## 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.

### MSC:

 11R23 Iwasawa theory 11S40 Zeta functions and $$L$$-functions 11R18 Cyclotomic extensions
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