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

Nguyen Quang Do, Thong

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

Keywords:

Leopoldt’s conjecture; Galois module structure; \(\Lambda \) -module structure; Weil pairing for curves; Iwasawa modules

Citations:

Zbl 0596.12008; Zbl 0653.00005; Zbl 0576.12010