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On the relation between the generalized Poincaré series and the Stöhr zeta function
Author(s):
Félix
Delgado
de la Mata;
Julio
José
Moyano Fernández
Journal:
Proc. Amer. Math. Soc.
137
(2009),
51-59.
MSC (2000):
Primary 11M38;
Secondary 14H20, 28A25
Posted:
August 7, 2008
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Abstract:
The aim of this paper is to show the relation between the zeta function introduced by Stöhr and the Poincaré series of a curve singularity introduced by Campillo, Delgado and Gusein-Zade for the complex case. The interpretation of the Stöhr zeta function in terms of integrals with respect to the (generalized) Euler characteristic over suitable subsets of the ring of functions (following the similar construction made by the previously named authors for subsets of the projectivization of the ring) provides the bridge between both subjects.
References:
-
- [CDG]
- A. Campillo, F. Delgado, S.M. Gusein-Zade: The Alexander polynomial of a plane curve singularity via the ring of functions on it. Duke Math. Journal 117 (1), 125-156 (2003). MR 1962784 (2004e:14047)
- [CDG-1]
- A. Campillo, F. Delgado, S.M. Gusein-Zade: The Alexander polynomial of a plane curve singularity and integrals with respect to the Euler characteristic. International Journal of Mathematics 14 (1), 47-54 (2003). MR 1955509 (2004a:14031)
- [CDG-2]
- A. Campillo, F. Delgado, S.M. Gusein-Zade: Multi-index filtrations and motivic Poincaré series. Monatshefte für Mathematik 150, 193-209 (2007). MR 2308548
- [Ki-Vi]
- K. Kiyek, J.L. Vicente: Resolution of Curve and Surface Singularities in Characteristic Zero. Kluwer, Dordrecht, 2004. MR 2106959 (2005k:14028)
- [St]
- K.O. Stöhr: Local and global zeta-functions of singular algebraic curves. Journal of Number Theory 71, 172-202 (1998). MR 1633801 (99e:11087)
- [Zu]
- W. Zúñiga: Zeta functions and Cartier divisors on singular curves over finite fields. Manuscripta Mathematica 94, 75-88 (1997). MR 1468935 (98h:11080)
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Additional Information:
Félix
Delgado
de la Mata
Affiliation:
Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Valladolid, Valladolid, Spain
Email:
fdelgado@agt.uva.es
Julio
José
Moyano Fernández
Affiliation:
Institut für Mathematik, Universität Osnabrück, Albrechtstrasse 28a, 49069-Osnabrück, Germany
Email:
moyano@agt.uva.es
DOI:
10.1090/S0002-9939-08-09469-0
PII:
S 0002-9939(08)09469-0
Keywords:
Zeta function,
Poincar\'e series,
integration with respect to Euler characteristic,
finite field
Received by editor(s):
May 30, 2006,
Received by editor(s) in revised form:
May 18, 2007, and December 27, 2007
Posted:
August 7, 2008
Additional Notes:
The authors were partially supported by MEC MTM2004-00958 and by Junta de CyL VA068/04 (Spain). The second author was also supported by FPU-AP2003-2755.
Communicated by:
Ted Chinburg
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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