Iwasawa’s -invariant conjecture for the cyclotomic extension

Establish that the Iwasawa -invariant of the p-primary class groups in the cyclotomic \mathbb{Z}_p-extension K_{\mathrm{cyc}}/K vanishes, namely, prove that \mu=0.

Background

For a number field K and an odd prime p, Iwasawa’s theorem associates invariants \lambda, \mu, and \nu to the growth of the p-primary parts of the class groups in a \mathbb{Z}p-extension K\infty/K. The invariant \mu measures the exponential-in-p^n component of this growth.

The cyclotomic \mathbb{Z}p-extension is the distinguished case K\infty=K_{\mathrm{cyc}}. The paper recalls Iwasawa’s conjecture that the corresponding \mu-invariant vanishes; the results of the paper concern local boundedness and variation of invariants across nearby \mathbb{Z}_p-extensions rather than proving this conjecture.