Note on Greenberg conjecture

TL;DR

This paper investigates Greenberg's conjecture on Iwasawa { extell}-invariants of totally real fields using logarithmic class groups, establishing new equivalences and conditions under Leopoldt's conjecture.

Contribution

It introduces a new perspective using logarithmic class groups, proving equivalences and providing explicit descriptions in special cases.

Findings

Greenberg's conjecture holds iff logarithmic classes principalize in cyclotomic extensions.

In the case where { extell} splits completely, the conjecture reduces to triviality of the logarithmic class group.

Explicit description of circular class groups in the abelian case related to the weak conjecture.

Abstract

We use logarithmic {\ell}-class groups to take a new view on Greenberg's conjecture about Iwasawa {\ell}-invariants of a totally real number field K. By the way we recall and complete some classical results. Under Leopoldt's conjecture, we prove that Greenberg's conjecture holds if and only if the logarithmic classes of K principalize in the cyclotomic Z{\ell}-extensions of K. As an illustration of our approach, in the special case where the prime {\ell} splits completely in K, we prove that the sufficient condition introduced by Gras just asserts the triviality of the logarithmic class group of K.Last, in the abelian case, we provide an explicit description of the circular class groups in connexion with the so-called weak conjecture.