Relative genus theory and the class group of 𝑙-extensions

Abstract

Abstract. The structure of the relative genus field is used to study the class group of relative /-extensions. Application to class field towers of cyclic /-extensions of the rationals are given.Given a number field E one tries to understand the unramified abelian extensions of E and so by Class Field Theory derive information about the class group of E, CE. One way of doing this is to ask for the maximal abelian unramified extension of E of the form EF^ where F" is an abelian extension of F C E. This field is called the relative genus field and we denote it by E*. If F -Q we speak of the absolute genus field. This paper will use relative genus theory to study the class group of certain /-extensions of Q. (Henceforth, / will always be an odd prime.) In particular we give applications to class field towers of such fields. We also study the /-ranks of the class group of number fields E containing a cyclic subextension of degree /. This will lead to a slight improvement on a bound of Iwasawa [13]. This technique is then used to give bounds on both the /-rank and the order of the class group of elementary abelian /-extensions of Q.

Counting non-uniform lattices

“…Gauss proved his celebrated theorem using the genus theory which he developed. Later on the method was applied to prove a similar result for biquadratic extensions of Q (see [C,Theorem 2]). We use these results in the proof of Theorem 1; the main restriction on the type of H which we impose there follows from the fact that in order for the genus theory to be applicable we need to assume that the center of the simply connected cover of the split form of H is a 2-group.…”

On Hilbert 2-class fields and 2-towers of imaginary quadratic number fields

“…Remark 3.8. In view of the group-theoretic success in (for instance) Koch [23], Maire [35], and Benjamin-Lemmermeyer-Snyder [4], it could potentially be enlightening to analyze the "near miss" or "borderline" cases Cl 2 (F ) (2,4) or Cl 2 (F ) (2,8). Proof of Lemma 3.6(2).…”

Counting maximal arithmetic subgroups

“…(i) if [l : Q] = 2, then ρ 2 (l) ≤ t l − 1 (by Gauss); (ii) if [k : Q] = 2 and [l : k] = 2, then ρ 2 (l) ≤ 2(t l +t k −1) (by [Cor,Theorem 2]…”