## Sur la torsion de certains modules galoisiens. II. (On the torsion of certain Galois modules. II).(French)Zbl 0687.12005

Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 271-297 (1988).
[For the entire collection see Zbl 0653.00005.]
Let K be an algebraic number field, p a fixed prime number, S a finite set of places of K containing the places above p. An interesting arithmetical object attached to K is the Galois group $${\mathcal X}^ S_ K$$ of the maximal abelian pro-p-extension of K which is unramified outside S. Its $${\mathbb{Z}}_ p$$-rank is given by Leopoldt’s conjecture, and its torsion subgroup $${\mathcal T}^ S_ K$$ is closely related to p-adic L- functions (when these are defined for K). In a previous paper [Ann. Inst. Fourier 36, No.2, 27-46 (1986; Zbl 0576.12010)] we studied the Galois module structure of $${\mathcal T}^ S_ K$$. In this one, we climb up a $${\mathbb{Z}}_ p$$-extension $$K_{\infty}/K$$ and study the $$\Lambda$$-module structure of $${\mathcal T}^ S_{\infty}=\lim_{\leftarrow}{\mathcal T}^ S_{K_ n}$$. First we get an expression of the defect $$\delta (K_{\infty})$$ of the “weak” Leopoldt conjecture for $$K_{\infty}$$. When $$K_{\infty}$$ is the cyclotomic $${\mathbb{Z}}_ p$$-extension, $$\delta (K_{\infty})=0$$ and we then construct an analogue of the Weil pairing for curves. Finally, we study some technical “Spiegelung” relations between $${\mathcal T}^ S_{\infty}$$ and related Iwasawa modules.
Reviewer: T.Nguyen Quang Do

### MSC:

 11R32 Galois theory 11R18 Cyclotomic extensions 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 11R70 $$K$$-theory of global fields

### Citations:

Zbl 0596.12008; Zbl 0653.00005; Zbl 0576.12010