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1 Zp-Extensions and Ideal Class Groups

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1 Zp-Extensions and Ideal Class Groups 1 Zp-extensions and ideal class groups. This chapter will present the theorems of Iwasawa concerning the growth of ∞ ClFn [p ], where Fn varies over the layers in a Zp-extension of a number field F . The main theorem was proved by Iwasawa in the mid 1950s and concerns the growth of the orders of these groups. However, we will also prove results of Iwasawa concerning their group structure. A key ingredient in the proof is to consider the inverse limit ∞ X = X ∞ = lim Cl [p ] F /F ←− Fn n as a module over the formal power series ring Λ = Zp[[T ]]. The inverse limit X is defined by the norm maps NFm/Fn for m ≥ n ≥ 0. It turns out to be a finitely generated, torsion Λ-module. We will be able to partially describe the structure of such modules, enough for a proof of the theorem. Finally, in the last section, we discuss the special case where F = Q(µp) for an odd prime p and F∞ = Q(µp∞ ). Then F∞/F is a Zp-extension. The n-th layer is Fn = Q(µpn+1 ). There is a lot that we can say about the various invariants and modules introduced in this chapter, a topic which will be continued in later chapters. We will also discuss the relationship between XF∞/F and the unramified cohomology groups associated to powers of the cyclotomic character, continuing the topic of section 1.6. 1.1 Introductory remarks about Zp-extensions. The theorem of Iwasawa alluded to above concerns a certain type of infinite extension K of a number field F . These infinite extensions were originally referred to as “Γ-extensions” by Iwasawa, but later he adopted the more descriptive term “Zp-extensions.” Let p be a fixed prime. A Galois extension K/F is called a Zp-extension if the topological group Gal(K/F ) is isomorphic to the additive group Zp of p-adic integers. Except for the trivial subgroup, all the closed subgroups of Zp have finite n index. Such a closed subgroup is of the form p Zp for some nonnegative integer n and the corresponding quotient group is cyclic of order pn. Thus, if K/F is a Zp-extension, the finite extensions of F which are contained in K form a tower F = F0 ⊂ F1 ⊂···⊂ Fn ⊂ · · · of Galois extensions of F such n that Gal(Fn/F )= Z/p Z for all n. Clearly K = n≥0 Fn. If one chooses any S 1 γo ∈ Gal(K/F ) such that γo|F1 is nontrivial, then the infinite cyclic subgroup generated by γo is dense in Gal(K/F ). We therefore say that Gal(K/F ) is a topologically cyclic group and that the element γo is a topological generator of Gal(K/F ). We will often use the notation F∞ for a Zp-extension of F . Let F be any number field and let p be a fixed prime. One important example of a Zp-extension of F is quite easy to construct. Let µp∞ denote the group of p-power roots of unity. The extension F (µp∞ )/F is an infinite Galois extension. At the beginning of section 1.6, we defined a continuous homomorphism × ∞ χ : Gal(F (µp )/F ) → Zp . This homomorphism is injective. Consequently, Gal(F (µp∞ )/F ) is isomor- × phic to an infinite closed subgroup of Zp . Such a group has a finite torsion subgroup and the corresponding quotient group will be isomorphic to Zp. ∞ ∼ Therefore, F (µp ) contains a unique subfield F∞ such that Gal(F∞/F ) = Zp. We refer to F∞ as the cyclotomic Zp-extension of F . In particular, we will let Q∞ denote the cyclotomic Zp-extension of Q. The cyclotomic Zp-extension of an arbitrary number field F is then F∞ = F Q∞. It is easy to show that the primes of F which are ramified in the cyclotomic Zp-extension F∞/F are precisely the primes lying over p. For an arbitrary Zp-extension, we have the following result. Proposition 2.1.1. Suppose that F∞/F is a Zp-extension. If v is a prime of F which is ramified in the extension F∞/F , then v lies over p. At least one such prime must be ramified in F∞/F . Proof. Let Γ = Gal(F∞/F ). Let Iv denote the inertia subgroup of Gal(F∞/F ) corresponding to a prime v of F . If v is ramified in F∞/F , then Iv is non- trivial. Hence Iv must be infinite. If v is an archimedian prime of F , then Iv would be of order 1 or 2, and so must be trivial. Consequently, archimedian primes of F split completely in F∞/F . If v is nonarchimedian, but lies over l, where l =6 p, then v is tamely ramified in F∞/F . It is known in general that if v is tamely ramified in any abelian extension of F , then its ramification index must divide N(v) − 1, where N(v) denotes the cardinality of the residue field for v. This can be proved either by using properties of ramification groups or by using local class field theory. (See reference.) Thus, Iv would be finite. Therefore, Iv must be trivial and v must be unramified in F∞/F . For the final assertion, we just remark that the maximal unramified, abelian extension of F (the Hilbert class field of F ) has finite degree over F . Thus, F∞/F must be ramified for at least one prime. 2 The existence and ramification properties of Zp-extensions of any number field F will be discussed in considerable detail in chapter 3. We will just make a few remarks now. If we take F = Q as the base field, then it is not hard to prove that there is only one Zp-extension, the cyclotomic Zp-extension Q∞ which was constructed above. To see this, one can use the Kronecker- Weber theorem which asserts that the maximal abelian extension Qab of Q is generated by all the roots of unity. Proposition 2.1.1 then implies that any Zp-extension of Q must be ramified only at p and therefore contained in Q(µp∞ ), and so must be Q∞. The cyclotomic Zp-extension of an arbitrary number field F is F∞ = F Q∞. If F is a totally real number field, then it should again be true that the cyclotomic Zp-extension is the only Zp-extension of F . This can be proved if F ⊂ Qab, but is an open question in general (a special case of “Leopoldt’s Conjecture”). If F is not totally real, then it turns out that there are infinitely many distinct Zp-extensions of F . We will discuss this matter in detail in chapter 3. In particular, theorem 3.3 gives a quantitative statement about the existence of Zp-extensions of an arbitrary number field F . One of the main results to be proved in this chapter is the following famous theorem of Iwasawa. Iwasawa’s Growth Formula. Suppose that F∞ = n≥0 Fn is a Zp-extension (p) en of a number field F . Let hn denote the class numberS of Fn and let hn = p denote the largest power of p dividing hn. Then there exists integers λ, µ, n and ν such that en = λn + µp + ν for all sufficiently large n. Iwasawa’s growth formula will be proved in section 2.4, based largely on the results of section 2.2 and 2.3. The integers λ and µ will be nonnegative. We will refer to them as the Iwasawa invariants for F∞/F , often denoting them by λ(F∞/F ) and µ(F∞/F ). Several interpretations of them will be given as we proceed. Proposition 1.1.4 implies one very simple special case of Iwasawa’s theo- rem, namely the following useful result. Proposition 2.1.2. Suppose that F is a number field and that p does not divide the class number of F . Let F∞ = n≥0 Fn be a Zp-extension of F and suppose that only one prime of F is ramifiedS in F∞/F . Then p does not divide the class number of Fn for any n ≥ 0. Therefore, Iwasawa’s growth formula is valid with λ = µ = ν = 0. Proof. Suppose that Iv denotes the inertia subgroup of Gal(F∞/F ) for the 3 one ramified prime v. It is clear that v must be totally ramified in F∞/F . Iv Otherwise, F∞ would be a nontrivial, unramified, cyclic p-extension of F , contradictict the assumption that p ∤ hF . Hence. for each n ≥ 0, the hy- potheses in proposition 1.1.4 are satisfied for the extension Fn/F . Therefore, the class number of Fn is not divisible by p. In particular, this result applies if F has only one prime lying above p and p ∤ hF . For example, p doesn’t divide the class number of Qn, the n-th layer in the cyclotomic Zp-extension Q∞/Q. Also, if we take F = Q(µp), where p is any odd regular prime, then it follows that the class number of Q(µpn ) will not be divisible by p for all n ≥ 1. The class number of Q(µ2n ) is 1 for n ≤ 2 and is odd for n> 2, again by proposition 2.1.2. Now suppose that F∞ = n≥0 Fn is any Zp-extension of F . For every n ≥ 0, let Ln denote the p-HilbertS class field of Fn. Let L∞ = n≥0 Ln. Then L∞ is an abelian extension of F∞.
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