Soit un nombre premier. Nous étudions certains groupes de cohomologie étale à coefficients associés à une représentation d’Artin -adique de groupe de Galois d’un corps des nombres . Ces coefficients sont munis d’un tordu à la Tate modifié avec un indice -adique. Ces groupes sont de type cofini, et nous déterminons la caractéristique d’Euler additive. Si est totalement réel et la représentation est paire, nous étudions la relation entre le comportement ou la valeur de la fonction -adique en le point de ce domaine et les groupes de cohomologie avec torsion -adique . 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 classiques. Pour nos résultats impliquant des fonctions -adiques dépendent d’une conjecture de la théorie d’Iwasawa.
Let be a prime number. We study certain étale cohomology groups with coefficients associated to a -adic Artin representation of the Galois group of a number field . These coefficients are equipped with a modified Tate twist involving a -adic index. The groups are cofinitely generated, and we determine the additive Euler characteristic. If is totally real and the representation is even, we study the relation between the behaviour or the value of the -adic -function at the point in its domain, and the cohomology groups with -adic twist . In certain cases this gives short proofs of a conjecture by Coates and Lichtenbaum, and the equivariant Tamagawa number conjecture for classical -functions. For our results involving -adic -functions depend on a conjecture in Iwasawa theory.
Classification : 11G40, 14F20, 11M41, 11S40, 14G10
Mots clés : corps de nombres, cohomologie étale, génération cofinie, caractéristique d’Euler, fonction 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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