C is r × d, etc. Therefore ! ! ABB AB 0 RK = ± det = ± det = ± det(AB) det(E − D). CDE CDE − D 2 Note that, for all u ∈ Uk, |u|wj = |u|vj for j = 1, . . . , d and |u|wj = |u|vj for j = d + 1, . . . , d + r. Therefore

d d det(AB) = det(log |ui|wj )1≤i,j≤d+r = 2 det(log |ui|vj )1≤i,j≤d+r = ±2 Rk, d and so RK /Rk = ±2 det(E − D). Let σ be the non-trivial element of G. Since wj+r =

σ·wj for j = d + 1, . . . , d + r,

det(E − D) = det(log |ui|σ·wj − log |ui|wj )d+r

Vr Let ε− be the image of u ∧ ... ∧ u in UK , so that d+r+1 d+2r gZ[G]

(r) eχ eχ λK (eχ ·ε−) = det log |ui |wj + log |ui |σ·wj σ d+r

Case 1: r(1G) > r(χ):

d (r) hK RK 2 hK In this case Θ = eχ = ± det(E − D)eχ, and so K/k hkRk hk

2dh ε = ± K e ·ε ∈ ^ r U ⊆ ^r U W χ − g [G] K [G],0 K hk Z Z by i) of lemma 5.6.1.

Case 2: r(1G) = r(χ):

In this case, (r) hK RK ΘK/k = hkRke1G + eχ hkRk and d = 0. Because NK/k(UK ) is a subgroup of the free abelian group Uk, it is free abelian, and we can ﬁndu ˇi ∈ UK so that {NK/k(ˇui): i = 1, . . . , r} is a Z-basis for 0 NK/k(UK ). By lemma A.7.2, we may assume that if Hb (G, UK ) 6= 0, thenu ˇ1 ∈ Uk. If Vr we let ε+ be the image ofu ˇ1 ∧ ... ∧ uˇr in UK , then gZ[G]

(r) σ λK (e1G ·ε+) = det((log |uˇi|wj + log |uˇi |wj σ)e1G )1≤i,j≤r ·xW 1+σ = det(log |uˇi |wj )1≤i,j≤re1G ·xW

= det(log |NK/k(ˇui)|wj )1≤i,j≤re1G ·xW ,

Note that det(log |NK/k(ˇui)|vj )1≤i,j≤r is the determinant of λk ◦ ι : NK/k(UK ) → RXk, where ι : NK/k(UK ) → Uk is the inclusion, taken with respect to Z-bases for NK/k(UK ) and Xk ⊆ RXk. Thus

(r) 0 λK (e1G ·ε+) = ±Rk[Uk : NK/k(UK )]e1G ·xW = ±Rk#Hb (G, UK )e1G ·xW , and hence ! (r) ±hk hK (r) λ e1 ·ε+ ± eχ ·ε− = Θ ·xW , K 0 G K/k #Hb (G, UK ) hk CHAPTER 5. THE RUBIN-STARK CONJECTURE 58

±hk hK εW = e1 ·ε+ ± eχ ·ε−. 0 G #Hb (G, UK ) hk ∼ ∼ r Since all places in S except one split completely, QUK = QXK = Q[G] , so we may r n r ﬁnd an embedding of UK in Z[G] with ﬁnite cokernel C . Since Hb (G, Z[G] ) = 0 for all n ∈ Z, the long exact Tate cohomology sequence applied to

r 0 → UK → Z[G] → C → 0

0 ∼ −1 1 ∼ 0 shows that Hb (G, UK ) = Hb (G, C ) and H (G, UK ) = Hb (G, C ). Since G is cyclic and 0 −1 0 1 C is ﬁnite, #Hb (G, UK ) = #Hb (G, C ) = #Hb (G, C ) = #H (G, UK ) (see appendix A.1.2). 0 Suppose Hb (G, UK ) 6= 0. Recall that we assume that NK/k(ur+1) = NK/k(ud+r+1) =

1 andu ˇ1 ∈ Uk; hence eχ ·ε− = ε− and e1G ·ε+ = ε+. Therefore ±h h ε = k ε ± K ε ∈ ^ r U ⊆ ^r U W 1 + − g [G] K [G],0 K #H (G, UK ) hk Z Z by lemma 5.6.1 i) and ii). 0 Now suppose Hb (G, UK ) = 0. Then by a theorem of Reiner (see the remarks after ∼ r theorem A.7.1 in appendix A.7), UK = Z[G] . Therefore {ui : i = r + 1,..., 2r} is a ‡ Z[G]-basis for UK , and {NK/k(ui): i = r + 1,..., 2r} is a Z-basis for NK/k(UK ) = Uk .

Since {NK/k(ˇui): i = 1, . . . , r} is also a Z-basis for Uk, it follows that e1G·ε− = ±e1G·ε+. Hence hK εW = ±hke1G ± eχ ·ε−. hk

Lemma 5.6.1 iii) shows that hk and hK /hk have the same parity, whence

^ r ^r εW ∈ UK ⊆ UK . g Z[G] Z[G],0

‡ r WLOG we may assume that, under the identiﬁcation of UK with Z[G] , ur+1 = (1, 0, 0,...), ur+2 = (0, 1, 0,...) etc., and that ui = (1 + σ)·ur+i for i = 1, . . . , r. Appendix A

Appendices

A.1 G-modules

A.1.1 Hom groups

We begin with some general nonsense. Let α : R → S be a homomorphism of commutative unital rings, and let R-Mod (resp. S-Mod) be the category of R-modules (resp. S-modules). Any S-module M can be given an R-module structure via α, which we call

αM, and this association gives a ‘forgetful’ functor from S-Mod to R-Mod.

If N is an R-module and M is an S-module, we deﬁne HomR(αM,N) to be the abelian group of R-homomorphisms from αM to N. This group has an S-module structure given by s·f : m 7→ f(s·m). The association N 7→ HomR(αS, N) deﬁnes a functor from R-Mod to S-Mod, and one can check that this is right adjoint to the forgetful functor. The natural isomorphism of Hom sets is given by

HomR(αM,N) → HomS(M, HomR(αS, N)) : f 7→ (m 7→ (s 7→ f(s·m))), (A.1.1) and one may verify that this is an isomorphism of S-modules. We can give S an S-R-bimodule structure, which allows us to deﬁne the S-module N N S R M, given an R-module M. The association M 7→ S R M deﬁnes a functor from R-Mod to S-Mod, and one can check that this is left adjoint to the forgetful functor. The natural isomorphism of Hom sets is given by

N HomR(M,α N) → HomS(S RM,N): f 7→ (s ⊗ m 7→ f(s·m)), (A.1.2) and again one may verify that this is an isomorphism of S-modules. Consider the case where H is a ﬁnite index subgroup of the abelian group G, A is a commutative unital ring, R = A[H], S = A[G] and α = ι is the inclusion. If M is an

59 APPENDIX A. APPENDICES 60

A[G]-module, there is an isomorphism of A[G]-modules

X −1 HomA[H](ιA[G],A[H]) → A[G]: f 7→ σ f(σ) σH∈G/H P P with inverse a 7→ (b 7→ prH (ab)), where prH ( σ∈G cσσ) = σ∈H cσσ. Thus for any A[G]-module M, equation A.1.1 gives an isomorphism of A[G]-modules

∼ HomA[H](ιM,A[H]) = HomA[G](M,A[G]). (A.1.3)

The case we will be interested in is where G is ﬁnite and H is trivial. Now consider the case where α = π : G → Γ is a surjective group homomorphism with ﬁnite kernel H (again we assume G to be abelian). Let s :Γ → G be a section of π. We also write π : Z[G] → Z[Γ] and s : Z[Γ] → Z[G] for the canonical associated maps. We wish to show that there is an isomorphism

HomA[G](πA[Γ],A[G]) → A[Γ].

If f ∈ HomA[G](A[Γ],A[G]), then for every σ ∈ H and every a ∈ A[Γ], σf(a) = f(σ·a) = H f(a). Thus f takes its values in A[G] = NH A[G], and so we can ﬁnd a function f0 :

A[Γ] → A[G] (not necessarily a homomorphism) such that f = NH·f0. The isomorphism sends f to π(f0(1)), and this is easily seen to be independent of the choice of f0. The inverse of this isomorphism sends b ∈ A[Γ] to the function in HomA[G](A[Γ],A[G]) which takes c to NH s(bc). Thus equation A.1.1 shows that for any A[Γ]-module M, there is an isomorphism of G-modules

∼ HomA[G](πM,A[G]) = HomA[Γ](M,A[Γ]). (A.1.4)

To be more explicit, this isomorphism takes f ∈ HomA[G](M,A[G]) to π◦f0 ∈ HomA[Γ](M,A[Γ]), where f = NH ·f0, as above. We will usually omit the subscripts ι and π.

A.1.2 Cohomology of G-modules

Let G be a finite group. We list here a collection of facts concerning the cohomology of G-modules. Proofs of these statements can be found in [AW67]. q Let M be a G-module. Define H (G, M)(q ∈ N) to be the right derived functors G of the left-exact functor M 7→ M , and define Hq(G, M)(q ∈ N) to be the left derived APPENDIX A. APPENDICES 61 functors of the right-exact functor M 7→ MG (see subsection 1.1.2 for the definition of

MG). The Tate cohomology groups are deﬁned as follows: For any G-module M there is a G map NG : MG → M : m + IG ·M 7→ NG ·m. The Tate cohomology groups are deﬁned to be  Hq(G, M) q > 0    cokerNG q = 0 Hb q(G, M) = (A.1.5) ker N q = −1  G   H−q−1(G, M) q < −1

The snake lemma applied to the following exact diagram

0 0 0 O O O

Hb 0(G, A) Hb 0(G, B) Hb 0(G, C) O O O

0 / AG / BG / CG / H1(G, A) / ... O O O

...H/ H1(G, C) / AG / BG / CG / 0 O O O

Hb −1(G, A) Hb −1(G, B) Hb −1(G, C) O O O

0 0 0 shows that the Tate cohomology groups ﬁt into a long exact sequence

... → Hb q−1(G, C) → Hb q(G, A) → Hb q(G, B) → Hb q(G, C) → Hb q+1(G, A) → ...

q ∼ q+2 If G is cyclic, Hb (G, M) = Hb (G, M) for all q ∈ Z. If, in addition, M is finite, #Hb 0(G, M) = #Hb 1(G, M). Consequently, if G is cyclic and M is finite, all Tate cohomology groups have the same order. G If H is a subgroup of the finite group G and M is an H-module, define IndH M = [G] N M. Shapiro’s lemma states that for all q ∈ , Hq G, IndG M =∼ Hq(H,M). Z Z[H] Z b H b A G-module M is said to be cohomologically trivial if Hb q(H,M) = 0 for all subgroups H of G, and all q ∈ Z. Proposition A.1.1. If 0 → A → B → C → 0 is a short exact sequence of G-modules and two of A, B or C are cohomologically trivial, then so is the third.

Proof. Follows from the long exact Tate cohomology sequence. APPENDIX A. APPENDICES 62

Corollary A.1.1. If a G-module M has a ﬁnite free resolution, it is cohomologically trivial.

Proof. We can split the resolution

0 → Fn → Fn−1 → ... → F0 → M → 0 up into short exact sequences 0 → Zi+1 → Fi → Zi → 0, i = 0, . . . , n−1, where Zn = Fn and Z0 = M. The result follows by induction.

A.2 Fitting ideals

Most of the results in this section can be found in the book by Northcott ([Nor]). We list some properties of Fitting ideals, and prove a lemma which is required in the proof of proposition 5.4.1. For the deﬁnition of the Fitting ideal, see subsection 4.2.2.

Proposition A.2.1. If 0 → A → B → C → 0 is a short exact sequence of ﬁnitely generated R-modules, then

FittR(A)FittR(C) ⊆ FittR(B) ⊆ FittR(C). (A.2.1)

0 Remark A.2.1. The proof relies on the fact that if A, B and C are as above, and FA → 0 FA → A → 0 and FC → FC → C → 0 are free presentations of A and C respectively, 0 L 0 L then there exists a free presentation of B of the form FA FC → FA FC → B → 0. If the short exact sequence splits, then the ﬁrst inclusion in equation A.2.1 is an equality, and so by induction

m ! m M Y FittR Mi = FittR (Mi) . i=1 i=1

The Fitting ideal is closely related to the annihilator ideal AnnR(M). If M has n n generators, then AnnR(M) ⊆ FittR(M) ⊆ AnnR(M). In particular, if M is cyclic,

FittR(M) = AnnR(M). It follows easily from the deﬁnition of the Fitting ideal that if S is a subring of R (both unital and commutative) and M is a ﬁnitely generated S, then

N ∼ FittR(R SM) = FittS(M)R (A.2.2)

A proof of the following lemma may be found in [CG98] (lemma 3, 462). APPENDIX A. APPENDICES 63

0 Lemma A.2.1. If FC → FC → C → 0 is a free presentation of a finitely generated 0 R-module C, where FC and FC have the same finite rank, then for any short exact sequence 0 → A → B → C → 0 of finitely generated R-modules, the first inclusion in equation A.2.1 is an equality.

The next lemma is needed in the proof of proposition 5.4.1.

Lemma A.2.2. Suppose

f g 0 / AAB/ BC/ C / 0, is a short exact sequence of ﬁnitely generated R-modules. Then for any r ∈ N, Fitt (C)·^r B ⊆ f (r) ^r A . (A.2.3) R R R

Proof. Let {c1, . . . , cn} be a set of n non-zero generators of C, and for each j = 1, . . . , n, choose bi ∈ B mapping to ci. Then B is generated by A ∪ {b1, . . . , bn} (we identify A Vr with its image in B). Thus R B is generated by monomials x1 ∧...∧xr, where at most n of the xi’s are in {b1, . . . , bn}, and the rest are in A. On the other hand, FittR(C) is Pn generated by det(rij), where j=1 rij ·cj = 0 for every i. Hence it will be suﬃcient to show that for such monomials and elements of FittR(C), det(r )·x ∧ ... ∧ x ∈ f (r) ^r A . (A.2.4) ij 1 r R Firstly, note that we may assume r ≥ n. To see this, choose n0 ≥ max{n, r}, and let D be a free R-module of rank n0 − r. Suppose we have shown that the theorem holds holds with r = n0 for all short exact sequences 0 → A0 → B0 → C0 → 0 where C0 has n generators. Since

L L f 1D g 0 0 / A L DBD / B L DCD / C / 0, is such a short exact sequence, we deduce that

n0 n0 M h 0 i M h 0 i Fitt (C)· ^i B N ^n −iD ⊆ f (i) ^i A N ^n −iD R R R R R R R i=1 i=1 (see 4.1.2). Looking at the r-th summand shows that

0 Fitt (C)·^r B ∼ Fitt (C)· ^r B N ^n −rD R R = R R R R 0 ⊆ f (r) ^r A N ^n −rD ∼ f (r) ^r A . R R R = R Secondly, we may assume r = n, since if A.2.4 holds in this case, it holds with r ≥ n

(recall that we assume at most n of x1, . . . , xr are in {b1, . . . , bn}). APPENDIX A. APPENDICES 64

m If m ∈ N, deﬁne [ n ] to be the set of all strictly increasing functions from {1, . . . , n} to {1, . . . , m}. For any β1, . . . , βm ∈ B and any γij ∈ R, we have the following equality Vn in R B:

 m   m  X X X  γ1j ·βj ∧ ... ∧  γnj ·βj = det(γiφ(j))·βφ(1) ∧ ... ∧ βφ(n). (A.2.5) m j=1 j=1 φ∈[ n ]

We will prove A.2.4 by induction on the number of the xi’s which are not in {b1, . . . , bn}.

Call this number `. When ` = 0, we may assume xj = bj for j = 1, . . . , n, and equation A.2.5 shows that

 n   n  X X det(rij)·b1 ∧ ... ∧ bn =  r1j ·bj ∧ ... ∧  rnj ·bj . j=1 j=1

Pn Pn Since g maps j=1 rij ·bj to j=1 rij ·cj = 0 for i = 1, . . . , n, this is an element of A, and we have proven A.2.4 for ` = 0. Now suppose d ≤ n−1 is a positive integer and A.2.4 holds for all ` ≤ d−1. We wish to show that A.2.4 holds with ` = d. Without loss of generality we may assume that xj = aj for j = 1, . . . , d and xj = bj for j = d + 1, . . . , n, where the aj’s are arbitrary elements of A. We apply equation A.2.5 with ( ( rij 1 ≤ j ≤ d aj 1 ≤ j ≤ d γij = βj = ri,j−d d + 1 ≤ j ≤ d + n bj−d d + 1 ≤ j ≤ d + n

Pd The left-hand side of A.2.5 is the wedge product of terms of the form j=1 rij ·aj + Pn j=1 rij ·bj, which are all in A. Any term in the sum on the right-hand side for which d+n φ ∈ [ n ] takes on the value i and i + d, for 1 ≤ i ≤ d, will be zero. By the pigeon-hole principle, all non-zero terms must correspond to functions φ which satisfy φ(j) = j + d for j = d + 1, . . . , n. Such a term can have at most d xj’s not in {b1, . . . , bn}, and in fact the only term which has exactly d xj’s not in {b1, . . . , bn} is

det(rij)·β1 ∧ ... ∧ βd ∧ β2d+1 ∧ ... ∧ βn+d = det(rij)·a1 ∧ ... ∧ ad ∧ bd+1 ∧ ... ∧ bn. By our inductive hypothesis, it must be in f (n) Vn A . Z[G]

A.3 An approximation theorem

A proof of the following proposition can be found in [Nar90] (proposition 2.1, pp 44-45). APPENDIX A. APPENDICES 65

Proposition A.3.1. Let F be a number ﬁeld, I an integral ideal of OF , and R an × element of OF /I. Then the restriction of the signature map sgnF : F → SgnF to R r {0} is onto. The following theorem can be thought of as a variation on the strong approximation theorem.

Theorem A.3.1. Let F be a number ﬁeld. Then for any ﬁnite set S of non-archimedean × places of F , any {av ∈ F : v ∈ S}, N ∈ N, and s ∈ SgnF , there exists x ∈ F such that:

• ordv(x − av) ≥ N for all v ∈ S,

• ordv(x) ≥ 0 for all non-archimedean places of F not in S,

• sgnF (x) = s.

We ﬁrst prove the theorem in a simple case:

Lemma A.3.1. The theorem is true if av = 1 for all v ∈ S.

Proof. Let I be an integral ideal satisfying ordv(I) ≥ N for all v ∈ S. By proposition

A.3.1, there exists a non-zero y ∈ 1 + I such that sgnF (y) = s, and such a y satisﬁes the conditions of the theorem.

Proof. (of theorem A.3.1) The strong approximation theorem implies that the first two conditions can be satisfied, say by y ∈ F . By the lemma above, we can find z ∈ F × such that:

• ordv(z − 1) ≥ N − ordv(y) for all v ∈ S,

• ordv(z) ≥ 0 for all non-archimedean places of F not in S,

• sgnF (z) = s sgnF (y).

Since ordv(yz − av) = ordv(y(z − 1) + (y − av)) ≥ N, x = yz satisﬁes the conditions of the theorem.

A.4 Unit and Picard groups of commutative rings

This section is not essential for the study of Stark’s conjectures, but some of the results will be used in the remaining appendices. Most of the results in this section can be found in the exercises in [Wei]. Let A be a commutative ring (possibly without a unit), and let R be a commutative ring with unit. If A has an R-algebra structure, we may form the R-algebra A +1 R, APPENDIX A. APPENDICES 66 whose underlying R-module is the direct sum of the underlying R-modules of A and R, but where multiplication is deﬁned by

(a, r)(a0, r0) = (aa0 + ra0 + r0a, rr0).

This is a unital R-algebra A+1 R, the unit being (0, 1R). The R-algebra homomorphism

R → A +1 R : r 7→ (0, r) has a left inverse A +1 R → R :(a, r) 7→ r, so under the units × × × functor R → (A +1 R) has a left inverse. We define the extended group of units A × × to be the cokernel of R → (A +1 R) , or equivalently (up to canonical isomorphism), × × the kernel of (A +1 R) → R . If A does have a unit, then there is an isomorphism of R-algebras M A +1 R → A R :(a, r) 7→ (a + r1A, r). L The composite A R → A +1 R → R is then projection onto R, and the units functor takes this to the projection of (A L R)× =∼ A× L R× onto R×. Hence A× as we have just defined it is isomorphic to the group of units of A, the isomorphism sending a to a + 1A. This justifies our notation. It is not hard to see that

A× =∼ {a ∈ A : ∃a0 ∈ A, aa0 + a + a0 = 0}, where the group operation is a∗b = ab+a+b. Thus A× is independent of the R-module structure on A. There are two special cases which we will make use of. Firstly, if A is an ideal I of R, then I× =∼ R× ∩ (1 + I). Secondly, if the product of any two elements of A is zero, then since a(−a) + a + (−a) = 0 and a ∗ b = ab + a + b = a + b, A× is isomorphic to the underlying additive group of A. We may deﬁne the extended Picard group Pic(A) in an analogous way to A×:

Pic(A) = ker[Pic(A +1 R) → Pic(R)], and one can verify that this is also independent of the R-algebra structure on A and coincides with the usual Picard group when A is unital. The (usual) Picard group of a finite commutative unital ring is zero ∗. In fact the same is true for the extended Picard group of a finite commutative ring. To see this, note that a finite ring A has non-zero characteristic, say n, so we may view it as a Z/nZ-algebra. Since Pic(A) is the kernel of Pic(A +1 (Z/nZ)) → Pic(Z/nZ), it is zero. Finally, one can show that if I is an ideal of R, there is an exact sequence

0 → I× → R× → (R/I)× → Pic(I) → Pic(R) → Pic(R/I). (A.4.1)

∗Finite rings are direct sums of local rings ([Lam01] p 340), and the Picard group of a local ring is trivial ([Mat86] p 166). APPENDIX A. APPENDICES 67

A.5 Algebraic number theory calculations

Our standing assumptions for the remaining three chapters are that K/k is an abelian extension of number ﬁelds with Galois group G, and that hypotheses H(K/k, S, m) are satisﬁed.

Let p be a prime ideal of Ok. We wish to calculate the Z[G]-module structure of OK /pOK .

Let P be a prime of OK dividing p, let p be the characteristic of the residue ﬁeld Ok/p, and let f = f(p/pZ)(p and f will always be deﬁned this way in this appendix). By the normal basis theorem ([Rom95] theorem 8.7.2, p 169), there exists a ∈ OK /P such that {σ(a): σ ∈ Dp} is a basis for the Ok/p-vector space OK /P. Let {b1, . . . , bf } be a Z/pZ-basis for Ok/p. Then {biσ(a): i = 1, . . . , f; σ ∈ Dp} is a Z/pZ-basis for OK /P, so {bia : i = 1, . . . , f} is a (Z/pZ)[Dp]-basis for OK /P. Therefore

f ∼ M OK /P = Z[Dp]/pZ[Dp], i=1 and so

f M O /pO =∼ [G]N (O /P) =∼ [G]/p [G] (A.5.1) K K Z Z[Dp] K Z Z i=1 (Note that these are isomorphisms of G-modules, not of rings).

Proposition A.5.1. Let QK,m be as deﬁned in chapter 4. Then QK,m is cohomologically trivial.

Proof. WLOG we may assume that m = vn for some non-complex place v of k, and + ∼ n ∈ N . If v is real, QK,m = Z[G]/2Z[G] is cohomologically trivial since it has a free resolution of length 2. So suppose v andv ¯ are non-archimedean, corresponding to the q prime ideals p and P of Ok and OK respectively. We ﬁrst show that Hb (G, QK,m) = 0 for all q ∈ Z (which is all we really need).

Q =∼ (O /pnO )× =∼ [G]N (O /Pn)×, K,m K K Z Z[Dp] K so by Shapiro’s lemma,

q ∼ q n × Hb (G, QK,m) = Hb (Dp, (OK /P ) ).

For each positive integer `, the units-Pic sequence A.4.1 applied to

` `+1 `+1 ` 0 → P /P → OK /P → OK /P → 0 APPENDIX A. APPENDICES 68 gives a short exact sequence of Dp-modules

× × × ` `+1 `+1 ` 0 → P /P → OK /P → OK /P → 0. (A.5.2)

Since the product of any two elements of P`/P`+1 is zero, the remarks in appendix A.4 ` `+1× ` `+1 show that P /P is isomorphic to the additive Dp-module P /P , which in turn † ∼ Lf is isomorphic to OK /P = i=1 Z[Dp]/pZ[Dp]. This has a free resolution of length 2, and is therefore cohomologically trivial. The sequence A.5.2 then shows that the Tate `× cohomology of OK /P is independent of `.

Since OK /P is a Galois ﬁeld extension of Ok/p with Galois group isomorphic to Dp, 1 × Hilbert’s theorem 90 ([Rom95] theorem 11.1.2, p 211) shows that Hb (Dp, (OK /P) ) = 0. × Since Dp is cyclic and (OK /P) is ﬁnite, all Tate cohomology groups are trivial. n × To show that (OK /p OK ) is cohomologically trivial, let H be a subgroup of G, H Q put L = K , and write pOL = i pi, where the pi’s are distinct primes of OL. Then for any integer q, ! q n × ∼ q M n × Hb (H, (OK /p OK ) ) = Hb H, (OK /pi OK ) i ∼ M q n × = Hb (Gal(K/L), (OK /pi OK ) ), i and the result follows since all the groups in the direct sum on the right are zero by the result proved above.

n A.5.1 A generator of FittZ[G](QK,v ) n We wish to show that if v is a modulus of k, then δ n generates Fitt (Q n ). We K,v Z[G] K,v will use the notation of subsection 5.4, where we deﬁned

( −1 n−1 (1 − σv Nv)Nv if v is non-archimedean δK,vn = 2 if v is archimedean

If v is archimedean, then Q ∼ [G]/2 [G], and clearly Fitt (Q ) is generated K,v = Z Z Z[G] K,v by δK,v = 2.

† ` `+1 The isomorphism OK /P =∼ P /P arises as follows: pick a ∈ k with ordp(a) = `;

then since p is unramiﬁed in K/k, ordP(a) = ordp(a) = `. The map

` `+1 `+1 OK /P → P /P : x + P 7→ ax + P

is an isomorphism of abelian groups (see [Nar90] p 11), and the fact that a is ﬁxed by Dp

shows that this is an isomorphism of Dp-modules. APPENDIX A. APPENDICES 69

Now suppose v is non-archimedean, corresponding to the prime ideal p. Since

n × N n × Q n =∼ (O /p O ) =∼ [G] (O /P ) K,v K K Z Z[Dv] K for any P lying above p, equation A.2.2 implies that it is suﬃcient to show that if n × n × Qˆ n = (O /P ) , Fitt (Qˆ n ) is generated by δ n . If n = 1, then (O /P ) K,v K Z[Dv] K,v K,v K × is cyclic and so Fitt (Qˆ n ) = Ann (O /P) ∼ (σ − Nv) [D ]. Thus Z[Dv] K,v Z[Dv] K = v Z v Fitt (Qˆ ) is generated by δ = 1 − σ−1Nv. If n > 1, we have the exact sequence Z[G] K,v K,v v A.5.2

f × × M `+1 ` 0 → Z[Dv]/pZ[Dv] → OK /P → OK /P → 0. (A.5.3) i=1

Lf ˆ ∼ −1 Since i=1 Z[Dv]/pZ[Dv] and Qv = Z[Dv]/(1 − σv Nv)Z[Dv] have free presentations with terms of equal rank, it follows by induction that the same is true of Qˆvn for all integers n > 1 (see remark A.2.1). By lemma A.2.1,

f ! M Fitt (Qˆ n ) = Fitt [D ]/p [D ] Fitt (Qˆ n−1 ) Z[Dv] K,v Z[Dv] Z v Z v Z[Dv] K,v i=1 f = p Fitt (Qˆ n−1 ) = Nv Fitt (Qˆ n−1 ) Z[Dv] K,v Z[Dv] K,v for all integers n > 1, and we see by induction that Fitt (Qˆ n ) is generated by Z[G] K,v −1 n−1 δK,vn = (1 − σv Nv)Nv .

A.6 Some exact sequences

In this appendix we will show how the exact sequences 4.2.2 and 5.6.3 arise.

A.6.1 The exact sequence 4.2.2

This is similar to the derivation of the exact sequence on page 472 of [Aok04]. A more direct proof can be found in [Coh00] (proposition 3.2.3, p 137). Let IK,S = IK,S,1 be the free abelian group on the places of K not in SK . We may also think of this as the group of fractional OS-ideals. Let FK,m be the free abelian group on supp(mK ). We give these groups their natural G-module structures. m Clearly K1 is in the kernel of the map

× X K → FK,m : x 7→ ordw(x)w,

w∈supp(mK ) APPENDIX A. APPENDICES 70

m × m so we have a map divK : K /K1 → FK,m (this is not to be confused with the map m m divK,S,m given in section 4.2). Since QK,m = K /K1 , it is clear that

× m 0 / QK,m / K /K1 / FK,m / 0,

× m is exact. We construct a splitting for K /K1 → FK,m as follows. For each v ∈ supp(m), × × choose av ∈ k with ordv(av) = 1. By theorem A.3.1, we can ﬁnd bv ∈ K such that

• ordw(bv − 1) ≥ ordw(mK ) for all w ∈ supp(mK ) r {v¯},

• ordv¯(bv − av) ≥ ordv¯(mK ) + 1,

• sgnK,m(bv) is the identity of SgnK,m. σ Note that for any σ ∈ Dv, the above remain true if bv is replaced by bv . Also note that ordw(bv) = 0 for all w ∈ supp(mK ) r {v¯} and ordv¯(bv) = ordv¯(av) = 1. Since

σ−1 σ σ ordw(bv − 1) = ordw(bv − bv) = ordw(bv − 1 − (bv − 1)) ≥ ordw(mK ) for all w ∈ supp(mK ) r {v¯} and

σ−1 σ σ ordv¯(bv − 1) = ordv¯(bv − bv) − 1 = ordw(bv − av − (bv − av)) − 1 ≥ ordw(mK ),

1−σ m we have bv ∈ K1 . Thus we may deﬁne a Z[G]-homomorphism [G]N → K×/Km : τ ⊗ n 7→ bnτ Km, Z Z[Dv]Z 1 v 1 ∼ L N where has trivial Dv action. Since FK,m = [G] , we obtain a Z v∈supp(m) Z Z[Dv] Z × m homomorphism FK,m → K /K1 , which is easily seen to be a splitting map. Hence there is a short exact sequence

m × L (A.6.1) 0 / K1 / K / QK,m FK,m / 0. Consider the commutative diagram with exact rows and columns: 0 0 0

0 / UK,S,m / UK,S / QK,m

m × L 0 / K1 / K / QK,m FK,m / 0

0 / IK,S,m / IK,S / FK,m / 0

AK,S,m / AK,S / 0 / 0

0 0 APPENDIX A. APPENDICES 71

The snake lemma then shows that there is an exact sequence

0 → UK,S,m → UK,S → QK,m → AK,S,m → AK,S → 0.

Remark A.6.1. In the case where m is not supported on any archimedean places (so that we may regard it as an integral Ok-ideal), this exact sequence is a case of the × units-Pic sequence A.4.1 with R = OK,S, I = mOK,S. It is clear that I = UK,S,m, × (R/I) = QK,m; and Pic(R/I) = 0 since the R/I is ﬁnite. Using Milnor’s patching theorem (see [Wei] theorem 2.7, p 11), one can show that Pic(I) = AK,S,m, but this is more diﬃcult.

A.6.2 The exact sequence 5.6.3

This derivation is adapted from results in a paper by Rim ([Rim65]). Suppose the extension K/k is cyclic.

1 Lemma A.6.1. H (G, IK,S,m) = 0 ∼ ` N Proof. Note that IK,S,m = [G] , where has the trivial Dv- v6∈S∪supp(m) Z Z[Dv] Z Z 1 ∼ ` 1 action. By Shapiro’s lemma, H (G, IK,S) = v6∈S∪supp(m) H (Dv, Z) = 0.

1 m Lemma A.6.2. H (G, K1 ) = 0

Proof. The exact sequence A.6.1, together with Hilbert’s theorem 90 and the fact that

QK,m is cohomologically trivial, implies that

0 × 0 1 m Hb (G, K ) → Hb (G, FK,m) → H (G, K1 ) → 0 is short exact. Note that for all x ∈ k×,

m X X X ‡ divK (x) = ordw(x)w = ordv(x) w ,

w∈supp(mK ) v∈supp(m) w|v

m P which is simply divk (x) under the identiﬁcation of v ∈ supp(m) with w|v w. Thus the m × m restriction of divK to k → Fk,m is divk , which is onto. Hence

0 × × × 0 Hb (G, K ) = k /NK/k(K ) → Fk,m/NG ·FK,m = Hb (G, FK,m)

1 m is onto, and so H (G, K1 ) = 0.

‡we use the fact that all v ∈ supp(m) are unramiﬁed. APPENDIX A. APPENDICES 72

For any Galois extension F/k, we have exact sequences of Gal(F/k)-modules

m m 0 → UF,S,m → F1 → F1 /UF,S,m → 0, (A.6.2) m 0 → F1 /UF,S,m → IF,m → AF,S,m → 0. (A.6.3)

(see 4.2.1.) From A.6.2 with F = K and lemma A.6.2 we obtain the exact sequence

m m G 1 0 → Uk,S,m → k1 → (K1 /UK,S,m) → H (G, UK,S,m) → 0, (A.6.4)

A A Recall the map i = iK/k : Ak,S,m → AK,S,m deﬁned in section 4.2. Clearly the image A G A A of i is contained in AK,S,m, and we again write i for i with codomain restricted to G AK,S,m. In the commutative diagram below, the top row is exact by A.6.3 with F = k, the bottom row is exact by lemma A.6.1, and the column on the left is exact by A.6.4.

0 0 ker(iA)

m 0 / k1 /Uk,S,m / Ik,S,m / Ak,S,m / 0

=∼ iA m G G G 1 m 0/ (K1 /UK,S,m) / IK,S,m / AK,S,m / H (G, K1 /UK,S,m) / 0

1 A H (G, UK,S,m) 0 coker(i )

0 0 Applying the snake lemma gives the exact sequence

A 1 A 1 m 0 → ker(i ) → H (G, UK,S,m) → 0 → coker(i ) → H (G, K1 /UK,S,m) → 0 and so under the resulting isomorphisms, the third column above becomes

1 G 1 m 0 → H (G, UK,S,m) → Ak,S,m → AK,S,m → H (G, K1 /UK,S,m) → 0.

Finally, A.6.2 and lemma A.6.2 show that

1 m 2 0 → H (G, K1 /UK,S,m) → H (G, UK,S,m) is exact, and splicing the last two exact sequences together shows that 5.6.3 is exact 2 ∼ 0 (H (G, UK,S,m) = Hb (G, UK,S,m) since G is assumed to be cyclic). APPENDIX A. APPENDICES 73

A.7 Some results required for the proof of the Rubin-Stark conjecture for quadratic extensions.

We will use the convention in subsection 5.6.3 that the subscripts S and m will be omitted. Our standing assumptions are that UK is torsion-free, that {ui : i = 1, . . . , d + 2r} and {ui : i = 1, . . . , d + r} are Z-bases for UK and Uk respectively, and that, in the case where r(1G) = r(χ), {NK/k(ˇui): i = 1, . . . , r} is a Z-basis for NK/k(UK ).

1 Lemma A.7.1. If H (G, UK ) 6= 0, we may assume that NK/k(ud+r+1) = 1.

1 1−σ Proof. ([Hay04] lemma 4.3, p 10) Since H (G, UK ) = ker(NK/k)/UK , we can ﬁnd 1−σ Q mi u ∈ ker(NK/k) r UK . Write u = ui , where ∈ Uk, mi ∈ Z, and the product runs 2 1+σ 1−σ Q ni from i = d + r + 1 to i = d + 2r. Since ui = ui ui , we can write u = 12 ui , 1−σ where 1 ∈ Uk, 2 ∈ UK , and ni = 0 or 1. If all the ni’s are zero, then 1 = NK/k(u) = 2 1−σ NK/k(1) = 1, and so 1 = 1 since Uk is torsion-free. But this implies u = 2 ∈ UK , a contradiction. Therefore, without loss of generality, we may assume nd+r+1 = 1, and −1 Q ni replacing ud+r+1 by u2 = 1 ui gives the desired basis.

0 Lemma A.7.2. If Hb (G, UK ) 6= 0, we may assume that uˇ1 ∈ Uk.

0 Proof. Since Hb (G, UK ) = Uk/NK/k(UK ), we can ﬁnd u ∈ Uk r NK/k(UK ). Write Q mi NK/k(u) = NK/k(ˇui) , where mi ∈ Z and the product runs from i = 1 to i = r. Q mi 2 1−σ 1+σ Then u = uˇi for some ∈ ker(NK/k). Sinceu ˇi =u ˇi uˇi , we may write Q ni u = 12 uˇi where 1 ∈ ker(NK/k), 2 ∈ NK/k(UK ) and ni = 0 or 1. If all the ni’s are 2 2 zero, then u = NK/k(u) = NK/k(2) = 2. Since Uk is torsion-free, u = 2 ∈ NK/k(UK ), a contradiction. Therefore, without loss of generality, we may assume n1 = 1, and −1 Q ni replacingu ˇ1 by u2 = 1 uˇi gives the desired basis.

We will need the following result of Reiner in proving the Rubin-Stark conjecture for quadratic extensions. Suppose G = hσi is a group of prime order p, with ζ a primitive p-th root of unity. Consider the following types of G-modules M:

Type 1) M = Z with trivial G-action. Type 2) M = U, an ideal of Z[ζ], with G-action given by σ·u = ζu. Type 3) Those M for which there exists a non-split extension

0 → U → M → Z → 0,

where U and Z are as in 2) and 1) respectively. APPENDIX A. APPENDICES 74

Theorem A.7.1 (Reiner). (see [Swa70] theorem 4.19, pp 73-74) Every ﬁnitely generated G-module is isomorphic to M M M1 M2 M3, where each Mi is a direct sum of G-modules of type i. Furthermore, the number of summands in each Mi is unique.

The case we are interested in is p = 2; in this case a module of type 2 is isomorphic to Z, with σ·n = −n, and it is not hard to see that any module of type 3 is isomorphic to 1 0 Z[G]. Note that if Mi is of type i (i = 1, 2), then H (G, M1) and Hb (G, M2) are both 0 1 Z/2Z. Thus if M is a ﬁnitely generated G-module with Hb (G, M) = H (G, M) = 0, M must be a direct sum of modules of type 3 only, hence free. Bibliography

[Aok04] Noboru Aoki. On Tate’s reﬁnement for a conjecture of Gross and its generalization. Journal de Th´eorie des Nombres de Bordeaux, 16:457–486, 2004.

[AW67] M. F. Atiyah and C. T. C. Wall. Cohomology of Groups. In J. W. S. Cassels & A. Fr¨olich., editor, ‘Algebraic Number Fields’, Pro- ceedings of the Brighton Symposium 1965. Academic Press Lon- don, 1967.

[Bly77] T. S. Blyth. Module Theory. Clarendon Press, 1977.

[Bra47] Richard Brauer. On Artin’s L-Functions with General Group Characters. The Annals of Mathematics, 48(2):502–514, April 1947.

[BS66] Z. I. Borevich and I. R. Shafarevich. Number Theory. Number 20 in Pure and Applied Mathematics: A Series of Monographs and Textbooks. Academic Press, 1966.

[Bur07] David Burns. Congruences between derivatives of abelian L- functions at s = 0. Inventiones Math., 169:451–499, 2007.

[Cas67] J. W. S. Cassels. Global Fields. In J. W. S. Cassels & A. Fr¨olich., editor, ‘Algebraic Number Fields’, Proceedings of the Brighton Symposium 1965. Academic Press London, 1967.

[CG98] P. Cornacchia and C. Greither. Fitting ideals of class groups of real ﬁelds with prime power conductor. J. Number Th., 73:459– 471, 1998.

75 BIBLIOGRAPHY 76

[Coh00] Henri Cohen. Advanced Topics in Computational Number Theory, volume 193 of Graduate Texts in Mathematics. Springer-Verlag, 2000.

[Coh03] P. M. Cohn. Basic Algebra: Groups, Rings and Fields. Springer- Verlag, 2003.

[Das] Samit Dasgupta. Stark’s Conjectures. Honours thesis, Harvard, 1999.

[DR80] P. Deligne and K. Ribet. Values of abelian L-functions at negative integers over totally real ﬁelds. Invent. Math., 59:227–286, 1980.

[Emm06] Caleb Emmons. Higher Order Integral Stark-Type Conjectures. PhD thesis, University of California, San Diego, 2006.

[Gra03] Georges Gras. Class Field Theory: From Theory to Practice. Springer Monographs in Mathematics. Springer-Verlag, 2003.

[Gro88] B. H. Gross. On the values of abelian L-functions at s = 0. J. Fac. Sci. Univ. Tokyo, 35:177–197, 1988.

[Hay04] A. Hayward. A class-number formula for higher derivatives of abelian L-functions. Compositio Mathematica, 140:99–129, 2004.

[JK81] Gordon James and Adalbert Kerber. The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Company, 1981.

[Lam01] Tsit-Yuen Lam. A First Course in Non-Commutative Rings. Number 131 in Graduate Texts in Mathematics. Springer-Verlag, 2 edition, 2001.

[Mar77] J. Martinet. Character Theory and Artin L-functions. In A. Fr¨olich., editor, ‘Algebraic Number Fields’, Proceedings of the Durham Symposium 1975. Academic Press London, 1977.

[Mat86] Hideyuki Matsumura. Commutative Ring Theory. Cambridge University Press, 1986.

[Nar90] W. Narkiewicz. Elementary and Analytic Theory of Algebraic Numbers. Springer-Verlag, second edition, 1990. BIBLIOGRAPHY 77

[Neu86] J¨urgen Neukirch. Class Field Theory, volume 280 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag, 1986.

[Nor] D. G. Northcott. Finite Free Resolutions, volume 71 of Cambridge Tracts in Mathematics. Cambridge University Press.

[Pop] Cristian Popescu. personal communication.

[Pop02] Cristian Popescu. Base change for Stark-type conjectures “over Z”. J. Reine Angew. Math, 524:85–111, 2002.

[Rim65] D. S. Rim. An Exact Sequence in Galois Cohomology. Proc. Amer. Math. Soc., 16(4):4837–840, 1965.

[Rom95] Steven Roman. Field Theory. Number 158 in Graduate Texts in Mathematics. Springer-Verlag, 1995.

[Rub92] K. Rubin. Stark units and Kolyvagin’s ‘Euler systems’. J. Reine Angew. Math, 452:141–154, 1992.

[Rub96] K. Rubin. A Stark Conjecture ‘over Z’ for Abelian L-functions with multiple zeros. Annales de l’institute Fourier, 46:33–62, 1996.

[San01] J. Sands. Stark’s question and Popescu’s conjecture for abelian L-functions. In M. Jutila and T. Mets¨ankyl¨a,editors, Number Theory: Proceedings of the Turku Symposium in Memory of K. Inkeri, pages 305–315. Walter de Gruyter, Berlin, 2001.

[Ser77] J.-P. Serre. Linear Representations of Finite Groups, volume 42 of Graduate Texts in Mathematics. Springer-Verlag, 1977.

[Ser79] J.-P. Serre. Local Fields. Number 67 in Graduate Texts in Math- ematics. Springer-Verlag, 1979.

[Shi76] T. Shintani. On the evaluation of zeta functions of totally real algebraic ﬁelds at non-positive integers. J. Fac. Sci. Univ. Tokyo, 23:393–417, 1976.

[Sie70] C. L. Siegel. Uber¨ die Fourierschen Koeﬃzienten von Modulfor- men. Nach. Akad. Wiss. G¨ottingen, 3:15–56, 1970. BIBLIOGRAPHY 78

[Sma] Charles Small. Arithmetic of Finite Fields, volume 148 of Pure and Applied Mathematics: A Program of Monographs, Textbooks, and Lecture Notes. Marcel Dekker, Inc.

[Sol02] D. Solomons. p-Adic Abelian Stark Conjectures at s = 1. Annales de l’institute Fourier, 52(2):379–417, 2002.

[Sta71] H. M. Stark. Values of L-Functions at s = 1. I. L-Functions for quadratic forms. Advances in Mathematics, 7:301–343, 1971.

[Sta75] H. M. Stark. L-Functions at s = 1. II. Artin L-Functions with Rational Characters. Advances in Mathematics, 17:60–92, 1975.

[Sta76] H. M. Stark. L-Functions at s = 1. III. Totally real ﬁelds and Hilbert’s twelfth problem. Advances in Mathematics, 22(1):64– 84, 1976.

[Sta80] H. M. Stark. L-Functions at s = 1. IV. First derivatives at s = 0. Advances in Mathematics, 35:197–235, 1980.

[Swa70] R. Swan. K-Theory of Finite Groups and Orders, volume 149 of Lecture Notes in Mathematics. Springer-Verlag, 1970.

[Tat67] J. Tate. Global Class Field Theory. In J. W. S. Cassels & A. Fr¨olich., editor, ‘Algebraic Number Fields’, Proceedings of the Brighton Symposium 1965. Academic Press London, 1967.

[Tat84] J. Tate. Les Conjectures de Stark sur les Fonctions L d’Artin en s = 0, volume 47 of Progress in Mathematics. Birkh¨auser, 1984.

[Was82] L. C. Washington. Introduction to Cyclotomic Fields. Number 83 in Graduate Texts in Mathematics. Springer-Verlag, 1982.

[Wei] C. Weibel. K-Theory (Chapter I). http://www.math.rutgers. edu/∼weibel/Kbook.html.