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Représentations $p$-adiques et normes universelles I. Le cas cristallin


Author: Bernadette Perrin-Riou
Journal: J. Amer. Math. Soc. 13 (2000), 533-551
MSC (2000): Primary 11S20, 11R23, 11G25
DOI: https://doi.org/10.1090/S0894-0347-00-00329-5
Published electronically: March 13, 2000
MathSciNet review: 1758753
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Abstract: Let $V$ be a crystalline $p$-adic representation of the absolute Galois group $G_K$ of an finite unramified extension $K$ of $\mathbb{Q}_p$, and let $T$ be a lattice of $V$ stable by $G_K$. We prove the following result: Let $\mathrm{Fil}^1V$ be the maximal sub-representation of $V$ with Hodge-Tate weights strictly positive and $\mathrm{Fil}^1T=T\cap \mathrm{Fil}^1V$. Then, the projective limit of $H^1_g(K(\mu_{p^n}), T)$ is equal up to torsion to the projective limit of $H^1(K(\mu_{p^n}), \mathrm{Fil} ^1T)$. So its rank over the Iwasawa algebra is $[K:\mathbb{Q}_p]\operatorname{dim}\mathrm{Fil}^1 V$.


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Additional Information

Bernadette Perrin-Riou
Affiliation: Département de Mathématiques, UMR 8628 du CNRS, bât 425, Université Paris-Sud, F-91405 Orsay Cedex, France
Email: bpr@geo.math.u-psud.fr

DOI: https://doi.org/10.1090/S0894-0347-00-00329-5
Received by editor(s): April 29, 1999
Received by editor(s) in revised form: January 10, 2000
Published electronically: March 13, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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