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

Gillard, Roland

J. Reine Angew. Math. 358, 76-91 (1985).

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.

Reviewer: Roland Gillard (Saint-Martin-d’Hères)

MSC:

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

Keywords:

vanishing of mu-invariant; cyclotomic extension; elliptic case; imaginary quadratic field; abelian extension; maximal abelian p-extension; vanishing of the \({\mathbb{Z}}_ p\) torsion subgroup; p-adic L-functions