Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-02-11T08:47:07.128Z Has data issue: false hasContentIssue false
  • English
  • Français

Motivic zeta functions for curve singularities

Published online by Cambridge University Press:  11 January 2016

J. J. Moyano-Fernández  and
W. A. Zúňiga-Galindo
J. J. Moyano-Fernández
Affiliation:
Institut für Mathematik, Universitát Osnabrück, Albrechtstrasse, 28a 49076 Osnabrück, Deutschlandjmoyano@mathematik.uni-osnabrueck.de
W. A. Zúňiga-Galindo
Affiliation:
Institut für Mathematik, Universitát Osnabrück, Albrechtstrasse, 28a 49076 Osnabrück, Deutschlandjmoyano@mathematik.uni-osnabrueck.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring Op,x at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if Op,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

Keywords

14H2014G1032S4011S40
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 198 , June 2010 , pp. 47 - 75
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

References

[1] A’Campo, N., La fonction zêeta d’une monodromie, Comment. Math. Helv. 50 (1975), 233–248.Google Scholar
[2] André, Y., An introduction to motivic zeta functions of motives, preprint, arXiv:0812.3920v1 [math.AG].Google Scholar
[3] Baldassarri, F., Deninger, C., and Naumann, N., “A motivic version of Pellikaan’s two variable zeta function,” in Diophantine Geometry, CRM Series 4, Ed. Norm., Pisa, 2007, 35–43.Google Scholar
[4] Campillo, A., Delgado, F., and Gusein-Zade, S. M., On the monodromy of a plane curve singularity and the Poincaré series of its ring of functions, Funct. Anal. Appl. 33 (1999), 56–57.Google Scholar
[5] Campillo, A., Delgado, F., and Gusein-Zade, S. M., The Alexander polynomial of a plane curve singularity and integrals with respect to the Euler characteristic, Internat. J. Math. 14 (2003), 47–54.Google Scholar
[6] Campillo, A., Delgado, F., and Gusein-Zade, S. M., The Alexander polynomial of a plane curve singularity via the ring of functions on it, Duke Math. J. 117 (2003), 125–156.Google Scholar
[7] Campillo, A., Delgado, F., and Gusein-Zade, S. M., Multi-index filtrations and generalized Poincaré series, Monatsh. Math. 150 (2007), 193–209.CrossRefGoogle Scholar
[8] Campillo, A., Delgado, F., and Kiyek, K., Gorenstein property and symmetry for one-dimensional local Cohen-Macaulay rings, Manuscripta Math. 83 (1994), 405–423.Google Scholar
[9] Mata, F. Delgado de la, The semigroup of values of a curve singularity with several branches, Manuscripta Math. 59 (1987), 347–374.CrossRefGoogle Scholar
[10] Mata, F. Delgado de la, Gorenstein curves and symmetry of the semigroup of values, Manuscripta Math. 61 (1988), 285–296.Google Scholar
[11] Mata, F. Delgado de la and Moyano-Fernández, J.-J., On the relation between the generalized Poincaré series and the Stöhr zeta function, Proc. Amer. Math. Soc. 137 (2009), 51–59.Google Scholar
[12] Denef, J. and Loeser, F., Caracteristiques de Euler-Poincaré, fonctions zeta locales et modifications analytiques, J. Amer. Math. Soc. 5 (1992), 705–720.Google Scholar
[13] Denef, J. and Loeser, F., Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201–232.Google Scholar
[14] Denef, J. and Loeser, F., “On some rational generating series occurring in arithmetic geometry,” in Geometric Aspects of Dwork Theory, Vol. I, II, de Gruyter, Berlin, 2004, 509–526.Google Scholar
[15] Galkin, V. M., Zeta-functions of certain one-dimensional rings, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 3–19.Google Scholar
[16] Green, B., Functional equations for zeta functions of non-Gorenstein orders in global fields, Manuscripta Math. 64 (1989), 485–502.Google Scholar
[17] Kapranov, M., The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups, preprint, arXiv:math/0001005 [math.AG].Google Scholar
[18] Loeser, F., Seattle Lectures on motivic integration, preprint, 2006.Google Scholar
[19] Moyano-Fernández, J.-J., Poincaré series associated with curves defined over perfect fields, Ph.D. dissertation, Universidad de Valladolid, Valladolid, Spain, 2008.Google Scholar
[20] Rosenlicht, M., Equivalence relations on algebraic curves, Ann. of Math. 56 (1952), 169–191.CrossRefGoogle Scholar
[21] Rosenlicht, M., Generalized Jacobian varieties, Ann. of Math. 59 (1954), 505–530.Google Scholar
[22] Serre, J.-P., Algebraic Groups and Class Fields, Grad. Texts Math. 117, Springer, New York, 1988.Google Scholar
[23] Stöhr, K.-O., On the poles of regular differentials of singular curves, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), 105–136.Google Scholar
[24] Stöhr, K.-O., Local and global zeta functions of singular algebraic curves, J. Number Theory 71 (1998), 172–202.CrossRefGoogle Scholar
[25] Veys, W., “Arc spaces, motivic integration and stringy invariants,” in Singularity Theory and Its Applications (Sapporo, Japan, 2003), Adv. Stud. Pure Math. 43, Math. Soc. Japan, 2006, 529–572.CrossRefGoogle Scholar
[26] Waldi, R., Wertehalbgruppe und Singularität einer ebenen algebroiden Kurve, Ph.D. dissertation, Universität Regensburg, Regensburg, Germany, 1972.Google Scholar
[27] von, H. Herzog, Jürgen and Kunz, E., Der kanonische Modul eines Cohen-Macaulay-Rings. Seminar über die lokale Kohomologietheorie von Grothendieck, Universitä t Regensburg, Wintersemester 1970/1971, Lecture Notes in Math. 238, Springer, Berlin, 1971.Google Scholar
[28] Zariski, O., The Moduli Problem for Plane Branches, Univ. Lecture Ser. 39, Amer. Math. Soc., Providence, 2006.Google Scholar
[29] Zún˜iga-Galindo, W. A., Zeta functions and Cartier divisors on singular curves over finite fields, Manuscripta Math. 94 (1997), 75–88.Google Scholar
[30] Zuún˜iga-Galindo, W. A., Zeta functions of singular curves over finite fields, Rev. Colombiana Mat. 31 (1997), 115–124.Google Scholar
[31] Zuún˜iga-Galindo, W. A., Zeta functions of singular algebraic curves over finite fields, Ph.D. dissertation, Instituto Nacional de Matemática Pure e Aplicada, Rio de Janeiro, Brazil, 1996.Google Scholar