Nonstandard point counting for algebraic varieties. (English) Zbl 1273.14011
Summary: For varieties over a field \(k\), we define motivic measures with values in ordered fields, using point counting over finite fields and taking the limits with respect to ultrafilters. Some properties and problems related to such measures are discussed. We define similar measures for algebraic dynamical systems by counting periodic points, and explain how they can be used to prove certain statements, such as the non-rationality of a motivic zeta function.
MSC:
| 14A99 | Foundations of algebraic geometry |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
References:
| [1] | Abramovich D., J. Amer. Math. Soc. 15 (3) pp 531– (2002) · Zbl 1032.14003 · doi:10.1090/S0894-0347-02-00396-X |
| [2] | André , Y. ( 2004 ). Une introduction aux motifs: Motifs purs, motifs mixtes, périodes, Panoramas et synthèses 17. SMF . |
| [3] | Batyrev , V. ( 1998 ). Stringy Hodge Numbers of Varieties with Gorenstein Canonical Singularities.Integrable Systems and Algebraic Geometry(Kobe/Kyoto 1997). River Edge, NJ: World Sci. Publ., pp. 1–32 . · Zbl 0963.14015 |
| [4] | Borel É., Bull. Sci. Math. 18 pp 22– (1894) |
| [5] | Deligne P., Inst. Hautes Études Sci. Publ. Math. 43 pp 273– (1974) · Zbl 0287.14001 · doi:10.1007/BF02684373 |
| [6] | Denef J., On Some Rational Generating Series Occurring in Arithmetic Geometry (2004) · Zbl 1061.11067 |
| [7] | Ellenberg J. S., Algebra Number Theory 4 (1) pp 105– (2010) · Zbl 1196.14021 · doi:10.2140/ant.2010.4.105 |
| [8] | Grothendieck , A. ( 1966 ). Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III.Inst. Hautes Études Sci. Publ. Math.(28) . |
| [9] | Gillet H., J. Reine Angew. Math. 478 pp 127– (1996) |
| [10] | Hasse H., Math. Ann. 162 pp 74– (1965) · Zbl 0135.10203 · doi:10.1007/BF01361933 |
| [11] | Kimura S., Math. Ann. 331 (1) pp 173– (2005) · Zbl 1067.14006 · doi:10.1007/s00208-004-0577-3 |
| [12] | Kollár J., Rational Curves on Algebraic Varieties (1996) · Zbl 0877.14012 · doi:10.1007/978-3-662-03276-3 |
| [13] | Larsen M., Compos. Math. 140 (6) pp 1537– (2004) |
| [14] | Lang S., Amer. J. Math. 76 pp 819– (1954) · Zbl 0058.27202 · doi:10.2307/2372655 |
| [15] | Macdonald I. G., Proc. Cambridge Philos. Soc. 58 pp 563– (1962) · doi:10.1017/S0305004100040573 |
| [16] | Schoutens H., Manuscripta Math. 111 (3) pp 379– (2003) · Zbl 1082.13005 · doi:10.1007/s00229-003-0380-6 |
| [17] | Silverman J. H., The Arithmetic of Dynamical Systems (2007) · Zbl 1130.37001 · doi:10.1007/978-0-387-69904-2 |
| [18] | Silverman J. H., New York J. Math. 14 pp 601– (2008) |
| [19] | Takahashi , N. ( 2011 ). Descending Chain Condition for Stringy Invariants. Higher dimensional algebraic geometry for stringy invariants. Higher dimensional algebraic geometry. RIMS K ky roku Bessatsu, B24,Res. Inst. Math. Sci.(RIMS), Kyoto, pp. 165–178 . · Zbl 1235.14018 |
| [20] | Veys W., Duke Math. J. 120 (3) pp 469– (2003) · Zbl 1089.14006 · doi:10.1215/S0012-7094-03-12031-1 |
| [21] | Wiertelak K., Acta Arith. 43 (2) pp 177– (1984) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
