On the \(\mu\)-invariant of \(p\)-adic \(L\)-functions attached to elliptic curves with complex multiplication. (English) Zbl 0615.12018

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 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)].


11R23 Iwasawa theory
11S40 Zeta functions and \(L\)-functions
11R18 Cyclotomic extensions
14H52 Elliptic curves
Full Text: DOI


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