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The MordellLang conjecture for function fields
 J. Amer. Math. Soc
, 1996
"... In [La65], Lang formulated a hypothesis including as special cases the Mordell conjecture concerning rational points on curves, and the ManinMumford conjecture on torsion points of Abelian varieties. Sometimes generalized to semiAbelian varieties, and to positive characteristic, this has been call ..."
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Cited by 88 (3 self)
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In [La65], Lang formulated a hypothesis including as special cases the Mordell conjecture concerning rational points on curves, and the ManinMumford conjecture on torsion points of Abelian varieties. Sometimes generalized to semiAbelian varieties, and to positive characteristic, this has been called the MordellLang conjecture;
Jet Spaces of Varieties over Differential and Difference Fields
, 2001
"... We give elementary proofs, using suitable jet spaces, of some old and new structural results concerning finitedimensional differential algebraic varieties (characteristic zero). We prove analogous results for difference algebraic varieties in characteristic zero. We also mention partial results and ..."
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Cited by 39 (10 self)
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We give elementary proofs, using suitable jet spaces, of some old and new structural results concerning finitedimensional differential algebraic varieties (characteristic zero). We prove analogous results for difference algebraic varieties in characteristic zero. We also mention partial results and problems in the positive characteristic case.
Some Model Theory of Compact Complex Spaces
 IN CONTEMPORARY MATHEMATICS 270, AMS
, 2000
"... We make several observations about the category C of compact complex manifolds, considered as a manysorted structure of finite Morley rank. We also point out that the MordellLang conjecture holds for complex tori: if A is a complex torus, # a finitely generated subgroup of A, and X an analyti ..."
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Cited by 15 (8 self)
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We make several observations about the category C of compact complex manifolds, considered as a manysorted structure of finite Morley rank. We also point out that the MordellLang conjecture holds for complex tori: if A is a complex torus, # a finitely generated subgroup of A, and X an analytic subvariety of A, then X # # is a finite union of translates of subgroups of A. This is implicit in the literature but we give an elementary reduction to the abelian variety case. We discuss analogies between C and the category of finite dimensional differential algebraic sets. (A brief survey of the MordellLang conjecture and modeltheoretic contributions is also included.)
The ManinMumford Conjecture: A Brief Survey
 Bull. London Math. Soc
, 1999
"... This is a survey paper on the ManinMumford conjecture for number fields with some emphasis on effectivity. It is based on the author's lecture at the Arizona Winter School on Arithmetical Algebraic Geometry (March 1999). We discuss some of the history of this conjecture (and of related conject ..."
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Cited by 12 (0 self)
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This is a survey paper on the ManinMumford conjecture for number fields with some emphasis on effectivity. It is based on the author's lecture at the Arizona Winter School on Arithmetical Algebraic Geometry (March 1999). We discuss some of the history of this conjecture (and of related conjectures) and some recent explicit results.
F structures and integral points on semiabelian varieties over finite fields
, 2003
"... Abstract. Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination and stability result for finitely generated modules over certain finite simple extensions of the integers given to ..."
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Cited by 9 (3 self)
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Abstract. Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination and stability result for finitely generated modules over certain finite simple extensions of the integers given together with predicates for cycles of the distinguished generator of the ring. 1. introduction The MordellLang conjecture asserts that for G a semiabelian variety over the complex numbers, X ⊂ G a subvariety, and Γ ≤ G(C) a finitely generated subgroup of the complex points, the set of points X(C) ∩ Γ is a finite union of cosets of subgroups of Γ. This fails when C is replaced by a field of positive characteristic. For example, suppose that X and G are defined over a finite field Fq and that F: G → G is the corresponding Frobenius morphism. Let K: = Fq(X) and let Γ ≤ G(K) be the Z[F]submodule generated by γ: = idX: X → X thought of as an element of X(K). Then X(K) ∩ Γ contains the infinite set {F n γ: n ∈ N}. If X
Division points on subvarieties of isotrivial semiabelian varieties
 Internat. Math. Res. Notices 19, 2006, Article ID
"... Abstract. The positive characteristic functionfield MordellLang conjecture for finite rank subgroups is resolved for curves as well as for subvarieties of semiabelian varieties defined over finite fields. In the latter case, the structure of the division points on such subvarieties is determined. ..."
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Cited by 7 (3 self)
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Abstract. The positive characteristic functionfield MordellLang conjecture for finite rank subgroups is resolved for curves as well as for subvarieties of semiabelian varieties defined over finite fields. In the latter case, the structure of the division points on such subvarieties is determined. 1.
The Absolute MordellLang Conjecture in Positive Characteristic
"... . We describe intersections of finitely generated subgroups of semiabelian varieties with subvarieties in characteristic p. 1. Introduction A version of the MordellLang Conjecture in characteristic zero asserts that if G is a semiabelian variety, # # G is a finitely generated group, and X # ..."
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Cited by 6 (3 self)
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. We describe intersections of finitely generated subgroups of semiabelian varieties with subvarieties in characteristic p. 1. Introduction A version of the MordellLang Conjecture in characteristic zero asserts that if G is a semiabelian variety, # # G is a finitely generated group, and X # G is a subvariety, then the Zariski closure of X # # is a finite union of cosets of algebraic groups. Easy examples constructed using the Frobenius endomorphism show that this statement cannot be true in positive characteristic. In [Hr], Hrushovski proved a relative version of the MordellLang Conjecture valid in all characteristics in which all counterexamples are shown to come from varieties defined over the algebraic closure of the prime field. By the work of Faltings [Fa], the MordellLang Conjecture is true in characteristic zero. In this paper, we supply a description of the exceptional sets in positive characteristic. In a sense to be made precise, the additional structure on the in...
The isotrivial case in the MordellLang theorem
, 2006
"... Abstract. We determine the structure of the intersection of a finitely generated subgroup of a semiabelian variety G defined over a finite field with a closed subvariety X ⊂ G. We also study a related question in the context of a power of the additive group scheme. 1. ..."
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Cited by 4 (1 self)
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Abstract. We determine the structure of the intersection of a finitely generated subgroup of a semiabelian variety G defined over a finite field with a closed subvariety X ⊂ G. We also study a related question in the context of a power of the additive group scheme. 1.
MordellLang for function fields in Characteristic Zero, Revisited
, 2003
"... We give an elementary proof of the function field (or geometric) case of the MordellLang conjecture in characteristic zero. ..."
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Cited by 3 (2 self)
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We give an elementary proof of the function field (or geometric) case of the MordellLang conjecture in characteristic zero.
Abelian Varieties and the Mordell–Lang Conjecture
, 2000
"... This is an introductory exposition to background material useful to appreciate various formulations of the Mordell–Lang conjecture (now established by recent spectacular work due to Vojta, Faltings, Hrushovski, Buium, Voloch, and others). It gives an exposition of some of the elementary and standar ..."
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Cited by 3 (0 self)
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This is an introductory exposition to background material useful to appreciate various formulations of the Mordell–Lang conjecture (now established by recent spectacular work due to Vojta, Faltings, Hrushovski, Buium, Voloch, and others). It gives an exposition of some of the elementary and standard constructions of algebrogeometric models (rather than modeltheoretic ones) with applications (for example, via the method of Chabauty) relevant to Mordell–Lang. The article turns technical at one point (the step in the proof of the Mordell–Lang Conjecture in characteristic zero which passes from number fields to general fields). Two different procedures are sketched for doing this, with more details given than are readily found in the literature. There is also some discussion of issues of