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

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, Volume 68 (2018) no. 5, pp. 2005-2067.

Abstract
Résumé

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.

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.

Received: 2011-03-04
Revised: 2016-10-27
Accepted: 2017-11-14
Published online: 2018-11-23

DOI: 10.5802/aif.3202

Classification: 11R58, 14F42, 19F27, 11G05
Keywords: K-theory, function field, elliptic curve, motivic cohomology

Author's affiliations:

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

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {First and second $K$-groups of an elliptic curve over a global field of positive characteristic},
     journal = {Annales de l'Institut Fourier},
     pages = {2005--2067},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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, Volume 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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