Iwasawa invariants of imaginary quadratic fields (Q1320380)

This paper studies the \(\mu\) invariant of Iwasawa in the easiest case not yet treated: for \(K/K_ \infty\) \(\mathbb{Z}_ p\)-extension of a principal quadratic imaginary field which is not totally ramified above \(p\). The ordinary hypothesis (\(p\) split in \(K\): \(p= {\mathfrak p}\overline {\mathfrak p}\)) is assumed. The vanishing of \(\mu\) is proven and the same result holds when looking at the extension unramified outside \({\mathfrak p}\) provided \(\overline{\mathfrak p}\) is not totally ramified in \(K/K_ \infty\). The fifth section gives the result that \(\lambda\) is bounded for varying \(K/K_ \infty\) provided that \(\mu=0\); the method is by proving a \(\mathbb{Z}_ p [[T]]\)-cyclicity for the main Iwasawa module.