Fonctions \(L\) \(p\)-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes. (French) Zbl 0551.12011

The aim of this paper is to prove the vanishing of \(\mu\) (the Iwasawa invariant) in the ordinary elliptic case: let \(K\) be an imaginary quadratic field and \(p\) a prime \(\neq 2,3\) which splits as \((p)={\mathfrak pp^-}\) in \(K\). Let \(F\) be a finite abelian extension of \(K\) and \(K_{\infty}\) the only extension of \(K\) with Galois group \({\mathbb{Z}}_ p\) which is unramified outside \({\mathfrak p}\). Let \(F_{\infty}\) be the composite \(FK_{\infty}\), and \(M_{\infty}\) the maximal abelian \(p\)-extension which is unramified outside \({\mathfrak p}\). The main result of this work is the vanishing of the \({\mathbb{Z}}_ p\) torsion subgroup in the Galois group of \(M_{\infty}/F_{\infty}\).
Unfortunately, one of the tools of the proof (the \(p\)-adic \(L\)-functions) did not exist with sufficient generality, so a large part of the paper is devoted to fill this gap (with an eye on the application to the \(\mu\)-invariant).
This paper generalizes recent work of the author (also found by L. Schneps) still to appear.


11R23 Iwasawa theory
11S40 Zeta functions and \(L\)-functions
11R18 Cyclotomic extensions
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