Functional equations for zeta functions of $ {\mathbb{F}}_{1}$-schemes

Lorscheid, Oliver

doi:10.1016/j.crma.2010.10.010

Number Theory/Algebraic Geometry

Functional equations for zeta functions of

F_{1}

-schemes

Presented by: Christophe Soulé

Lorscheid, Oliver ¹

¹ The City College of New York, Department of Mathematics, 160 Convent Avenue, New York, NY 10031, USA

Comptes Rendus. Mathématique, Volume 348 (2010) no. 21-22, pp. 1143-1146

Abstract (Original language)
Résumé

For a scheme X whose $F_{q}$ -rational points are counted by a polynomial $N (q) = \sum a_{i} q^{i}$ , the $F_{1}$ -zeta function is defined as $ζ_{X} (s) = \prod {(s - i)}^{- a_{i}}$ . Define $χ = N (1)$ . In this paper we show that if X is a smooth projective scheme, then its $F_{1}$ -zeta function satisfies the functional equation $ζ_{X} (n - s) = {(- 1)}^{χ} ζ_{X} (s)$ . We further show that the $F_{1}$ -zeta function $ζ_{G} (s)$ of a split reductive group scheme G of rank r with N positive roots satisfies the functional equation $ζ_{G} (r + N - s) = {(- 1)}^{χ} (ζ_{G} {(s))}^{{(- 1)}^{r}}$ .

Pour un schéma X dont les points $F_{q}$ -rationnels sont comptés par un polynôme $N (q) = \sum a_{i} q^{i}$ , la fonction zêta sur $F_{1}$ est définie par $ζ_{X} (s) = \prod {(s - i)}^{- a_{i}}$ . Posons $χ = N (1)$ . Dans cette Note nous montrons que si X est un schéma projectif lisse, alors sa fonction zêta sur $F_{1}$ satisfait l'équation fonctionnelle $ζ_{X} (n - s) = {(- 1)}^{χ} ζ_{X} (s)$ . Nous montrons aussi que la fonction zêta $ζ_{G} (s)$ sur $F_{1}$ d'un schéma en groupes réductif déployé G de rang r avec N racines positives satisfait l'équation fonctionnelle $ζ_{G} (r + N - s) = {(- 1)}^{χ} (ζ_{G} {(s))}^{{(- 1)}^{r}}$ .

Received: 2010-10-08
Accepted: 2010-10-11
Published online: 2010-10-28

DOI: 10.1016/j.crma.2010.10.010

Author's affiliations:

Lorscheid, Oliver ¹

¹ The City College of New York, Department of Mathematics, 160 Convent Avenue, New York, NY 10031, USA

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     title = {Functional equations for zeta functions of $ {\mathbb{F}}_{1}$-schemes},
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Lorscheid, Oliver. Functional equations for zeta functions of $ {\mathbb{F}}_{1}$-schemes. Comptes Rendus. Mathématique, Volume 348 (2010) no. 21-22, pp. 1143-1146. doi: 10.1016/j.crma.2010.10.010

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