On 2-adic Lie iterated extensions of number fields arising from a Joukowski map
The article provides some technical developments in Iwasawa theory, which could have important consequences, particularly in the study of ideal class numbers. The main objects of study in the paper are dyadic (2-adic) Lie extensions of an algebraic number field, and their associated Iwasawa modules. These modules are of great consequence within number theory, since the dyadic growth of the class number of such a Lie extension is controlled by the structure of Iwasawa modules and their associated Galois action. The authors are particularly interested in the notion of ``pseudo-nullity'' of Iwasawa modules, which in this context simply means finiteness, and they take steps towards advancing a conjecture of Ralph Greenberg that the Iwasawa module is pseudo-null if the underlying field is totally real, and the extension is cyclotomic. The main results of the article establish conditions on a base field \(k\) and an extension \(K_{\infty}\) that result in the Iwasawa module being pseudo-null. The extensions in question are constructed using a ``Joukowski map'', i.e., a complex map of the form \(J(x)=\frac{x+x^{-1}}{2}\). Taking \(\phi\) to be a specific conjugate of \(J\), the main theorems in the paper consider a 2-adic extension \(K_{\infty}\) of \(k\) arising from a union of fields \(K_n\) of the form \(K_n=k(\phi^{-n}(b_0))\), where \(b_0\) is a canonically chosen element of \(k\). The first result of the paper, Theorem 1, states that the Iwasawa module \(X(K_{\infty})\) is pseudo-null under certain appropriate conditions which would follow from Greenberg's theorem, and Theorem 2 then states a simple number theoretic criterion for \(b_0\) which also ensures that \(X(K_{\infty})\) is pseudo-null. The structure of the paper is very straightforward. After outlining some motivation and stating the main results in section 1, it goes on in section 2 to outline some preliminary definitions and concepts used throughout the paper, mainly concerning constructions of extension fields and some number theoretic ideas. In section 3, the most technical section of the article, the authors go on to prove Theorem 1 by exploring the geometric properties of Joukowski maps and their conjugates. Finally, in section 4, they give a short proof of Theorem 2 before going on to outline some more general examples of field extensions with trivial or pseudo-null Iwasawa modules. Overall, this paper is a short, technical account of some illustrative examples of Iwasawa modules, which are fundamental objects in number theory, and it illustrates properties of these modules which lend credence to an important conjecture in the field. This article should be of interest to anyone who works in Iwasawa theory or ideal class theory, and while its results may not be ground-breaking, it is certainly important for the community to be aware of them.
scientific article; zbMATH DE number 7394481
scientific article