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103
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 102 (14 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 21 (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.
to use finite fields for problems concerning infinite fields. Arithmetic, geometry, cryptography and coding theory
, 2009
"... As the title indicates, the purpose of the present lecture is to show how to use finite fields for solving problems on infinite fields. This can be done on two different levels: the elementary one uses only the fact that most algebraic geometry statements involve only finitely many data, hence come ..."
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Cited by 18 (0 self)
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As the title indicates, the purpose of the present lecture is to show how to use finite fields for solving problems on infinite fields. This can be done on two different levels: the elementary one uses only the fact that most algebraic geometry statements involve only finitely many data, hence come from geometry over a finitely generated ring, and the residue fields of such a ring are finite; the examples we give in §§14 are of that type. A different level consists in using Chebotarev’s density theorem and its variants, in order to obtain results over nonalgebraically closed fields; we give such examples in §§56. The last two sections were only briefly mentioned in the actual lecture; they explain how cohomology (especially the étale one) can be used instead of finite fields; the proofs are more sophisticated 1, but the results have a wider range. 1. Automorphisms of the affine nspace Let us start with the following simple example: Theorem 1.1. Let σ be an automorphism of the complex affine nspace C n, viewed as an algebraic variety. Assume that σ 2 = 1. Then σ has a fixed point. Surprisingly enough this theorem can be proved by “replacing C by a finite field”. More generally: Theorem 1.2. Let G be a finite pgroup acting algebraically on the affine space A n over an algebraically closed field k with char k ̸ = p. Then the action of G has a fixed point. Proof of Theorem 1.2 a) The case k = Fℓ, where ℓ is a prime number ̸ = p We may assume that the action of G is defined over some finite extension Fℓm of Fℓ. Then the group G acts on the product Fℓm × · · · ×Fℓm. However, G is a pgroup and the number of elements of Fℓm × · · ·×Fℓ m is not divisible by p. Hence there is an orbit consisting of one element, i.e. there is a fixed point for the action of G. b) Reduction to the case k = Fℓ