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by: Addison Beer


Addison Beer
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Class Notes
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This 2 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 135 at University of Washington taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/192101/math-135-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15
Motivic and p adic integration by Francois Loeser Outline of the lectures My goal is to explain the basics of p adic integration and motivic integration and to discuss some connections with Model Theory Lecture 1 p adic integration We will explain the basics of p adic integration on smooth varieties its relation with number of points of reductions modulo 1 Oesterl 7s Theorem and applications to rationality of Poincare series work of lgusa and Denef We shall conclude by presenting Denef7s results on the measure of definable sets Lecture 2 Motivic integration Arc spaces Grothendieck rings of varieties Construction of motivic measures and basic properties Change of variable formula Ap plications to rationality results Lecture 3 Assigning virtual motives t0 de nable sets We shall explain first Chow motives Galois stratifications and quantifier elimination for pseudo finite fields Then we will be able to assign a virtual motive to definable sets We shall explain how it relates to counting points Lecture 4 Arithmetic motivic integration Using results from the previous lecture we shall round the loop by explaining how one can construct motivic integrals that spe cialize to p adic ones If time allows we shall show how this fits in a much more general framework Prerequisites Familiarity with the language of Algebraic Geometry as developed in Hartshorne7s book and with the most elementary part of Model Theory References Lecture 1 l Jlgusa An introduction to the theory of local zeta functions AMSlP Studies in Ad vanced Mathematics 14 American Mathematical Society Providence RI International Press Cambridge MA 2000 2 JDenef Arithmetic and geometric applications of quantifier elimination for valued fields Model theory algebra and geometry 1737198 Math Sci Res Inst Publ 39 Cambridge Univ Press Cambridge 2000 Lecture 2 3 JDenef F Loeser Geometry on arc spaces of algebraic varieties European Congress of Mathematics Vol I Barcelona 2000 3277348 Progr Math 201 Birkhuser Basel 2001 4 JDenef F Loeser Germs of arcs on singular algebraic varieties and motivic integration lnvent Math 135 1999 no 1 2017232 Lecture 3 4 5 MFried MJarden Field arithmetic Ergebnisse der Mathematik und ihrer Grende biete 3 ll Springer Verlag Berlin 1986 6 JDenef FLoeser Definable sets motives and p adic integrals J Amer Math Soc 4 2001 no 2 4297469 7 THales Can p adic integrals be computed mathRT0209001 8 JDenef FLoeser Motivic integration and the Grothendieck group of pseudo finite elds AG0207163 9 JDenef FLoeser On some rational generating series occuring in arithmetic geometry available at httpwwwdmaensfr loeser Project 1 The quanti er elimination Theorem of Fried and Sacerdote plays a basic role in Lecture 3 The proof presented in the book 5 Proposition 259 of is given in a very algebraic language and could be translated in more geometric terms using basic knowledge of Algebraic Geometry such as Galois Theory for coverings of schemes The project has 2 steps the first is to present a neat self contained geometric proof of Proposition 259 of the second is to find and to prove a generalization of that result over a more general base than the spectrum of a field Suggested readings for the project Familiarity with the relevant chapters of 5 and learning about geometric aspects Galois covers in 13 Project 2 The complete proof of Theorem 641 in 9 the main result in lecture 3 is scattered between 3 places 6 8 and The project would be to rearrange the arguments given or sketched in these papers to be able to write down a self contained direct proof of the Theorem Suggested readings for the project Learn basics about Chow motives in 10 A Scholl Classical motives Motives Seattle WA 1991 1637187 Proc Sym pos Pure Math 55 Part 1 Amer Math Soc Providence RI 1994 Get a look to the paper without going through proofs 11 Sdel Bao Rollin VNavarro Aznar On the motive of a quotient variety Collect Math 49 1998 no 2 3 203726 The standard modern reference for coverings of varieties and schemes is SGA1 avail able on the arXiv 12 AGrothendieck MRaynaud SGA 1 Revetements tales et groupe fondamental mathAG0206203 but this may seem somewhat arid for most A more leisurely introduction can be found in 13 J PSerre Topics in Galois theory Research Notes in Mathematics 1 Jones and Bartlett Publishers Boston MA 1992


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