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

Presented by: 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.

Abstract (Original language)
Résumé

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.

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.

Received: 2015-01-19
Accepted: 2015-04-22
Published online: 2015-05-13

DOI: 10.1016/j.crma.2015.04.018

Author's affiliations:

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/

References
Cited by

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

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

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

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

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

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

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

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

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

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