Théorie d’Iwasawa des représentations \(p\)-adiques semi-stables. (Iwasawa theory of semi-stable \(p\)-adic representations). (French) Zbl 1031.11064

Perrin-Riou, Bernadette

Mém. Soc. Math. Fr., Nouv. Sér. 84, vi, 111 p. (2001).

This work is part of a set of papers studying Iwasawa’s theory of \(p\)-adic geometric representations on a finite extension \(K\) of the field of \(p\)-adic numbers \({\mathbb Q}_p\) and \(p\)-adic functions associated to a \(p\)-adic geometric representation of a number field. In a previous paper [Invent. Math. 115, 81-149 (1994; Zbl 0838.11071)] the author considers the case of crystalline \(p\)-adic representations generalizing results of Iwasawa, Coleman, Rubin and Perrin-Riou herself among others. These representations generalize those of the Tate module of any Lubin-Tate group.
In this paper the author turns her attention to the case of semi-stable \(p\)-adic representations. It is constructed in the case of a \(p\)-adic Galois semi-stable representation \(V\) of an unramified finite extension of \({\mathbb Q}_p\), an Iwasawa theory for \(V\) and the \({\mathbb Z}_p\)-cyclotomic extension and a logarithm from the Iwasawa module associated to the Galois cohomology in an explicit module on an algebra generated by analytic functions on the annulus \(\{p ^{-1/(p-1)} <|x|<1\}\) and \(\log x\). It is shown, for semi-stable representations, that the author’s conjecture stated in [J. Am. Math. Soc. 13, 533-551 (2000; Zbl 1024.11069)] about the rank of universal norms holds. The author uses Colmez’s reciprocity law. The paper is organized as follows. The first chapter studies certain function rings and their structure as \(G_\infty\)-modules, where \(G_\infty = \text{Gal}({\mathbb Q}_p(\mu _{p^\infty})/ {\mathbb Q}_p)\). The next three chapters are dedicated to the study of certain \(G_\infty\)-modules. In Chapter 5, Galois cohomology is introduced and the exponential built. The last chapter deals with universal norms.

Reviewer: Gabriel D.Villa-Salvador (México)

MSC:

11R23	Iwasawa theory
11S25	Galois cohomology
11E95	\(p\)-adic theory
11R34	Galois cohomology

Keywords:

\(p\)-adic representations; semi-stable representations; universal norms; Galois cohomology; Iwasawa theory; Iwasawa module; logarithm map

Citations:

Zbl 0838.11071; Zbl 1024.11069