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

de Jeu, Rob; Navilarekallu, Tejaswi

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, p. 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},
     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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