Iwasawa invariants and linking numbers of primes

Abstract

For an odd prime number $p$ and a number field $k$ which is an elementary abelian $p$-extension of the rationals, we prove the equivalence between the vanishing of all Iwasawa invariants of the cyclotomic $\mathbb{Z}_p$-extension of $k$ and an arithmetical condition described by the linking numbers of primes from a viewpoint of analogies between pro-$p$ Galois groups and link groups. A criterion of Greenberg's conjecture for $k$ of degree $p$ is also described in terms of linking matrices.