Cyclotomic Units and Class Groups in Z_p extensions of real abelian fields

TL;DR

This paper investigates the relationship between class groups and units in Z_p extensions of real abelian fields, assuming Greenberg's conjecture, and determines the kernel and cokernel of a key map.

Contribution

It provides a detailed analysis of the kernel and cokernel of a map between class groups and units in the context of cyclotomic Z_p-extensions, assuming Greenberg's conjecture.

Findings

Kernel and cokernel of the map are explicitly determined.

Results depend on Greenberg's conjecture about the lambda-invariant.

Enhances understanding of class groups and units in cyclotomic extensions.

Abstract

For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension.