Pour un schéma X dont les points -rationnels sont comptés par un polynôme , la fonction zêta sur est définie par . Posons . Dans cette Note nous montrons que si X est un schéma projectif lisse, alors sa fonction zêta sur satisfait l'équation fonctionnelle . Nous montrons aussi que la fonction zêta sur d'un schéma en groupes réductif déployé G de rang r avec N racines positives satisfait l'équation fonctionnelle .
For a scheme X whose -rational points are counted by a polynomial , the -zeta function is defined as . Define . In this paper we show that if X is a smooth projective scheme, then its -zeta function satisfies the functional equation . We further show that the -zeta function of a split reductive group scheme G of rank r with N positive roots satisfies the functional equation .
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@article{CRMATH_2010__348_21-22_1143_0, author = {Lorscheid, Oliver}, title = {Functional equations for zeta functions of $ {\mathbb{F}}_{1}$-schemes}, journal = {Comptes Rendus. Math\'ematique}, pages = {1143--1146}, publisher = {Elsevier}, volume = {348}, number = {21-22}, year = {2010}, doi = {10.1016/j.crma.2010.10.010}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2010.10.010/} }
TY - JOUR AU - Lorscheid, Oliver TI - Functional equations for zeta functions of $ {\mathbb{F}}_{1}$-schemes JO - Comptes Rendus. Mathématique PY - 2010 SP - 1143 EP - 1146 VL - 348 IS - 21-22 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2010.10.010/ DO - 10.1016/j.crma.2010.10.010 LA - en ID - CRMATH_2010__348_21-22_1143_0 ER -
%0 Journal Article %A Lorscheid, Oliver %T Functional equations for zeta functions of $ {\mathbb{F}}_{1}$-schemes %J Comptes Rendus. Mathématique %D 2010 %P 1143-1146 %V 348 %N 21-22 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2010.10.010/ %R 10.1016/j.crma.2010.10.010 %G en %F CRMATH_2010__348_21-22_1143_0
Lorscheid, Oliver. Functional equations for zeta functions of $ {\mathbb{F}}_{1}$-schemes. Comptes Rendus. Mathématique, Tome 348 (2010) no. 21-22, pp. 1143-1146. doi : 10.1016/j.crma.2010.10.010. http://archive.numdam.org/articles/10.1016/j.crma.2010.10.010/
[1] On the notion of geometry over , 2008 (preprint) | arXiv
[2] Remarks on zeta functions and K-theory over , Japan Academy. Proceedings. Series A. Mathematical Sciences, Volume 82 (2006)
[3] Motivic zeta functions of motives, Pure and Applied Mathematics Quarterly, Volume 5 (2009) no. 1, pp. 507-570
[4] Zeta functions over , Japan Academy. Proceedings. Series A. Mathematical Sciences, Volume 81 (2005) no. 10, pp. 180-184
[5] J. López Peña, O. Lorscheid, Torified varieties and their geometries over , Online first at Mathematische Zeitschrift (2009).
[6] J. López Peña, O. Lorscheid, Mapping -land: an overview over geometries over the field with one element, in: Proceeding of the Conferences on , 2009, , in press. | arXiv
[7] Algebraic groups over the field with one element, 2009 | arXiv
[8] Moduli of representations of quivers. Trends in representation theory of algebras and related topics, EMS Series of Congress Reports (2008), pp. 589-637
[9] Les variétés sur le corps à un élément, Moscow Mathematical Journal, Volume 4 (2004), pp. 217-244
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