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What is motivic measure?
Author:
Thomas C. Hales
Journal:
Bull. Amer. Math. Soc. 42 (2005), 119-135
MSC (2000):
Primary 14G20
Posted:
January 28, 2005
MathSciNet review:
2133307
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: This article gives an exposition of the theory of arithmetic motivic measure, as developed by J. Denef and F. Loeser.
- 1.
Raf
Cluckers and François
Loeser, Fonctions constructibles et intégration motivique.
I, C. R. Math. Acad. Sci. Paris 339 (2004),
no. 6, 411–416 (French, with English and French summaries). MR 2092754
(2005f:14049), http://dx.doi.org/10.1016/j.crma.2004.06.026
- 2.
A. Craw, An introduction to motivic integration, 1999.
- 3.
Jan
Denef and François
Loeser, Motivic Igusa zeta functions, J. Algebraic Geom.
7 (1998), no. 3, 505–537. MR 1618144
(99j:14021)
- 4.
Jan
Denef and François
Loeser, Germs of arcs on singular algebraic varieties and motivic
integration, Invent. Math. 135 (1999), no. 1,
201–232. MR 1664700
(99k:14002), http://dx.doi.org/10.1007/s002220050284
- 5.
Jan
Denef and François
Loeser, Definable sets, motives and
𝑝-adic integrals, J. Amer. Math.
Soc. 14 (2001), no. 2, 429–469 (electronic). MR 1815218
(2002k:14033), http://dx.doi.org/10.1090/S0894-0347-00-00360-X
- 6.
J.
Denef and F.
Loeser, Motivic integration and the Grothendieck group of
pseudo-finite fields, (Beijing, 2002) Higher Ed. Press, Beijing,
2002, pp. 13–23. MR 1957016
(2004f:14040)
- 7.
J. Denef and F. Loeser, On some rational generating series occurring in arithmetic geometry, to appear.
- 8.
William
Fulton, Intersection theory, 2nd ed., Ergebnisse der
Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in
Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series
of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin,
1998. MR
1644323 (99d:14003)
- 9.
G. van der Geer and B. Moonen, Abelian Varieties, preliminary version of Chapter XI. The Fourier transform and the Chow ring, July 2003, http://turing.wins.uva.nl/bmoonen/boek/BookAV.html.
- 10.
M. Kontsevich, lecture at Orsay, December 1995.
- 11.
François
Loeser and Julien
Sebag, Motivic integration on smooth rigid varieties and invariants
of degenerations, Duke Math. J. 119 (2003),
no. 2, 315–344. MR 1997948
(2004g:14026), http://dx.doi.org/10.1215/S0012-7094-03-11924-9
- 12.
Eduard
Looijenga, Motivic measures, Astérisque
276 (2002), 267–297. Séminaire Bourbaki, Vol.
1999/2000. MR
1886763 (2003k:14010)
- 13.
Ernesto
Lupercio and Mainak
Poddar, The global McKay-Ruan correspondence via motivic
integration, Bull. London Math. Soc. 36 (2004),
no. 4, 509–515. MR 2069013
(2005c:14026), http://dx.doi.org/10.1112/S002460930300290X
- 14.
Johan
Pas, Uniform 𝑝-adic cell decomposition and local zeta
functions, J. Reine Angew. Math. 399 (1989),
137–172. MR 1004136
(91g:11142), http://dx.doi.org/10.1515/crll.1989.399.137
- 15.
A.
J. Scholl, Classical motives, Motives (Seattle, WA, 1991)
Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI,
1994, pp. 163–187. MR 1265529
(95b:11060)
- 16.
Julien
Sebag, Intégration motivique sur les schémas
formels, Bull. Soc. Math. France 132 (2004),
no. 1, 1–54 (French, with English and French summaries). MR 2075915
(2005e:14017)
- 1.
- R. Cluckers and F. Loeser, Fonctions constructibles et intégration motivique I, II, C. R. Math. Acad. Sci. Paris 339 (2004), no. 6, 411-416. MR 2092754
- 2.
- A. Craw, An introduction to motivic integration, 1999.
- 3.
- J. Denef and F. Loeser, Motivic Igusa functions, Journal of Algebraic Geometry 7, 505-537 (1998). MR 1618144 (99j:14021)
- 4.
- J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Inventiones Mathematicae 135, 201-232 (1999). MR 1664700 (99k:14002)
- 5.
- J. Denef and F. Loeser, Definable sets, motives and
 -adic integrals, JAMS 14, No. 2, 429-469, 2000. MR 1815218 (2002k:14033)
- 6.
- J. Denef and F. Loeser, Motivic integration and the Grothendieck group of pseudo-finite fields, Proc. ICM, Vol. II (Beijing, 2002), 13-23, Higher Education Press, 2002. MR 1957016 (2004f:14040)
- 7.
- J. Denef and F. Loeser, On some rational generating series occurring in arithmetic geometry, to appear.
- 8.
- W. Fulton, Intersection Theory, second edition, Springer, 1998. MR 1644323 (99d:14003)
- 9.
- G. van der Geer and B. Moonen, Abelian Varieties, preliminary version of Chapter XI. The Fourier transform and the Chow ring, July 2003, http://turing.wins.uva.nl/bmoonen/boek/BookAV.html.
- 10.
- M. Kontsevich, lecture at Orsay, December 1995.
- 11.
- F. Loeser and J. Sebag, Motivic integration on smooth rigid varieties and invariants of degenerations, Duke Mathematical Journal 119, 315-344 (2003). MR 1997948 (2004g:14026)
- 12.
- E. Looijenga, Motivic measures. Séminaire Bourbaki, Vol. 1999/2000. Astérisque No. 276 (2002), 267-297. MR 1886763 (2003k:14010)
- 13.
- E. Lupercio and M. Poddar, The global McKay-Ruan correspondence via motivic integration, Bull. London Math. Soc. 36 (2004), no. 4, 509-515. MR 2069013
- 14.
- J. Pas, Uniform
 -adic cell decomposition and local zeta functions, J. reine angew. Math. 399 (1989), 137-172. MR 1004136 (91g:11142)
- 15.
- A. Scholl, Classical motives, in Motives (U. Jannsen, S. Kleiman, J-P. Serre, Eds.), Proc. Symp. Pure Math., Vol. 55, Part 1 (1994), 163-187, Amer. Math. Soc. MR 1265529 (95b:11060)
- 16.
- J. Sebag, Intégration motivique sur les schémas formels, Bull. Soc. Math. France 132 (2004), no. 1, 1-54. MR 2075915
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Additional Information
Thomas C. Hales
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
DOI:
http://dx.doi.org/10.1090/S0273-0979-05-01053-0
PII:
S 0273-0979(05)01053-0
Received by editor(s):
June 1, 2003
Posted:
January 28, 2005
Additional Notes:
Work supported by the NSF
This work is licensed under the Creative Commons Attribution License. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA
Article copyright:
© Copyright 2005 Thomas C. Hales
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