Cyclotomic conjecture and semi-simplicity of S-split T-ramified Iwasawa modules

TL;DR

This paper explores the cyclotomic conjecture's implications on the structure of Iwasawa modules, establishing its relation to classical conjectures in number theory, especially in CM-fields.

Contribution

It demonstrates that the cyclotomic conjecture influences the Zℓ-rank of fixed point submodules and proves equivalences with Leopoldt and Gross-Kuz'min conjectures in CM-fields.

Findings

Cyclotomic conjecture governs the Zℓ-rank of fixed point submodules.

In CM-fields, the weak and strong conjectures are equivalent to Leopoldt and Gross-Kuz'min conjectures.

The results connect Iwasawa theory with classical conjectures in number theory.

Abstract

We show that the cyclotomic conjecture on the characteristic polynomial of T-ramified S-split Iwasawa modules introduced in a previous paper and satisfied by abelian fields governs the Z${\ell}$-rank of the submodule of fixed points for all finite disjoint sets S and T of places.Last, in the CM-case we prove that the weak and the strong versions of the cyclotomic conjecture both are equivalent to the conjunction of the classical conjectures of Leopoldt and Gross-Kuz'min.