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57
Germs of Arcs on Singular Algebraic Varieties and Motivic Integration
, 1999
"... Introduction Let k be a field of characteristic zero. We denote by M the Grothendieck ring of algebraic varieties over k (i.e. reduced separated schemes of finite type over k). It is the ring generated by symbols [S], for S an algebraic variety over k, with the relations [S] = [S 0 ] if S is iso ..."
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Cited by 133 (19 self)
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Introduction Let k be a field of characteristic zero. We denote by M the Grothendieck ring of algebraic varieties over k (i.e. reduced separated schemes of finite type over k). It is the ring generated by symbols [S], for S an algebraic variety over k, with the relations [S] = [S 0 ] if S is isomorphic to S 0 , [S] = [S n S 0 ] + [S 0 ] if S 0 is closed in S and [S \Theta S 0 ] = [S] [S 0 ]. Note that, for S an algebraic variety over k, the mapping S 0 7! [S 0 ] from the
Geometry on Arc Spaces of Algebraic Varieties
 Proceedings of the Third European Congress of Mathematics, Barcelona 2000, Progr. Math. 201 (2001
, 2001
"... This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants. ..."
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Cited by 57 (6 self)
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This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants.
Definable sets, motives and padic integrals
 J. Amer. Math. Soc
, 2001
"... 0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Zp) ofitsZprational points. For every n in N, there is a natural map πn: X(Zp) → X(Z/pn+1) assigning to a Zprational point its class modulo pn+1. The image Yn,p of X(Zp) byπn is ..."
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Cited by 43 (9 self)
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0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Zp) ofitsZprational points. For every n in N, there is a natural map πn: X(Zp) → X(Z/pn+1) assigning to a Zprational point its class modulo pn+1. The image Yn,p of X(Zp) byπn is exactly the set of
The CasselsTate Pairing On Polarized Abelian Varieties
 Ann. of Math
, 1998
"... . Let (A; ) be a principally polarized abelian variety dened over a global eld k, and let X(A) be its Shafarevich{Tate group. Let X(A) nd denote the quotient of X(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing X(A) nd X(A) nd ! Q=Z : If A is an ellip ..."
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Cited by 42 (14 self)
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. Let (A; ) be a principally polarized abelian variety dened over a global eld k, and let X(A) be its Shafarevich{Tate group. Let X(A) nd denote the quotient of X(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing X(A) nd X(A) nd ! Q=Z : If A is an elliptic curve, then by a result of Cassels' the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent denitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on X(A) nd . These criteria are expressed in terms of an element c 2 X(A) nd that is canonically associated to the polarization . In the case that A is the Jacobian of some curve, a downtoearth version of the result allows us to determine eectively whether #X(A) (if nite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperell...
Motivic integration and the grothendieck group of pseudofinite fields
 Proceedings of the International Congress of Mathematicians (ICM 2002
"... Motivic integration is a powerful technique to prove that certain quantities associated to algebraic varieties are birational invariants or are independent of a chosen resolution of singularities. We survey our recent work on an extension of the theory of motivic integration, called arithmetic motiv ..."
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Cited by 16 (4 self)
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Motivic integration is a powerful technique to prove that certain quantities associated to algebraic varieties are birational invariants or are independent of a chosen resolution of singularities. We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how padic integrals of a very general type depend on p. Quantifier elimination plays a key role.
Analytic cell decomposition and analytic motivic integration
 ANN. SCI. ÉCOLE NORM. SUP
, 2006
"... ..."
OneDimensional PAdic Subanalytic Sets
 J. London Math. Soc
"... Introduction In this article we extend two theorems from [2] on padic subanalytic sets, where p is a fixed prime number, Q p is the field of padic numbers and Z p is the ring of padic integers. One of these theorems, 3.32 in [2], says that each subanalytic subset of Z p is semialgebraic. This is ..."
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Cited by 13 (0 self)
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Introduction In this article we extend two theorems from [2] on padic subanalytic sets, where p is a fixed prime number, Q p is the field of padic numbers and Z p is the ring of padic integers. One of these theorems, 3.32 in [2], says that each subanalytic subset of Z p is semialgebraic. This is extended here as follows. Theorem A Let S ` Z m+1 p be a subanalytic set. Then there is a semialgebraic set S 0 ` Z m 0 +1 p such that for each x 2 Z m p there is x 0 2 Z m 0 p with S x = S 0 x 0 . Hence the semialgebraic c
Analytic padic cell decomposition and integrals
 Trans. Amer. Math. Soc
"... Abstract. We prove a conjecture of Denef on parameterized padic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), ..."
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Cited by 12 (12 self)
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Abstract. We prove a conjecture of Denef on parameterized padic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection. 1.