Book contents
- Frontmatter
- Contents
- List of contributors
- 1 Introduction
- 2 Introductory notes on the model theory of valued fields
- 3 On the definition of rigid analytic spaces
- 4 Topological rings in rigid geometry
- 5 The Grothendieck ring of varieties
- 6 A short course on geometric motivic integration
- 7 Motivic invariants of rigid varieties, and applications to complex singularities
- 8 Motivic integration in mixed characteristic with bounded ramification: a summary
- References
1 - Introduction
Published online by Cambridge University Press: 07 October 2011
Book contents
- Frontmatter
- Contents
- List of contributors
- 1 Introduction
- 2 Introductory notes on the model theory of valued fields
- 3 On the definition of rigid analytic spaces
- 4 Topological rings in rigid geometry
- 5 The Grothendieck ring of varieties
- 6 A short course on geometric motivic integration
- 7 Motivic invariants of rigid varieties, and applications to complex singularities
- 8 Motivic integration in mixed characteristic with bounded ramification: a summary
- References
Summary
Since its creation by Kontsevich in 1995, motivic integration has developed quickly in several directions and has found applications in various domains, such as singularity theory and the Langlands program. In its development, it incorporated tools from model theory and nonarchimedean geometry. The aim of the present book is to give an introduction to different theories of motivic integration and related topics.
Motivic integration is a theory of integration for various classes of geometric objects. One term in this “trade-name” seems unusual: motivic. What is a motive and in what sense is this theory of integration motivic? These questions lie at the heart of the theory. We will try to answer them in Section 4. First, we briefly introduce the two other protagonists of the present book: model theory and non-archimedean geometry. We will explain below how they interact with the theory of motivic integration.
Model theory
A central notion in model theory is language. A language ℒ is a collection of symbols, divided into three types: function symbols, relation symbols, and constant symbols. For every function symbol and relation symbol, one specifies the number of arguments. A formula in the language ℒ is built from these symbols, variables, Boolean combinations and quantifiers.
- Type
- Chapter
- Information
- Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry , pp. 1 - 34Publisher: Cambridge University PressPrint publication year: 2011
References
[1] . Éléments de Géométrie Rigide. Volume I. Construction et étude géométrique des espaces rigides. to appear in Progress in Mathematics, Birkhäuser.
[2] , and (eds.). Séminaire de Géométrie Algébrique du Bois Marie (1963-64), Théorie des topos et cohomologie étale des schémas. Vol. 1. Volume 269 of Lecture notes in mathematics.Springer-Verlag, Berlin, 1972.
[3] . Relative elimination of quantifiers for Henselian valued fields. Ann. Pure Appl. Logic 53(1):51–74, 1991.Google Scholar
[4] . Birational Calabi-Yau n-folds have equal Betti numbers. In: New trends in algebraic geometry (Warwick, 1996), pages 1–11. Volume 264 of London Math. Soc. Lecture Note Series.Cambridge Univ. Press, Cambridge, 1999.Google Scholar
[5] . Spectral theory and analytic geometry over nonarchimedean fields. Volume 33 of Mathematical Surveys and Monographs. AMS, 1990.Google Scholar
[6] and . Formal and rigid geometry. I. Rigid spaces. Math. Ann., 295(2):291–317, 1993.Google Scholar
[7] and . Formal and rigid geometry. II. Flattening techniques. Math. Ann., 296(3):403–429, 1993.Google Scholar
[8] , and . Formal and rigid geometry. III. The relative maximum principle. Math. Ann., 302(1):1–29, 1995.Google Scholar
[9] , and . Formal and rigid geometry. IV. The reduced fibre theorem. Invent. Math., 119(2):361–398, 1995.Google Scholar
[10] , and . Transfer Principle for the fundamental lemma. to appear in: (ed.). Stabilisation de la formule des traces, variétés de Shimura, et applications arithmétiques.
[11] and . Constructible motivic functions and motivic integration. Invent. Math. 173(1):23–121, 2008.Google Scholar
[12] and . Constructible exponential functions, motivic Fourier transform and transfer principle. Ann. Math. 171:1011–1065, 2010.Google Scholar
[13] and (eds.) Correspondance Grothendieck-Serre. Volume 2 of Documents Mathématiques. Société Mathématique de France, Paris, 2001.
[14] . The rationality of the Poincaré series associated to the p-adic points on a variety. Invent. Math. 77:1–23, 1984.Google Scholar
[16] and . Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 165:201–232, 1999.Google Scholar
[17] and . Definable sets, motives and p-adic integrals. J. Am. Math. Soc., 14(2):429–469, 2001.Google Scholar
[18] and . Field arithmetic, 3rd edition. Volume 11 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.Springer-Verlag, Berlin, 2008.Google Scholar
[19] and . Integration in valued fields. In: (ed.). Algebraic geometry and number theory, pages 261–405. Volume 253 of Progress in Mathematics, Birkhäuser, Basel, 2006.Google Scholar
[21] and . Non-archimedean tame topology and stably dominated types. arXiv:1009.0252.
[22] . Étale cohomology of rigid analytic varieties and adic spaces. Volume E30 of Aspects of Mathematics. Friedr. Vieweg & Sohn, Braun-schweig, 1996.Google Scholar
[24] . Seattle lectures on motivic integration. In: et al. (eds.). Algebraic geometry—Seattle 2005. Part 2, pages 745–784. Volume 80 of Proceedings of Symposia in Pure Mathematics. American Mathematical Society, Providence, RI, 2009.Google Scholar
[25] and . Motivic integration on smooth rigid varieties and invariants of degenerations. Duke Math. J., 119:315–344, 2003.Google Scholar
[26] . Motivic measures. Séminaire Bourbaki, Vol. 1999/2000. Astérisque No. 276, pages 267–297, 2002.Google Scholar
[27] . Relative motives and the theory of pseudofinite fields. Int. Math. Res. Pap. 2007:rpm 001, 69 pages, 2010.Google Scholar
[28] . A trace formula for rigid varieties, and motivic weil generating series for formal schemes. Math. Ann., 343(2):285–349, 2009.Google Scholar
[29] . An introduction to p-adic and motivic zeta functions and the monodromy conjecture. In: , and (eds.). Algebraic and analytic aspects of zeta functions and L-functions. Volume 21 of MSJ Memoirs, pages 115–140. Mathematical Society of Japan, Tokyo, 2010.Google Scholar
[30] . A trace formula for varieties over a discretely valued field. J. Reine Angew. Math. 650:193–238, 2011.Google Scholar
[31] . Geometric criteria for tame ramification. preprint, arXiv: 0910.3812.
[32] and . The motivic Serre invariant, ramification, and the analytic Milnor fiber. Invent. Math., 168(1):133–173, 2007.Google Scholar
[33] . Géométrie analytique rigide d'après Tate, Kiehl,…. Mémoires de la S.M.F., 39-40:319–327, 1974.Google Scholar
[35] . Intégration motivique sur les schémas formels. Bull. Soc. Math. France, 132(1):1–54, 2004.Google Scholar
[36] . Classification des variétés analytiques p-adiques compactes. Topology, 3:409–412, 1965.Google Scholar
[37] . Rigid analytic spaces. Private notes, reproduced with(out) his permission by I.H.É.S. (1962). Reprinted in Invent. Math. 12:257–289, 1971.Google Scholar
[38] . Motivic integration over Deligne-Mumford stacks. Adv. Math. 207(2):707–761, 2006.Google Scholar
[39] . Arc spaces, motivic integration and stringy invariants. In: et al. (eds.), Singularity Theory and its applications. Volume 43 of Advanced Studies in Pure Mathematics, pages 529–572. Mathematical Society of Japan, Tokyo, 2006.Google Scholar
[40] , with an appendix by . The fundamental lemma of Jacquet-Rallis in positive characteristics. To appear in Duke Math. J., arXiv:0901.0900, appendix arXiv:1005.0610.