General SectionOn iterated extensions of number fields arising from quadratic polynomial maps
Introduction
Let k be a number field, and a rational function of degree . Let be the n-fold iterate of ϕ, that is, the n-fold composition of ϕ with itself. Fix . The set 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 commutes with ϕ, we obtain a map where is the group of tree automorphisms of . 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 (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 . Let E be an elliptic curve defined over a number field k, p a rational prime. Then we obtain the p-adic Galois representation . Serre [21] proved that for a given elliptic curve E without complex multiplication the image of has finite index in for all p.
In this paper, we consider the case where the size of the image of ρ is small (i.e. ). 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 is ramified above only finitely many primes of k (see [1], [4]).
An example of post-critically finite ϕ is . Then the iterated extension for is the cyclotomic -extension of . Also, if ϕ is the x-coordinate of the multiplication-by-p map of an elliptic curve E, then for some base point β (see [4, Section 5]). In particular, and are both p-adic Lie extensions, and the latter is studied in Iwasawa theory. For a number field k of finite degree, let denote a -extension of k. For each integer , there exists a unique intermediate field of with . Let be the p-Sylow subgroup of the ideal class group of . In [11], Iwasawa proved that, for all sufficiently large n, where , , are rational integers (called the Iwasawa invariants of ). Greenberg [9] conjectured that for the cyclotomic -extension of any totally real number field k. On the other hand, the similar phenomenon sometimes occurs for non-cyclotomic -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 the p-Sylow subgroup of the ideal class group of k. Let be a post-critically finite rational function with and let be a sequence satisfying and for . Problem 1.1 Put and . Then how large can be as ? Moreover, is the projective limit 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 , , and . If we put and , since , the iterated extensions associated to and are the same, and is the splitting field of the polynomial .
For an algebraic extension , we denote by 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 the set of ideal classes in such that is a product of primes lying over T. We put . By class field theory, 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 . Put and , where is a root of , such that Let denote the splitting field of over k. Let and let . If and , then the following three conditions hold true. for all , is unramified outside 2 if γ is an algebraic integer. If is not Galois, then is a 2-adic Lie group of dimension 2.
Moreover, if we put , where is a prime number, and if γ is an algebraic integer, then the following also hold; Let and be two distinct primes in k lying over 2. If ramifies in and is inert in , then for all , for all , and is an infinite quotient of on which acts trivially. (Then, in particular, is a pseudo-null -module.) If, moreover, ramifies in and , then for all and for any . The latter implies that .
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 -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 -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 -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 , , , and , 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 be a -extension of number fields, and let be the subextension of degree p. Let S be the set of all ramified primes of F in . Suppose and , where is inert in and totally ramifies in . Let T be such that . If , then . Remark 2.2 Since is the p-Sylow subgroup of the T-ideal class group , if and only if 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 . Put . We choose an end of the tree , in other words, we have and for . We obtain the following for iterated extensions . Theorem 3.1 Let k be a number field and let . Let . Let be a sequence satisfying and for , and let . If
Galois groups of 2-adic Lie extensions
Let k be a number field and let . First, we compute the splitting field of the n-th iterate of over k in the situation of Theorem 3.1. For , the splitting fields of were already known by Gottesman and Tang [8]. If we assume and , then there is only one value of γ satisfying the condition of Theorem 3.1: . We treat the general case of a -iteration arising from f over k based on the idea of [8]. Note that for any choice of γ, becomes a subquotient of
2-Class groups of iterated extensions
Combining the results in Section 2 for , 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 , and the class number of k is odd. Let be a -iteration arising from with over k. For , let be the splitting field of over k. Then and for all . Proof Let be a prime ideal of k lying above 2, and let be a prime ideal of lying above for . By assumption,
Calculation of examples
Suppose , where is a prime number with . We shall construct -iterations over k when γ is an algebraic integer in k. Put . Then is a -extension. Let be the roots of unity in K and let be the unit index of . Suppose α is an algebraic integer of K such that . Then Since 2 is unramified in and is ramified in , is ramified only at the prime above 2. So we have . Therefore for some and
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