Étale cohomology, cofinite generation, and $p$-adic $L$-functions

de Jeu, Rob; Navilarekallu, Tejaswi

doi:10.5802/aif.2989

Étale cohomology, cofinite generation, and

p

-adic

L

-functions [ Cohomologie étale, engendrement cofini, et fonctions

L

p

-adiques ]

de Jeu, Rob ; Navilarekallu, Tejaswi

Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2331-2383.

Résumé
Abstract

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},
     pages = {2331--2383},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     doi = {10.5802/aif.2989},
     language = {en},
     url = {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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