On the cohomology of semi-stable <i>p</i>-adic Galois representations

Yi Ouyang; Jinbang Yang

doi:10.1016/j.crma.2014.04.008

Number theory

On the cohomology of semi-stable p-adic Galois representations

Presented by: Jean-Marc Fontaine

Yi Ouyang ¹; Jinbang Yang ¹

¹ Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR China

Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 557-561.

Abstract
Résumé

Let K be a field of characteristic 0 complete with respect to a non-trivial discrete valuation with perfect residue field k of characteristic $p > 0$ . Let V be a p-adic representation of the absolute Galois group of K. We compute explicitly Kato's filtration on the continuous cohomology group $H^{1} (K, V)$ . When k is finite, we give a simple proof of Hyodo's celebrated result $H_{g}^{1} (K, V) = H_{st}^{1} (K, V)$ when V is a potentially semi-stable Galois representation.

Soit K un corps de caractéristique 0 complet pour une valuation discrète non triviale à corps résiduel parfait k de caractéristique $p > 0$ . Soit V une représentation p-adique du groupe de Galois absolu de K. On calcule explicitement la filtration de Kato sur le groupe de cohomologie continue $H^{1} (K, V)$ . Lorsque k est fini, on en déduit une preuve simple du résultat bien connu de Hyodo qui dit que, si V est potentiellement semi-stable, alors $H_{g}^{1} (K, V) = H_{st}^{1} (K, V)$ .

Received: 2013-12-07
Accepted: 2014-04-17
Published online: 2014-05-16

DOI: 10.1016/j.crma.2014.04.008

Author's affiliations:

Yi Ouyang ¹; Jinbang Yang ¹

¹ Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, PR China

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     author = {Yi Ouyang and Jinbang Yang},
     title = {On the cohomology of semi-stable \protect\emph{p}-adic {Galois} representations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {557--561},
     publisher = {Elsevier},
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     doi = {10.1016/j.crma.2014.04.008},
     language = {en},
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Yi Ouyang; Jinbang Yang. On the cohomology of semi-stable p-adic Galois representations. Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 557-561. doi : 10.1016/j.crma.2014.04.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.04.008/

References
Cited by

[1] L. Berger Représentations p-adiques et équations différentielles, Invent. Math., Volume 148 (2002), pp. 219-284

[2] S. Bloch; K. Kato L-functions and Tamagawa numbers of motives (P. Cartier; L. Illusie; N.M. Katz; G. Laumon; Y.I. Manin; K.A. Ribet, eds.), The Grothendieck Festschrift, vol. I, Prog. Math., vol. 86, Birkhäuser, Boston, MA, USA, 1990, pp. 333-400

[3] J.-M. Fontaine Le corps des périodes p-adiques (J.-M. Fontaine, ed.), Périodes p-Adiques, Astérisque, vol. 223, Société mathématique de France, Paris, 1994, pp. 59-111

[4] J.-M. Fontaine; Y. Ouyang Theory of p-adic Galois representations http://staff.ustc.edu.cn/~yiouyang/galoisrep.pdf (preprint)

[5] J.-M. Fontaine; B. Perrin-Riou Autour des conjectures de Bloch et Kato : cohomologie galoisienne et valeurs de fonctions L, Motives (Part 1), Proc. Symp. Pure Math., vol. 55, American Mathematical Society, Providence, RI, USA, 1994, pp. 599-706

[6] O. Hyodo, $H_{g}^{1} = H_{st}^{1}$ , unpublished.

[7] J. Tate Relations between $K_{2}$ and Galois cohomology, Invent. Math., Volume 36 (1976), pp. 257-274

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