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

##### MSC:
 11R23 Iwasawa theory 11S25 Galois cohomology 11E95 $$p$$-adic theory 11R34 Galois cohomology
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