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

de Jeu, Rob; Navilarekallu, Tejaswi

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.

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