Motivic zeta functions for curve singularities
J. J. Moyano-Fernández and W. A. Zúñiga-Galindo
Source: Nagoya Math. J. Volume 198
(2010), 47-75.
Abstract
Let $X$ be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic $p$ big enough. Given a local ring $\mathcal{O}_{P,X}$ 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 $\mathcal{O}_{P,X}$ 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.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.nmj/1273496985
Digital Object Identifier: doi:10.1215/00277630-2009-007
Mathematical Reviews number (MathSciNet): MR2666577
Zentralblatt MATH identifier: 05735944
References
[1] N. A’Campo, La fonction zêta d’une monodromie, Comment. Math. Helv. 50 (1975), 233–248.
[2] Y. André, An introduction to motivic zeta functions of motives, preprint, arXiv:0812.3920v1 [math.AG].
Mathematical Reviews (MathSciNet): MR2115000
[3] F. Baldassarri, C. Deninger, and N. Naumann, “A motivic version of Pellikaan’s two variable zeta function,” in Diophantine Geometry, CRM Series 4, Ed. Norm., Pisa, 2007, 35–43.
Mathematical Reviews (MathSciNet): MR2349645
Zentralblatt MATH: 1151.14017
[4] A. Campillo, F. Delgado, and S. M. Gusein-Zade, On the monodromy of a plane curve singularity and the Poincaré series of its ring of functions, Funct. Anal. Appl. 33 (1999), 56–57.
Mathematical Reviews (MathSciNet): MR1711890
[5] A. Campillo, F. Delgado, and S. M. Gusein-Zade, The Alexander polynomial of a plane curve singularity and integrals with respect to the Euler characteristic, Internat. J. Math. 14 (2003), 47–54.
Mathematical Reviews (MathSciNet): MR1955509
Zentralblatt MATH: 1056.14035
Digital Object Identifier: doi:10.1142/S0129167X03001703
[6] A. Campillo, F. Delgado, and S. M. Gusein-Zade, The Alexander polynomial of a plane curve singularity via the ring of functions on it, Duke Math. J. 117 (2003), 125–156.
Mathematical Reviews (MathSciNet): MR1962784
Zentralblatt MATH: 1028.32013
Digital Object Identifier: doi:10.1215/S0012-7094-03-11712-3
Project Euclid: euclid.dmj/1085598340
[7] A. Campillo, F. Delgado, and S. M. Gusein-Zade, Multi-index filtrations and generalized Poincaré series, Monatsh. Math. 150 (2007), 193–209.
Mathematical Reviews (MathSciNet): MR2308548
[8] A. Campillo, F. Delgado, and K. Kiyek, Gorenstein property and symmetry for one-dimensional local Cohen-Macaulay rings, Manuscripta Math. 83 (1994), 405–423.
Mathematical Reviews (MathSciNet): MR1277539
Zentralblatt MATH: 0822.13011
Digital Object Identifier: doi:10.1007/BF02567623
[9] F. Delgado de la Mata, The semigroup of values of a curve singularity with several branches, Manuscripta Math. 59 (1987), 347–374.
Mathematical Reviews (MathSciNet): MR909850
Zentralblatt MATH: 0611.14025
Digital Object Identifier: doi:10.1007/BF01174799
[10] F. Delgado de la Mata, Gorenstein curves and symmetry of the semigroup of values, Manuscripta Math. 61 (1988), 285–296.
Mathematical Reviews (MathSciNet): MR949819
Zentralblatt MATH: 0692.13017
Digital Object Identifier: doi:10.1007/BF01258440
[11] F. Delgado de la Mata and J.-J. Moyano-Fernández, On the relation between the generalized Poincaré series and the Stöhr zeta function, Proc. Amer. Math. Soc. 137 (2009), 51–59.
[12] J. Denef and F. Loeser, Caracteristiques de Euler-Poincaré, fonctions zeta locales et modifications analytiques, J. Amer. Math. Soc. 5 (1992), 705–720.
Mathematical Reviews (MathSciNet): MR1151541
JSTOR: links.jstor.org
[13] J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201–232.
Mathematical Reviews (MathSciNet): MR1664700
Zentralblatt MATH: 0928.14004
Digital Object Identifier: doi:10.1007/s002220050284
[14] J. Denef and F. Loeser, “On some rational generating series occurring in arithmetic geometry,” in Geometric Aspects of Dwork Theory, Vol. I, II, de Gruyter, Berlin, 2004, 509–526.
Mathematical Reviews (MathSciNet): MR2099079
[15] V. M. Galkin, Zeta-functions of certain one-dimensional rings, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 3–19.
Mathematical Reviews (MathSciNet): MR332729
[16] B. Green, Functional equations for zeta functions of non-Gorenstein orders in global fields, Manuscripta Math. 64 (1989), 485–502.
Mathematical Reviews (MathSciNet): MR1005249
Zentralblatt MATH: 0723.11058
Digital Object Identifier: doi:10.1007/BF01170941
[17] M. Kapranov, The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups, preprint, arXiv:math/0001005 [math.AG].
[18] F. Loeser, Seattle Lectures on motivic integration, preprint, 2006.
Mathematical Reviews (MathSciNet): MR2483954
Zentralblatt MATH: 1181.14017
[19] J.-J. Moyano-Fernández, Poincaré series associated with curves defined over perfect fields, Ph.D. dissertation, Universidad de Valladolid, Valladolid, Spain, 2008.
[20] M. Rosenlicht, Equivalence relations on algebraic curves, Ann. of Math. 56 (1952), 169–191.
Mathematical Reviews (MathSciNet): MR48856
Digital Object Identifier: doi:10.2307/1969773
JSTOR: links.jstor.org
[21] M. Rosenlicht, Generalized Jacobian varieties, Ann. of Math. 59 (1954), 505–530.
Mathematical Reviews (MathSciNet): MR61422
Digital Object Identifier: doi:10.2307/1969715
JSTOR: links.jstor.org
[22] J.-P. Serre, Algebraic Groups and Class Fields, Grad. Texts Math. 117, Springer, New York, 1988.
Mathematical Reviews (MathSciNet): MR918564
[23] K.-O. Stöhr, On the poles of regular differentials of singular curves, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), 105–136.
Mathematical Reviews (MathSciNet): MR1224302
Zentralblatt MATH: 0788.14020
Digital Object Identifier: doi:10.1007/BF01231698
[24] K.-O. Stöhr, Local and global zeta functions of singular algebraic curves, J. Number Theory 71 (1998), 172–202.
Mathematical Reviews (MathSciNet): MR1633801
Zentralblatt MATH: 0940.14016
Digital Object Identifier: doi:10.1006/jnth.1998.2240
[25] W. Veys, “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.
Mathematical Reviews (MathSciNet): MR2325153
Zentralblatt MATH: 1127.14004
[26] R. Waldi, Wertehalbgruppe und Singularität einer ebenen algebroiden Kurve, Ph.D. dissertation, Universität Regensburg, Regensburg, Germany, 1972.
[27] H. von Jürgen Herzog and E. Kunz, Der kanonische Modul eines Cohen-Macaulay-Rings. Seminar über die lokale Kohomologietheorie von Grothendieck, Universität Regensburg, Wintersemester 1970/1971, Lecture Notes in Math. 238, Springer, Berlin, 1971.
[28] O. Zariski, The Moduli Problem for Plane Branches, Univ. Lecture Ser. 39, Amer. Math. Soc., Providence, 2006.
Mathematical Reviews (MathSciNet): MR2273111
Zentralblatt MATH: 1107.14021
[29] W. A. Zúñiga-Galindo, Zeta functions and Cartier divisors on singular curves over finite fields, Manuscripta Math. 94 (1997), 75–88.
[30] W. A. Zúñiga-Galindo, Zeta functions of singular curves over finite fields, Rev. Colombiana Mat. 31 (1997), 115–124.
[31] W. A. Zúñiga-Galindo, Zeta functions of singular algebraic curves over finite fields, Ph.D. dissertation, Instituto Nacional de Matemática Pure e Aplicada, Rio de Janeiro, Brazil, 1996.
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