A consequence of Greenberg's generalized conjecture on Iwasawa invariants of ${\mathbb{Z}}_p$-extensions

Abstract

For a prime number $p$ and a number field $k$, let $\tilde{k}$ be the compositum of all ${\mathbb{Z}}_p$-extensions of $k$. Greenberg's Generalized Conjecture (GGC) claims the pseudo-nullity of the unramified Iwasawa module $X(\tilde{k})$ of $\tilde{k}$. It is known that, when $k$ is an imaginary quadratic field, GGC has a consequence on the Iwasawa invariants associated to ${\mathbb{Z}}_p$-extensions of $k$. In this paper, we partially generalize it to arbitrary number fields $k$.

arXiv comment28 pages