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

Andrade, Julio

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

Andrade, Julio ^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, Tome 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 :

Andrade, Julio ^{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

@article{CRMATH_2015__353_8_677_0,
     author = {Andrade, Julio},
     title = {A simple proof of the mean value of $ |{K}_{2}(\mathcal{O})|$ in function fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {677--682},
     publisher = {Elsevier},
     volume = {353},
     number = {8},
     year = {2015},
     doi = {10.1016/j.crma.2015.04.018},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.04.018/}
}

TY  - JOUR
AU  - Andrade, Julio
TI  - A simple proof of the mean value of $ |{K}_{2}(\mathcal{O})|$ in function fields
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 677
EP  - 682
VL  - 353
IS  - 8
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2015.04.018/
DO  - 10.1016/j.crma.2015.04.018
LA  - en
ID  - CRMATH_2015__353_8_677_0
ER  -

%0 Journal Article
%A Andrade, Julio
%T A simple proof of the mean value of $ |{K}_{2}(\mathcal{O})|$ in function fields
%J Comptes Rendus. Mathématique
%D 2015
%P 677-682
%V 353
%N 8
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2015.04.018/
%R 10.1016/j.crma.2015.04.018
%G en
%F CRMATH_2015__353_8_677_0

Andrade, Julio. A simple proof of the mean value of $ |{K}_{2}(\mathcal{O})|$ in function fields. Comptes Rendus. Mathématique, Tome 353 (2015) no. 8, pp. 677-682. doi : 10.1016/j.crma.2015.04.018. http://archive.numdam.org/articles/10.1016/j.crma.2015.04.018/

Bibliographie
Cité par

[1] J.C. Andrade, Rudnick and Soundararajan's theorem for function fields, preprint, 2014.

[2] Andrade, J.C.; Keating, J.P. The mean value of $L (\frac{1}{2}, χ)$ in the hyperelliptic ensemble, J. Number Theory, Volume 132 (2012), pp. 2793-2816

[3] Andrade, J.C.; Keating, J.P. Conjectures for the integral moments and ratios of L-functions over function fields, J. Number Theory, Volume 142 (2014), pp. 102-148

[4] Faifman, D.; Rudnick, Z. Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field, Compos. Math., Volume 146 (2010), pp. 81-101

[5] Goldfeld, D.; Hoffstein, J. Eisenstein series of $\frac{1}{2}$ -integral weight and the mean value of real Dirichlet L-series, Invent. Math., Volume 80 (1985), pp. 185-208

[6] Hoffstein, J.; Rosen, M. Average values of L-series in function fields, J. Reine Angew. Math., Volume 426 (1992), pp. 117-150

[7] Hsu, C. Estimates for coefficients of L-functions for function fields, Finite Fields Appl., Volume 5 (1999), pp. 76-88

[8] Jutila, M. On the mean value of $L (\frac{1}{2}, χ)$ for real characters, Analysis, Volume 1 (1981), pp. 149-161

[9] Quillen, D. On the cohomology and K-theory of the general linear group over a finite field, Ann. Math. (2), Volume 96 (1972), pp. 552-586

[10] Rosen, M. Average value of $| K_{2} (O) |$ in function fields, Finite Fields Appl., Volume 1 (1995), pp. 235-241

[11] Rosen, M. Number Theory in Function Fields, Graduate Texts in Mathematics, vol. 210, Springer-Verlag, New York, 2002

[12] Tate, J. Symbols in arithmetic, International Congress of Mathematics, vol. 1, Gauthier-Villars, Paris, 1971, pp. 201-211

Cité par Sources :