First and second $K$-groups of an elliptic curve over a global field of positive characteristic

Kondo, Satoshi; Yasuda, Seidai

doi:10.5802/aif.3202

First and second

K

-groups of an elliptic curve over a global field of positive characteristic
[Sur les premier et second

K

-groupes d’une courbe elliptique sur un corps global de caractéristique positive]

Kondo, Satoshi ^1,² ; Yasuda, Seidai ³

¹ Kavli Institute for the Physics and Mathematics of the Universe University of Tokyo 5-1-5 Kashiwanoha Kashiwa, 277-8583 (Japan)
² National Research University Higher School of Economics Usacheva St., 7, Moscow 119048 (Russia)
³ Department of Mathematics, Osaka University Toyonaka, Osaka 560-0043 (Japan)

Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 2005-2067.

Résumé
Abstract

On démontre que les plus grands sous-groupes divisibles desgroupes $K_{1}$ et $K_{2}$ d’une courbe elliptique $E$ sur un corps global de caractéristique positive sont uniquement divisibles et on décrit explicitement les $K$ -groupes modulo leurs plus grands sous-groupes divisibles. On calcule également la cohomologie motivique du modèle minimal de $E$ qui est une surface elliptique sur un corps fini.

In this paper, we show that the maximal divisible subgroup of groups $K_{1}$ and $K_{2}$ of an elliptic curve $E$ over a function field is uniquely divisible. Further those $K$ -groups modulo this uniquely divisible subgroup are explicitly computed. We also calculate the motivic cohomology groups of the minimal regular model of $E$ , which is an elliptic surface over a finite field.

Reçu le : 2011-03-04
Révisé le : 2016-10-27
Accepté le : 2017-11-14
Publié le : 2018-11-23

DOI : 10.5802/aif.3202

Classification : 11R58, 14F42, 19F27, 11G05
Keywords: K-theory, function field, elliptic curve, motivic cohomology
Mot clés : K-théorie, corps de fonctions, courbe elliptique, cohomologie motivique

Affiliations des auteurs :

Kondo, Satoshi ^{1, 2} ; Yasuda, Seidai ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Kondo, Satoshi; Yasuda, Seidai. First and second $K$-groups of an elliptic curve over a global field of positive characteristic. Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 2005-2067. doi : 10.5802/aif.3202. https://aif.centre-mersenne.org/articles/10.5802/aif.3202/

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