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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 187 (21 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 101 (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.
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 67 (15 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...
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 60 (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 ERGODIC THEORY OF LATTICE SUBGROUPS
, 2007
"... Abstract. We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or Salgebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice Γ in G, we use the ergodic theorems for G to solve the latti ..."
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Cited by 31 (11 self)
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Abstract. We prove mean and pointwise ergodic theorems for general families of averages on a semisimple algebraic (or Salgebraic) group G, together with an explicit rate of convergence when the action has a spectral gap. Given any lattice Γ in G, we use the ergodic theorems for G to solve the lattice point counting problem for general domains in G, and prove mean and pointwise ergodic theorems for arbitrary measurepreserving actions of the lattice, together with explicit rates of convergence when a spectral gap is present. We also prove an equidistribution theorem in arbitrary isometric actions of the lattice. For the proof we develop a general method to derive ergodic theorems for actions of a locally compact group G, and of a lattice subgroup Γ, provided certain natural spectral, geometric and regularity conditions are satisfied by the group G, the lattice Γ, and the domains where the averages are supported. In particular, we
Analytic cell decomposition and analytic motivic integration
 ANN. SCI. ÉCOLE NORM. SUP
, 2006
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Representation Growth for Linear Groups
"... Abstract. Let Γ be a group and rn(Γ) the number of its ndimensional irreducible complex representations. We define and study the associated representation zeta function ZΓ(s) = ∞∑ rn(Γ)n−s. When Γ is an arithmetic n=1 group satisfying the congruence subgroup property then ZΓ(s) has an “Euler facto ..."
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Cited by 23 (2 self)
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Abstract. Let Γ be a group and rn(Γ) the number of its ndimensional irreducible complex representations. We define and study the associated representation zeta function ZΓ(s) = ∞∑ rn(Γ)n−s. When Γ is an arithmetic n=1 group satisfying the congruence subgroup property then ZΓ(s) has an “Euler factorization”. The “factor at infinity ” is sometimes called the “Witten zeta function ” counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups U of the associated simple group G over the associated local field K. Here we show a surprising dichotomy: if G(K) is compact (i.e. G anisotropic over K) the abscissa of convergence goes to 0 when dim G goes to infinity, but for isotropic groups it is bounded away from 0. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa. 1.
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 22 (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.