On the \(\mu\)-invariant of \(p\)-adic \(L\)-functions attached to elliptic curves with complex multiplication
The aim of this paper is to prove the vanishing of \(\mu\) (the Iwasawa invariant) in the ordinary elliptic case; let \(K\) be a principal imaginary quadratic field and \(p\) a prime \(\ne 2,3\) which splits as \(p=\mathfrak pp^-_{\infty}\) be obtained by adjoining to \(K\) all the \({\mathfrak p}^ n\)-division points on \(E\) \((n=1,\ldots)\) and \(M_{\infty}\) the maximal abelian \(p\)-extension of \(F_{\infty}\) unramified outside \({\mathfrak p}\); the result means that \(\mathrm{Gal}(M_{\infty}/F_{\infty})\) has no \({\mathbb{Z}}_ p\)-torsion. Its proof uses a result of algebraic independence for the elliptic curve. The method is similar (in this much more difficult case) to the one introduced by \textit{W. Sinnott} [Invent. Math. 75, 273--283 (1984; Zbl 0531.12004)]. The result has been generalized by the reviewer [J. Reine Angew. Math. 358, 76--91 (1985; Zbl 0551.12011)].
scientific article; zbMATH DE number 3995832
scientific article