On small Iwasawa invariants and imaginary quadratic fields

Abstract:

If $p$ is an odd prime that does not divide the class number of the imaginary quadratic field $k$, and the cyclotomic ${\mathbb {Z}_p}$-extension of $k$ has $\lambda$-invariant less than or equal to two, we prove that every totally ramified ${\mathbb {Z}_p}$-extension of $k$ has $\mu$-invariant equal to zero and $\lambda$-invariant less than or equal to two. Combined with a result of Bloom and Gerth, this has the consequence that $\mu = 0$ for every ${\mathbb {Z}_p}$-extension of $k$, under the same assumptions. In the principal case under consideration, Iwasawa’s formula for the power of $p$ in the class number of the $n$th layer of a ${\mathbb {Z}_p}$-extension becomes valid for all $n$ , and is completely explicit.