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97
A Survey on the Model Theory of Difference Fields
, 2000
"... We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conject ..."
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Cited by 101 (16 self)
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We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conjecture over a number field. We conclude with some other applications.
The inverse Galois problem and rational points on moduli spaces
 MATH. ANNALEN
, 1991
"... We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P(x) with L regular over any PAC field P of characteristic zer ..."
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Cited by 80 (29 self)
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We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P(x) with L regular over any PAC field P of characteristic zero. A special case of this implies that G is a Galois group over Fp(x) for almost all primes p.
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 58 (10 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 elementary theory of the Frobenius automorphisms
, 2004
"... We lay down some elements of a geometry based on difference equations. Various constructions of algebraic geometry are shown to have meaningful analogs: dimensions, blowingup, moving lemmas. Analogy aside, the geometry of difference equations has two quite different functorial connections with ordin ..."
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Cited by 48 (0 self)
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We lay down some elements of a geometry based on difference equations. Various constructions of algebraic geometry are shown to have meaningful analogs: dimensions, blowingup, moving lemmas. Analogy aside, the geometry of difference equations has two quite different functorial connections with ordinary algebraic geometry. On the one hand, a difference scheme is determined by geometric data, including principally a proalgebraic scheme. On the other hand, for each prime power p m, one has a functor into algebraic schemes over Fp, where the structure endomorphism becomes Frobenius. Transformal zerocycles have a rich structure in the new geometry. In particular, the Frobenius reduction functors show that they encapsulate data described in classical cases by zeta or Lfunctions. A theory of rational and algebraic equivalence of 0cycles is initiated, via a study of the transformal analog of discrete valuation rings. The central application and motivation is the determination of the elementary theory of the class of Frobenius difference fields (algebraically closed fields of characteristic p> 0,
The embedding problem over a Hilbertian PACfield
 ANNALS OF MATH
, 1992
"... We show that the absolute Galois group of a countable Hilbertian P(seudo)A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A). As a consequence, G ( ¯ Q/Q) is the extension of groups with a fairly simple structure (e.g., ∏ ∞ n=2 Sn) by ..."
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Cited by 37 (18 self)
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We show that the absolute Galois group of a countable Hilbertian P(seudo)A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A). As a consequence, G ( ¯ Q/Q) is the extension of groups with a fairly simple structure (e.g., ∏ ∞ n=2 Sn) by a countably free group. In addition, we characterize those PAC fields over which every finite group is a Galois group as those with the RGHilbertian property (Theorem B).
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 20 (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.
On Some Rational Generating Series Occuring in Arithmetic Geometry
"... this paper, by a variety over a ring R, we mean a reduced and separated scheme of finite type over Spec R ..."
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Cited by 18 (4 self)
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this paper, by a variety over a ring R, we mean a reduced and separated scheme of finite type over Spec R