Comptes Rendus
no. 8
Number theory
A simple proof of the mean value of |K2(O)| in function fields
[Une démonstration simple de la valeur moyenne de |K2(O)| dans des corps de fonctions]

Présenté par : Jean-Pierre Serre

Julio Andrade1, 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.

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∈H, we denote by OD the integral closure of A in k(D). In this Note, we give a simple proof for the average value of the size of the groups K2(OD) 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.

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∈H, on note par OD la clóture intégrale de A en k(D). Dans cette Note, nous donnons une preuve simple de la valeur moyenne de la taille des groupes K2(OD) 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.04.018

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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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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