IWASAWA THEORY Romyar Sharifi Contents Introduction 3 Chapter 1. Class groups and units7 1.1. Notation and background7 1.2. Regulators9 1.3. Finite Galois extensions 11 1.4. Kummer theory 20 1.5. Leopoldt’s conjecture 24 Chapter 2. Module theory 33 2.1. Pseudo-isomorphisms 33 2.2. Power series rings 39 2.3. Completed group rings 42 2.4. Invariants of L-modules 46 2.5. Pontryagin duality 54 2.6. Iwasawa adjoints 56 2.7. The group ring of a cyclic p-group 61 2.8. Eigenspaces 63 Chapter 3. Iwasawa theory 69 3.1. Zp-extensions 69 3.2. Limits of class groups 71 3.3. The p-ramified Iwasawa module 76 3.4. CM fields 82 3.5. Kida’s formula 85 Chapter 4. Cyclotomic fields 91 4.1. Dirichlet L-functions 91 4.2. Bernoulli numbers 95 4.3. Cyclotomic units 101 4.4. Reflection theorems 103 3 4 CONTENTS 4.5. Stickelberger theory 106 4.6. Distributions 109 4.7. Sinnott’s theorem 112 Chapter 5. Kubota-Leopoldt p-adic L-functions 119 5.1. p-adic measures 119 5.2. p-adic L-functions 122 5.3. Iwasawa power series 127 5.4. Coleman theory 131 Chapter 6. The Iwasawa main conjecture 145 6.1. Semi-local units modulo cyclotomic units 145 6.2. The Ferrero-Washington theorem 148 6.3. The main conjecture over Q 152 6.4. The Euler system of cyclotomic units 154 6.5. The main conjecture via Euler systems 161 6.6. Geometry of modular curves 165 Appendix A. Duality in Galois cohomology 171 Bibliography 175 Introduction The class group ClF of a number field F is an object of central importance in number theory. It is a finite abelian group, and its order hF is known as the class number. In general, the explicit determination of hF , let alone the structure of ClF as a finite abelian group, can be a difficult and computationally intensive task. In the late 1950’s, Iwasawa initiated a study of the growth of class groups in certain towers of number fields. Given a tower F = F0 ⊂ F1 ⊂ F2 ⊂ ··· of Galois extensions of F, one asks if there is any regularity to the growth of hFn . The knowledge of this growth, in turn, can be used to say something about the structure of ClFn as a finite abelian group. Iwasawa was concerned with towers ∼ S such that Gal(F¥=F) = Zp for some prime p, where F¥ = n Fn, known as Zp-extensions. He set n G = Gal(F¥=F) and Gn = Gal(Fn=F), and let us suppose that Fn is chosen to be (cyclic) of degree p over F. For example, for odd p, the cyclotomic Zp-extension F¥ of F is the largest subextension of F(mp¥ )=F with pro-p Galois group. The question of how hFn grows in the tower defined by a Zp-extension is quite difficult, in particular as the order away from p of ClFn has little to do with the order away from p of ClFn+1 , other than the fact that the latter order is a multiple of the former. On the other hand, if we concentrate on the order h(p) of the Sylow p-sugroup A of F , we have the following theorem of Iwasawa. Fn n n THEOREM (Iwasawa). There exist nonnegative integers l and m and an integer n such that n h(p) = pnl+p m+n Fn for all sufficiently large n. In the case that F¥ is the cyclotomic Zp-extension, Iwasawa conjectured that the invariant m in the theorem is 0. Ferrero and Washington later proved this result for abelian extensions of Q. We have maps between the p-parts of class group in the tower in both directions jn : An ! An+1, which takes the class of an ideal a to the class of the ideal it generates, and Nn : An+1 ! An, which takes the class of an ideal to the class of its norm. Iwasawa considered the direct and inverse limits A = A X = A ¥ −!lim n and ¥ lim− n n n 5 6 INTRODUCTION under the jn and Nn, respectively. As each An has the structure of a finite Zp[Gn]-module through the _ standard action of Gn on ideal classes, both X¥ and the Pontryagin dual A¥ = Homcts(A¥;Qp=Zp) of A¥ are finitely generated torsion modules over the competed Zp-group ring of G: = [ ]: Zp G lim− Zp Gn J K n The ring L = Zp G is known as the Iwasawa algebra, and it has a very simple structure. In fact, a J K choice of a topological generator g of G gives rise to an isomorphism ∼ Zp T −! L; T 7! g − 1: J K The following result on the structure of L-modules allowed Serre to rephrase the theorem of Iwa- sawa. THEOREM (Serre). For any finitely generated torsion L-module M, there exists a homomorphism of L-modules s t M ki M ` j M ! L= fi(T) L ⊕ L=p L; i=1 j=1 with finite kernel and cokernel, for some nonnegative integers s and t, irreducible fi(T) 2 Zp[T] with deg f fi(T) ≡ T i mod p, and positive integers ki and ` j. From Serre’s theorem, we are able to deduce several important invariants of a finitely generated L-module M. For instance, in the notation of the theorem, let us set s t l(M) = ∑ ki deg fi and m(M) = ∑ ` j: i=1 j=1 _ These are known as the l and m-invariants of M. Serre showed that these invariants for X¥ and A¥ agree with the l and m of Iwasawa’s theorem. An even more interesting invariant of M is its characteristic ideal, given by s ! m(M) ki charL M = p ∏ fi(T) L; i=1 which we shall consider in a specific case shortly. It is worth remarking here that one usually thinks of X¥ as a Galois group. Recall that the Artin reciprocity map provides an isomorphism between An and the Galois group of the Hilbert p-class field S Ln of Fn, which is to say the maximal unramified abelian p-extension of Fn. Setting L¥ = n Ln, ∼ we have a canonical isomorphism X¥ = Gal(L¥=F¥). The resulting action on G on Gal(L¥=F¥) is a conjugation action, given by a lift of G to a subsgroup of Gal(L¥=F). Let us focus now on the specific case that F = Q(mp), and let us take F¥ to be the cyclotomic Zp-extension of F for an odd prime p. In this setting, Iwasawa proved that his m = m(X¥) is zero. INTRODUCTION 7 × We define the Teichmuller¨ character w : D ! Zp by setting w(d) for d 2 D to be the unique (p − 1)st root of unity in Zp such that w(d) d(zp) = zp for any primitive pth root of unity zp. As with G, the Galois group D = Gal(F=Q) will act on X¥. For any i, we may consider the (i) i eigenspace X¥ of X¥ on which every d 2 D acts through multiplication by w (d). We have the fol- lowing theorem of Herbrand and Ribet. (1−k) THEOREM (Herbrand-Ribet). Let k be an even with 2 ≤ k ≤ p − 3. Then X¥ 6= 0 if and only if p divides the Bernoulli number Bk. The interesting fact is that Bernoulli numbers and their generalizations appear as values of L- functions. Kubota and Leopoldt showed how that the L-values of certain characters at negative integers can be interpolated, in essence, by a function of Zp, denoted Lp(c;s) and known as a p-adic L-function. Let us fix the particular generator g of G such that g(z) = z 1+p for every p-power root of unity z, and in particular the isomorphism of L with Zp T . Iwasawa made the following conjecture on the J K characteristic ideal of an eigenspace of X, which was later proven by Mazur and Wiles. THEOREM (Main conjecture of Iwasawa theory, Mazur-Wiles). Let k be an even integer. Then (1−k) charL X¥ = ( fk); s k where fk((1 + p) − 1) = Lp(w ;s) for all s 2 Zp. In fact, Mazur and Wiles proved a generalization of this to abelian extensions F of Q, and Wiles proved a further generalization to abelian extensions of totally real fields. This line of proof was primarily geometric in nature, and came by studying the action of the absolute Galois group of F on the cohomology groups of modular curves. Rubin gave a proof of a rather different nature of a main conjecture for abelian extensions of imaginary quadratic fields, following work of Kolyvagin and Thaine, using a Galois cohomological tool known as an Euler system. Let us end this introduction by mentioning the two of the major directions in which Iwasawa theory has expanded over the years. As a first and obvious course of action, one can replace our limits of class _ groups with more general objects. Via class field theory, we note that the Pontryagin dual X¥ may be identified with the kernel of the map 1 M 1 ker H (GF¥;S;Qp=Zp) ! H (Iv;Qp=Zp) ; v2S where GF¥;S denotes the Galois group of the maximal extension of F¥ unramified outside S and S in this case is the set of primes of F¥ lying over p, and where Iv is the inertia group at v 2 S in the absolute 8 INTRODUCTION _ Galois group of F¥.