no. 6
Étale cohomology, cofinite generation, and p-adic L-functions
[Cohomologie étale, engendrement cofini, et fonctions L p-adiques]
de Jeu, Rob1 ; Navilarekallu, Tejaswi1
1 Faculteit der Exacte Wetenschappen Afdeling Wiskunde VU University Amsterdam De Boelelaan 1081a 1081 HV Amsterdam (The Netherlands)
Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2331-2383.

Soit p un nombre premier. Nous étudions certains groupes de cohomologie étale à coefficients associés à une représentation d’Artin p-adique de groupe de Galois d’un corps des nombres k. Ces coefficients sont munis d’un tordu à la Tate modifié avec un indice p-adique. Ces groupes sont de type cofini, et nous déterminons la caractéristique d’Euler additive. Si k est totalement réel et la représentation est paire, nous étudions la relation entre le comportement ou la valeur de la fonction L p-adique en le point e de ce domaine et les groupes de cohomologie avec torsion p-adique 1-e. Dans certains cas, ceci donne une preuve courte d’une conjecture de Coates et Lichtenbaum, et de la conjecture équivariante des nombres de Tamagawa pour les fonctions L classiques. Pour p=2 nos résultats impliquant des fonctions L p-adiques dépendent d’une conjecture de la théorie d’Iwasawa.

Let p be a prime number. We study certain étale cohomology groups with coefficients associated to a p-adic Artin representation of the Galois group of a number field k. These coefficients are equipped with a modified Tate twist involving a p-adic index. The groups are cofinitely generated, and we determine the additive Euler characteristic. If k is totally real and the representation is even, we study the relation between the behaviour or the value of the p-adic L-function at the point e in its domain, and the cohomology groups with p-adic twist 1-e. In certain cases this gives short proofs of a conjecture by Coates and Lichtenbaum, and the equivariant Tamagawa number conjecture for classical L-functions. For p=2 our results involving p-adic L-functions depend on a conjecture in Iwasawa theory.

DOI : 10.5802/aif.2989
Classification : 11G40, 14F20, 11M41, 11S40, 14G10
Keywords: number field, étale cohomology, cofinite generation, Euler characteristic, Artin $L$-function, $p$-adic $L$-function
Mot clés : corps de nombres, cohomologie étale, génération cofinie, caractéristique d’Euler, fonction $ L $ d’Artin
de Jeu, Rob 1 ; Navilarekallu, Tejaswi 1

1 Faculteit der Exacte Wetenschappen Afdeling Wiskunde VU University Amsterdam De Boelelaan 1081a 1081 HV Amsterdam (The Netherlands) 
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de Jeu, Rob; Navilarekallu, Tejaswi. Étale cohomology, cofinite generation, and $p$-adic $L$-functions. Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2331-2383. doi : 10.5802/aif.2989. https://aif.centre-mersenne.org/articles/10.5802/aif.2989/

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