A simple proof of the mean value of $ |{K}_{2}(\mathcal{O})|$ in function fields

Julio Andrade

doi:10.1016/j.crma.2015.04.018

Number theory

A simple proof of the mean value of

| K_{2} (O) |

in function fields
[Une démonstration simple de la valeur moyenne de

| K_{2} (O) |

dans des corps de fonctions]

Présenté par : Jean-Pierre Serre

Julio Andrade ^1,²

¹ Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
² Depto. Matematica, PUC-Rio, Rio De Janeiro, RJ, Brazil

Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 677-682.

Résumé
Abstract

Soit F un corps fini de cardinalité impaire q, $A = F [T]$ l'anneau de polynômes sur F, $k = F (T)$ le corps des fonctions rationnelles sur F et $H$ l'ensemble des polynômes unitaires et sans facteur carré en A de degré impair. Si $D \in H$ , on note par $O_{D}$ la clóture intégrale de A en $k (\sqrt{D})$ . Dans cette Note, nous donnons une preuve simple de la valeur moyenne de la taille des groupes $K_{2} (O_{D})$ quand D varie dans l'ensemble $H$ et quand q est maintenu fixe. La preuve est basée sur des estimations des sommes de caractères et sur l'utilisation de l'hypothèse de Riemann pour les courbes sur les corps finis.

Let F be a finite field of odd cardinality q, $A = F [T]$ the polynomial ring over F, $k = F (T)$ the rational function field over F and $H$ the set of square-free monic polynomials in A of degree odd. If $D \in H$ , we denote by $O_{D}$ the integral closure of A in $k (\sqrt{D})$ . In this Note, we give a simple proof for the average value of the size of the groups $K_{2} (O_{D})$ as D varies over the ensemble $H$ and q is kept fixed. The proof is based on character sums estimates and on the use of the Riemann hypothesis for curves over finite fields.

Reçu le : 2015-01-19
Accepté le : 2015-04-22
Publié le : 2015-05-13

DOI : 10.1016/j.crma.2015.04.018

Affiliations des auteurs :

Julio Andrade ^{1, 2}

¹ Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
² Depto. Matematica, PUC-Rio, Rio De Janeiro, RJ, Brazil

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     title = {A simple proof of the mean value of $ |{K}_{2}(\mathcal{O})|$ in function fields},
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     pages = {677--682},
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     doi = {10.1016/j.crma.2015.04.018},
     language = {en},
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Julio Andrade. A simple proof of the mean value of $ |{K}_{2}(\mathcal{O})|$ in function fields. Comptes Rendus. Mathématique, Volume 353 (2015) no. 8, pp. 677-682. doi : 10.1016/j.crma.2015.04.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.04.018/

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