On the Invariants of some Zℓ-Extensions

Author: Bloom, John Roll

Year: 1977

Degree: Dissertation (Ph.D.)

Advisor: Kisilevsky, Hershy Harry

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/s74r-gx65

Abstract

Let k be a number field, l a prime, kck1ck2c...cK, and kcm1cm2c...cM two Zl-extensions of k. The structure of the galois group of a certain extension of MK is studied, and it is shown how, in some cases, the l-parts of the class groups of the intermediate fields mikj. can be obtained from this group.

This galois group is a module over Zl[[S,T]], the power series ring in two variables over the l-adic integers, but the structure theory of such modules is not well developed. The main results come from studying the structure of this group as a Zl[[S]] or Zl[[T]] module. Necessary and sufficient conditions are given for this group to be a Noetherian module over Zl[[T]], and thus it has a well known structure. Sufficient conditions are given for the module to be a torsion module.

The structure of this group is then used to obtain information on the Iwasawa invariants μ and λ of the Zl-extensions kmi/mi and Mkj/kj. In suitable situations it is shown that μ(K/k)=O implies that μ(Kmi/mi)=0 for all i, and λ(Kmi/mi)=rli + iΣj=0 cjφ(lj), with cj=0 for all j>n0 and it is shown that r=0 iff the above module is torsion.

In certain situations, this group is also used to study the invariants of all Zl-extensions of k contained in MK. With suitable hypotheses, it is shown that at most one Zl-extension has μ≠0.

Some examples are computed.

Files