Summary
We develop a theory of integration over valued fields of residue characteristic zero. In particular, we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of Denef, Loeser, and Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a precise description of the Grothendieck semigroup of such sets in terms of related groups over the residue field and value group. This yields new invariants of all definable bijections, as well as invariants of measure-preserving bijections.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
J. T. Baldwin and A. H. Lachlan, On strongly minimal sets, J. Symbolic Logic, 36 (1971), 79–96.
Ş. A. Basarab and F. V. Kuhlmann, An isomorphism theorem for Henselian algebraic extensions of valued fields, Manuscripta Math., 77-2-3 (1992), 113–126.
V. V. Batyrev, Birational Calabi-Yau n-folds have equal Betti numbers, in New Trends in Algebraic Geometry (Warwick, 1996) London Mathematical Society Lecture Note Series, Vol. 264, Cambridge University Press, Cambridge, UK, 1999, 1–11.
I. N. Bernstein, Analytic continuation of generalized functions with respect to a parameter, Funk. Anal. Priložen., 6-4 (1972), 26–40.
Z. Chatzidakis and E. Hrushovski, Model theory of difference fields, Trans. Math. Soc. Amer., 351-8 (1999), 2997–3071.
R. Cluckers and D. Haskell, Grothendieck rings of Z-valued fields, Bull. Symbolic Logic, 7-2 (2001), 262–226.
R. Cluckers and F. Loeser, Fonctions constructibles et intégration motivique I, II, math.AG/0403350 and math.AG/0403349, 2004; also available online from http://www.dma.ens.fr/~loeser/.
R. Cluckers and F. Loeser, Fonctions constructibles exponentiel les, transformation de Fourier motivique et principe de transfert, math.NT/0509723, 2005; C. R. Acad. Sci. Paris Sér. I Math., to appear.
C. C. Chang and H. J. Keisler, Model Theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, Vol. 73, North-Holland, Amsterdam, 1990.
J. Denef and F. Loeser, Definable sets, motives and p-adic integrals, J. Amer. Math. Soc., 14 (2001), 429–469.
H. B. Enderton, A Mathematical Introduction to Logic 2nd ed., Harcourt/Academic Press, Burlington, MA, 2001.
I. Fesenko and M. Kurihara, eds., Invitation to Higher Local Fields, Geometry and Topology Monographs, Vol. 3, Mathematics Institute, University of Warwick, Coventry, UK, 2000.
M. Gromov, Endomorphisms of symbolic algebraic varieties, J. European Math. Soc., 1-2 (1999), 109–197.
R. Cluckers, L. Lipshitz, and Z. Robinson, Analytic cell decomposition and analytic motivic integration, math.AG/0503722, 2005; Ann. Sci. École Norm. Sup., to appear.
D. Haskell, and D. Macpherson, Cell decompositions of C-minimal structures, Ann. Pure Appl. Logic, 66-2 (1994), 113–162.
D. Haskell, E. Hrushovski, and H. D. Macpherson, Definable sets in algebraically closed valued fields, Part I: Elimination of imaginaries, preprint, 2002; Crelle, to appear.
D. Haskell, E. Hrushovski, and H. D. Macpherson, Stable domination and independence in algebraically closed valued fields, math.LO/0511310, 2005.
E. Hrushovski, Elimination of imaginaries for valued fields, preprint.
P. T. Johnstone, Notes on Logic and Set Theory, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, UK, 1987.
M. Kageyama and M. Fujita, Grothendieck rings of o-minimal expansions of ordered Abelian groups, math.LO/0505331, 2005.
D. Kazhdan, An algebraic integration, in Mathematics: Frontiers and Perspectives, American Mathematical Society, Providence, RI, 2000, 93–115.
M. Larsen and V. A. Lunts, Motivic measures and stable birational geometry, Moscow Math. J., 3-1 (2003), 85–95.
L. Lipshitz, Rigid subanalytic sets, Amer. J. Math., 115-1 (1993), 77–108.
L. Lipshitz and Z. Robinson, One-dimensional fibers of rigid subanalytic sets, J. Symbolic Logic, 63-1 (1998), 83–88.
F. Loeser and J. Sebag, Motivic integration on smooth rigid varieties and invariants of degenerations, Duke Math. J., 119-2 (2003), 315–344.
J. Maříková, Geometric Properties of Semilinear and Semibounded Sets, M.A. thesis, Charles University, Prague, 2003; preprint.
T. Mellor, talk, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, 2005.
A. Pillay, An Introduction to Stability Theory, Oxford Logic Guides, Oxford University Press, Oxford, UK, 1983.
A. Pillay, Geometric Stability Theory, Oxford Logic Guides, Oxford University Press, Oxford, UK, 1996.
J. Pas, Uniform p-adic cell decomposition and local zeta functions, J. Reine Angew. Math., 399 (1989), 137–172.
B. Poizat, Cours de Théorie des modeles: Nur al mantiq wal ma’arifah (A Course in Model Theory: An Introduction to Contemporary Mathematical Logic), Universitext, Springer-Verlag, New York, 2000 (translated from the French by M. Klein and revised by the author).
B. Poizat, Une théorie de Galois imaginaire, J. Symbolic Logic, 48-4 (1983), 1151–1170.
A. Robinson, Complete Theories, North-Holland, Amsterdam, 1956.
H. H. Crapo and G.-C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries, preliminary ed., M.I.T. Press, Cambridge, MA, London, 1970.
S. Shelah, Classification Theory and the Number of Nonisomorphic Models, 2nd ed., Studies in Logic and the Foundations of Mathematics 92, North-Holland, Amsterdam, 1990.
L. van den Dries, Dimension of definable sets, algebraic boundedness and Henselian fields, Ann. Pure Appl. Logic, 45-2 (1989), 189–209.
L. van den Dries, Tame Topology and o-Minimal Structures, London Mathematical Society Lecture Note Series, Vol. 248, Cambridge University Press, Cambridge, UK, 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Boston
About this chapter
Cite this chapter
Hrushovski, E., Kazhdan, D. (2006). Integration in valued fields. In: Ginzburg, V. (eds) Algebraic Geometry and Number Theory. Progress in Mathematics, vol 253. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4532-8_4
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4532-8_4
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4471-0
Online ISBN: 978-0-8176-4532-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)