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36
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 136 (20 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
Pseudoreductive groups
, 2010
"... Why go beyond reductive groups? The theory of connected reductive groups over a general field, and its applications over arithmetically interesting fields, constitutes one of the most beautiful topics within pure mathematics. However, it does sometimes happen that one is confronted with linear algeb ..."
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Cited by 32 (6 self)
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Why go beyond reductive groups? The theory of connected reductive groups over a general field, and its applications over arithmetically interesting fields, constitutes one of the most beautiful topics within pure mathematics. However, it does sometimes happen that one is confronted with linear algebraic groups that are
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 17 (5 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.
Motivic integration on smooth rigid varieties and invariants of degenerations
 Duke Math. J
"... We develop a theory of motivic integration for smooth rigid varieties. As an application we obtain a motivic analogue for rigid varieties of Serre’s invariant for padic varieties. Our construction provides new geometric birational invariants of degenerations of algebraic varieties. For degeneration ..."
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Cited by 12 (1 self)
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We develop a theory of motivic integration for smooth rigid varieties. As an application we obtain a motivic analogue for rigid varieties of Serre’s invariant for padic varieties. Our construction provides new geometric birational invariants of degenerations of algebraic varieties. For degenerations of CalabiYau varieties, our results take a stronger form. 1.
Additive Polynomials and Their Role in the Model Theory of Valued Fields, Model Theory of Valued Fields
 Logic in Tehran, Proceedings of the Workshop and Conference on Logic, Algebra, and Arithmetic, held October 1822, 2003. Lecture Notes in Logic 26
, 2006
"... We discuss the role of additive polynomials and ppolynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of valued fields of positive characteristic as modules over such ..."
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Cited by 10 (5 self)
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We discuss the role of additive polynomials and ppolynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of valued fields of positive characteristic as modules over such rings. We prove the existence of Frobeniusclosed bases of algebraic function fields F K in one variable and deduce that F/K is a free module over the ring of additive polynomials with coefficients in K. Finally, we prove that every minimal purely wild extension of a henselian valued field is generated by a ppolynomial. 1
The Brauer–Manin obstruction for subvarieties of abelian varieties over function fields
, 2008
"... We prove that for a large class of subvarieties of abelian varieties over global function fields, the BrauerManin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with ..."
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Cited by 9 (2 self)
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We prove that for a large class of subvarieties of abelian varieties over global function fields, the BrauerManin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.
Jet schemes and singularities
"... The study of singularities of pairs is fundamental for higher dimensional birational geometry. The usual approach to invariants of such singularities is via divisorial valuations, as in [Kol]. In this paper we give a selfcontained presentation of an alternative approach, via contact loci in spaces ..."
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Cited by 9 (1 self)
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The study of singularities of pairs is fundamental for higher dimensional birational geometry. The usual approach to invariants of such singularities is via divisorial valuations, as in [Kol]. In this paper we give a selfcontained presentation of an alternative approach, via contact loci in spaces of arcs. Our main application is