Elsevier

Journal of Number Theory

Volume 209, April 2020, Pages 289-311
Journal of Number Theory

General Section
On iterated extensions of number fields arising from quadratic polynomial maps

https://doi.org/10.1016/j.jnt.2019.08.022Get rights and content

Abstract

A post-critically finite rational map ϕ of prime degree p and a base point β yield a tower of finitely ramified iterated extensions of number fields, and sometimes provide an arboreal Galois representation with a p-adic Lie image. In this paper, we take ϕ to be the monic Chebyshev polynomial x22, and we examine the size of the 2-part of the ideal class group of extensions in the resulting tower. In some cases, we prove an analogue of Greenberg's conjecture from Iwasawa theory. A key tool is a general theorem on p-indivisibility of class numbers of relative cyclic extensions of degree p2.

Introduction

Let k be a number field, and ϕk(x) a rational function of degree d2. Let ϕn be the n-fold iterate of ϕ, that is, the n-fold composition of ϕ with itself. Fix βP1(k). The set T=n0ϕn(β) has the structure of a tree with root β by assigning edges according to the action of ϕ. Since any element of the absolute Galois group Gal(k/k) commutes with ϕ, we obtain a mapρ:Gal(k/k)Aut(T), where Aut(T) is the group of tree automorphisms of T. The map ρ is called the arboreal Galois representation (see [2], [15], etc.). Much recent work has concerned the question of when the image of ρ is a large subgroup of Aut(T) (see [13], [15], [16], [20] etc.).

In the well-studied case of p-adic Galois representations arising from elliptic curves, there is an analog for the case [Aut(T):Im(ρ)]<. Let E be an elliptic curve defined over a number field k, p a rational prime. Then we obtain the p-adic Galois representation ρE:Gal(k/k)Aut(E[p])GL2(Zp). Serre [21] proved that for a given elliptic curve E without complex multiplication the image of ρE has finite index in GL2(Zp) for all p.

In this paper, we consider the case where the size of the image of ρ is small (i.e. [Aut(T):Im(ρ)]=). By [15, Theorem 3.1], if ϕ is post-critically finite, i.e., if the union of all forward orbits of the critical points of ϕ is a finite set, then the size of the image of ρ is small. Moreover, when ϕ is post-critically finite, the extension k(T)/k is ramified above only finitely many primes of k (see [1], [4]).

An example of post-critically finite ϕ is ϕ(x)=x22. Then the iterated extension for T=n0ϕn(0) is the cyclotomic Z2-extension of Q. Also, if ϕ is the x-coordinate of the multiplication-by-p map of an elliptic curve E, then k(T)k(E[p]) for some base point β (see [4, Section 5]). In particular, k(T)/k and k(E[p])/k are both p-adic Lie extensions, and the latter is studied in Iwasawa theory. For a number field k of finite degree, let k denote a Zp-extension of k. For each integer n0, there exists a unique intermediate field kn of k/k with [kn:k]=pn. Let A(kn) be the p-Sylow subgroup of the ideal class group of kn. In [11], Iwasawa proved that, for all sufficiently large n,|A(kn)|=pλ(k/k)n+μ(k/k)pn+ν(k/k), where λ(k/k), μ(k/k), ν(k/k) are rational integers (called the Iwasawa invariants of k/k). Greenberg [9] conjectured that λ(k/k)=μ(k/k)=0 for the cyclotomic Zp-extension of any totally real number field k. On the other hand, the similar phenomenon sometimes occurs for non-cyclotomic Zp-extensions of imaginary quadratic fields (see [7], [10]).

We now consider the following problem as an analogue of Greenberg's conjecture. For a number field k, we denote by A(k) the p-Sylow subgroup of the ideal class group of k. Let ϕk(x) be a post-critically finite rational function with degϕ=p and let {βn} be a sequence satisfying β0=β and ϕ(βn+1)=βn for n0.

Problem 1.1

Put kn=k(βn) and Kn=k(ϕn(β)). Then how large can A(kn) be as n? Moreover, is the projective limit limA(Kn) with respect to the norm maps also “small” as a Galois module? In particular, is it “pseudo-null” if ρ has a p-adic Lie image (cf. [23])?

For this problem, we consider the case where p=2, ϕ(x)=x22, and β0=β=γ. If we put f(x)=gϕg1(x)=(xγ)2+γ2 and g(x)=x+γ, since gPGL2(k), the iterated extensions associated to n0ϕn(β) and n0fn(0) are the same, and Kn is the splitting field of the polynomial fn(x).

For an algebraic extension K/Q, we denote by Kur,ab the maximal unramified abelian p-extension of K. For a number field K, let T be a set of primes of a subfield of K. We denote by DT(K) the set of ideal classes [a] in A(K) such that a is a product of primes lying over T. We put AT(K)=A(K)/DT(K). By class field theory, AT(K) is isomorphic to the Galois group of the maximal unramified abelian p-extension of K in which all primes lying over T split completely. The main result of this paper is the following.

Theorem 1.2

Let k be a number field and let γk. Put f(x)=(xγ)2+γ2k[x] and kn=k(αn), where αn=βn+γ is a root of fn(x), such thatk=k0k1kn. Let Kn=k(fn(0)) denote the splitting field of fn over k. Let k=n0kn and let K=n1Kn. If 2+γ(k×)2 and (2+γ)(2γ)(k×)2, then the following three conditions hold true.

  • Gal(kn+2/kn)C4 for all n0,

  • k/k is unramified outside 2 if γ is an algebraic integer.

  • If k/k is not Galois, then Gal(K/k)Z2Z2 is a 2-adic Lie group of dimension 2.

Moreover, if we put k=Q(q), where q7(mod8) is a prime number, and if γ is an algebraic integer, then the following also hold; Let p1 and p2 be two distinct primes in k lying over 2. If p1 ramifies in k1/k and p2 is inert in k1/k, then A{p2}(kn)=0 for all n0, A{p1,p2}(Kn)=0 for all n1, and Gal(Kur,ab/K)limA(Kn) is an infinite quotient of Z22 on which Gal(K/k) acts trivially. (Then, in particular, limA(Kn) is a pseudo-null Z2[[Gal(K/k)]]-module.) If, moreover, p2 ramifies in k2/k1 and A(k2)=0, then A(kn)=0 for all n0 and A{p1}(Kn)=0 for any n1. The latter implies that Gal(Kur,ab/K)limA(Kn)Z2.

Proof

This is a combination of Theorem 3.1, Theorem 4.3, Theorem 5.3, Corollary 5.6, which are proved in the following sections. 

An important ingredient in the proof of Theorem 1.2 is a result on p-indivisibility of class numbers of relative Cp2-extensions. This result is given in Section 2 (see Theorem 2.1), and it may be regarded as a partial generalization of Fukuda's theorem [6]. In Section 3, we consider the sequence of number fields in Theorem 1.2 (call it C4-iteration, see Theorem 3.1 below). In Section 4, we give Galois-theoretic results. In Section 5, we obtain some results for Problem 1.1. In Section 6, we examine the construction of C4-iterations over an imaginary quadratic field and give some computations.

Notations

For a number field K the unit group, the ring of integers, the ideal class group, and the class number are denoted by EK, OK, Cl(K), and hK, respectively.

Acknowledgments

I would like to thank my advisor, Yasushi Mizusawa, for his patient guidance and useful suggestions, and for supporting this work by JSPS KAKENHI Grant Number JP17K05167. The author also thanks Satoshi Fujii, Sohei Tateno, and Takashi Hara for helpful comments. The author also thanks the referee for constructive suggestions for the improvement of this paper.

Section snippets

p-indivisibility of class numbers

Theorem 2.1

Let p be a prime number, let K/F be a Cp2-extension of number fields, and let M/F be the subextension of degree p. Let S be the set of all ramified primes of F in K/F. Suppose S and S=S1S2, where S1={pS|p is inert in M/F} and S2={pS|p totally ramifies in K/F}. Let T be such that S1TS. If AT(M)=0, then AT(K)=0.

Remark 2.2

Since AT(K) is the p-Sylow subgroup of the T-ideal class group ClT(K), AT(K)=0 if and only if ClT(K) has order indivisible by p.

Proof

Let L be the maximal unramified abelian p-extension

Ramification in iterated extensions

A partial result on ramification is obtained by [1], however we discuss in detail for the convenience of the reader. Let k be a number field and let γk. Put f(x)=(xγ)2+γ2. We choose an end (α0,α1,α2,) of the tree T=n0fn(0), in other words, we have α0=0 and f(αn+1)=αn for n0. We obtain the following for iterated extensions {k(αn)}.

Theorem 3.1

Let k be a number field and let γk. Let f(x)=(xγ)2+γ2k[x]. Let {αn} be a sequence satisfying α0=0 and f(αn+1)=αn for n0, and let kn=k(αn). If 2+γ(k×)2

Galois groups of 2-adic Lie extensions

Let k be a number field and let γk. First, we compute the splitting field of the n-th iterate of f(x)=(xγ)2+γ2 over k in the situation of Theorem 3.1. For k=Q, the splitting fields of fn were already known by Gottesman and Tang [8]. If we assume k=Q and γZ, then there is only one value of γ satisfying the condition of Theorem 3.1: γ=0. We treat the general case of a C4-iteration arising from f over k based on the idea of [8]. Note that for any choice of γ, Gal(K/k) becomes a subquotient of

2-Class groups of iterated extensions

Combining the results in Section 2 for p=2, Theorem 3.1, and Proposition 4.1, we obtain the following results.

Theorem 5.1

Let k be a totally imaginary number field such that 2 does not split in k/Q, and the class number of k is odd. Let {kn} be a C4-iteration arising from f(x)=(xγ)2+γ2 with γOk over k. For n1, let Kn be the splitting field of fn over k. Then 2hkn and 2hKn for all n1.

Proof

Let p be a prime ideal of k lying above 2, and let pn be a prime ideal of kn lying above p for n1. By assumption, p

Calculation of examples

Suppose k=Q(q), where q3(mod4) is a prime number with q3. We shall construct C4-iterations over k when γ is an algebraic integer in k. Put K=k(1). Then K/Q is a C2×C2-extension. Let μK be the roots of unity in K and let q(K) be the unit index of K/Q. Suppose α is an algebraic integer of K such that |NK/kα|=4. ThenNK/k(αOK)=(2Ok)2. Since 2 is unramified in k/Q and is ramified in Q(1)/Q, K/k is ramified only at the prime above 2. So we have αOK=2OK. Thereforeα=2ζxuy for some x{0,1,2,3} and y

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