On Galois groups of unramified pro-p extensions

TL;DR

This paper investigates the structure of Galois groups related to unramified pro-p extensions of cyclotomic fields, providing bounds, conditions for abelianity, and confirming Greenberg's conjecture for primes less than 1000.

Contribution

It establishes a lower bound for the annihilator height of an Iwasawa module and offers a necessary and sufficient condition for Galois group abelianity based on cyclotomic p-units.

Findings

Greenberg's conjecture holds for p < 1000.

G is abelian under mild Bernoulli number assumptions.

Bounds are expressed via special values of cup product pairings.

Abstract

Let p be an odd prime satisfying Vandiver's conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z_p-extensions of the pth cyclotomic field and the Galois group G of the unramified pro-p extension of the cyclotomic field of all p-power roots of unity. We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for G to be abelian. The bound and the condition in the two results are given in terms of the special values of a cup product pairing on cyclotomic p-units. We obtain, in particular, that for p less than 1000, Greenberg's conjecture on the pseudo-nullity of X holds and G is in fact abelian.