Étale cohomology, cofinite generation, and p-adic L-functions  [ Cohomologie étale, engendrement cofini, et fonctions L p-adiques ]
Annales de l'Institut Fourier, Tome 65 (2015) no. 6, p. 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 : https://doi.org/10.5802/aif.2989
Classification:  11G40,  14F20,  11M41,  11S40,  14G10
Mots clés: corps de nombres, cohomologie étale, génération cofinie, caractéristique d’Euler, fonction L d’Artin
@article{AIF_2015__65_6_2331_0,
     author = {de Jeu, Rob and Navilarekallu, Tejaswi},
     title = {\'Etale cohomology, cofinite generation, and $p$-adic $L$-functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     pages = {2331-2383},
     doi = {10.5802/aif.2989},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_6_2331_0}
}
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. http://www.numdam.org/item/AIF_2015__65_6_2331_0/

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