Model Theory of Differential Fields

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62 DAVID MARKERvariety of A is isomorphic to an abelian variety defined over k, Γ is a finite ranksubgroup of A, and X is a subvariety of A such that X ∩ Γ is Zariski dense inX. Then X is a finite union of cosets of abelian subvarieties of A.I will sketch the ideas of the proof; for full details see [Hrushovski 1996], [Bouscaren1998] or [Pillay 1997]. First I give the full proof in one easy case.Theorem 5.2. Suppose K ⊃ k are algebraically closed fields of characteristiczero, A is a simple abelian variety defined over K that is not isomorphic to anabelian variety defined over k, Γ is the torsion points of A, and X is a propersubvariety of A. Then X ∩ Γ is finite.Proof. The main idea is to move to a differential field setting where we mayapply model-theoretic tools. In doing so we will replace the group Γ by A # , asmall (that is, finite-dimensional) differential algebraic group.The first step is to define a derivation δ : K → K such that k = {x ∈ K :δ(x) = 0}. One can show that if ˆK is the differential closure of K then the fieldof constants of ˆK is still k. Thus without loss of generality we may assume thatK is a differentially closed field and k is the constant field of K.Let A # be the δ-closure of the torsion points of A. It suffices to show thatA # ∩ X is finite. Suppose not. Since A # is strongly minimal, A # \ X is finite.But then the Zariski closure of A # is contained in X ∪ A # \ X, which is properlycontained in A. This is a contradiction since the torsion points are Zariski dense.□The proof above does not explicitly use the fact that A # is locally modular(though this does come into the proof that it is strongly minimal). Local modularityplays more of a role if we consider larger groups Γ. Suppose K, k, A andX are as above and Γ is a finite rank subgroup of A. Let µ : A → K n be adefinable homomorphism with kernel A # . Since Γ has finite rank, the image ofΓ under µ is contained in V ⊂ K n which is a finite dimensional k-vector space.Consider G = µ −1 (V ). This is a definable finite Morley rank subgroup of A.Some analysis of this group allows us to conclude that it is 1-based (see [Hart2000] for a definition: basically this means that all of the strongly minimal setsare locally modular). Hrushovski and Pillay showed that in an 1-based group Gany definable subset of G n is a boolean combination of cosets of definable subgroups.In particular X ∩G will be a finite union of cosets of definable subgroupsof G. If any of these subgroups is infinite, then its Zariski closure is an algebraicsubgroup and must hence be the whole group. This would contradict the factthat the Zariski closure would be contained in X.To prove Theorem 5.1 in general, we use the fact that every abelian variety isisogenous to a direct sum of simple abelian subvarieties together with a numberof techniques from finite Morley rank group theory.
MODEL THEORY OF DIFFERENTIAL FIELDS 63References[Bouscaren 1998] E. Bouscaren, “Proof of the Mordell–Lang conjecture for functionfields”, pp. 177–196 in Model theory and algebraic geometry, edited by E. Bouscaren,Lecture Notes in Math. 1696, Springer, Berlin, 1998.[Buium 1994] A. Buium, Differential algebra and Diophantine geometry, Actualitésmath., Hermann, Paris, 1994.[Hart 2000] B. Hart, “Stability theory and its variants”, pp. 131–148 in Model theory,algebra and geometry, edited by D. Haskell et al., Math. Sci. Res. Inst. Publ. 39,Cambridge Univ. Press, New York, 2000.[Hrushovski 1996] E. Hrushovski, “The Mordell–Lang conjecture for function fields”,J. Amer. Math. Soc. 9:3 (1996), 667–690.[Hrushovski and Sokolovic ≥ 2001] E. Hrushovski and Z. Sokolovic, “Minimal subsetsof differentially closed fields”. To appear in Trans. Amer. Math. Soc.[Hrushovski and Zilber 1996] E. Hrushovski and B. Zilber, “Zariski geometries”, J.Amer. Math. Soc. 9:1 (1996), 1–56.[Kaplansky 1957] I. Kaplansky, An introduction to differential algebra, Actualités Sci.Ind. 1251, Hermann, Paris, 1957.[Kolchin 1973] E. R. Kolchin, Differential algebra and algebraic groups, Pure andApplied Mathematics 54, Academic Press, New York, 1973.[Magid 1994] A. R. Magid, Lectures on differential Galois theory, Amer. Math. Soc.,Providence, RI, 1994.[Marker 2000] D. Marker, “Introduction to model theory”, pp. 15–35 in Model theory,algebra and geometry, edited by D. Haskell et al., Math. Sci. Res. Inst. Publ. 39,Cambridge Univ. Press, New York, 2000.[Marker et al. 1996] D. Marker, M. Messmer, and A. Pillay, Model theory of differentiablefields, Lecture Notes in Logic 5, Springer, Berlin, 1996.[Pierce and Pillay 1998] D. Pierce and A. Pillay, “A note on the axioms for differentiallyclosed fields of characteristic zero”, J. Algebra 204:1 (1998), 108–115.[Pillay 1996] A. Pillay, Differential algebraic groups and the number of countabledifferentially closed fields, Lecture Notes in Logic 5, Springer, Berlin, 1996.[Pillay 1997] A. Pillay, “Model theory and Diophantine geometry”, Bull. Amer. Math.Soc. (N.S.) 34:4 (1997), 405–422. Erratum in 35:1 (1998), 67.[Pong ≥ 2001] W. Y. Pong, “Some applications of ordinal dimensions to the theory ofdifferentially closed fields”. To appear in J. Symb. Logic.David MarkerUniversity of Illinois at ChicagoDepartment of Mathematics, Statistics, and Computer Science (M/C 249)851 S. Morgan St.Chicago, IL 60613United Statesmarker@math.uic.edu
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