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The Mordell-Lang conjecture for function fields. (English) Zbl 0864.03026
The Mordell-Lang conjecture is a hypothesis formulated by Lang and extending the Mordell conjecture on rational points of curves, as well as the Manin-Mumford conjecture on torsion points of Abelian varieties. In characteristic 0, the Mordell-Lang conjecture was proved in a series of papers by Raynaud, Faltings and Vojta. In positive characteristic, only partial cases had been solved before this paper. Hrushovski presents here a uniform proof valid in any characteristic and using a model theoretic approach. Here is the statement of the main theorem. One deals with an algebraically closed field \(k\) and a field extension \(K/k\). Let \(S\) be a semi-Abelian variety over \(K\) and \(X\) be a subvariety of \(S\). Take a subgroup \(\Gamma\) of \(S\) such that \({\mathbb Q}_p \otimes \Gamma\) is finitely generated as a \({\mathbb Q}_p\)-module (here \({\mathbb Q}_p\) denotes \({\mathbb Q}\) if \(p=0\), and \(\{ m/n \in {\mathbb Q} : n\) prime to \(p \}\) otherwise). Suppose that \(X \cap \Gamma\) is Zariski dense in \(X\). Then there are a semi-Abelian variety \(S_0\) defined over \(k\), a subvariety \(X_0\) of \(S_0\) also defined over \(k\), and a rational homomorphism \(h\) from a group subvariety of \(S\) into \(S_0\) such that \(X\) is a translate of \(h^{-1}(X_0)\). As a consequence, when \(S\) has \(K/k\) trace 0, then \(X \cap \Gamma\) is a finite union of cosets of subgroups of \(\Gamma\).
The core of the Hrushovski approach does not depend on the characteristic \(p\). But, just to explain the idea, let us treat before the case \(p=0\). The argument in this case uses differential algebra and was inspired by some recent papers of Buium. Without loss of generality, one can assume that \(K\) is algebraically closed and has infinite transcendence degree over \(k\). Accordingly, one can equip \(K\) with a derivation \(D\) making \(K\) a differential field, and even a differentially closed field (so an \(\omega\)-stable structure) with constant field \(k\). Model theory assigns an ordinal dimension to every non-empty set definable in \((K, D)\) (its Morley rank); in particular, the Morley rank agrees with the usual dimension for varieties. \(X\) is definable and its Morley rank is an integer. By using the Manin homomorphism in \((K, D)\), one sees that, with no loss of generality, \(\Gamma\) is definable, too, and its Morley rank is again an integer. Let us apply a dichotomy theorem of Hrushovski and Zil’ber to certain groups related to \(X\) and \(\Gamma\). It comes out that such a group is either a module over a suitable local ring, with no additional structure, or an algebraic group over an algebraically closed field. A theorem of Sokolovic says that the only algebraically closed field definable in \((K, D)\) is – up to definable isomorphism – the constant field \(k\). So the previous dichotomy just leads to the subvarieties mentioned in the main theorem: group subvarieties, or subvarieties defined over \(k\). And actually the analysis of the two cases accomplishes the proof.
What happens when \(p>0\)? Here the plan is similar, but needs some changes. Most notably, differential algebra can be avoided now. Separably closed fields \(F\) of characteristic \(p\) such that \(|F : F^p|\) is finite and \(>1\) are enough. Their theory is not \(\omega\)-stable, but only stable, so Morley rank cannot be used in this setting. However some suitable modifications let the previous machinery work; for instance, the role of the Manin homomorphism is played now by the projection modulo \(\bigcap_n p^n S\) (which is not a definable set, but is \(\infty\)-definable).
The paper is well written and well worth reading for several reasons. As recalled before, it gives the first proof of the conjecture (in the general form stated above). Moreover it provides a very brilliant and ingenious application of Model Theory to a question of Diophantine Geometry, using some deep model theoretic tools (like differentially and separably closed fields, or the Hrushovski-Zil’ber Dichotomy Theorem) to obtain the positive solution. So the paper is a significant step within the increasing connection between Model Theory and Algebraic Geometry.

03C60 Model-theoretic algebra
14G05 Rational points
11G10 Abelian varieties of dimension \(> 1\)
03C45 Classification theory, stability, and related concepts in model theory
Full Text: DOI
[1] Dan Abramovich and José Felipe Voloch, Toward a proof of the Mordell-Lang conjecture in characteristic \?, Internat. Math. Res. Notices 5 (1992), 103 – 115. · Zbl 0787.14026
[2] A. Buium, Intersections in jet spaces and a conjecture of S. Lang, Ann. of Math. (2) 136 (1992), no. 3, 557 – 567. · Zbl 0817.14021
[3] Alexandru Buium, Effective bound for the geometric Lang conjecture, Duke Math. J. 71 (1993), no. 2, 475 – 499. · Zbl 0812.14029
[4] Alexandru Buium and José Felipe Voloch, Integral points of abelian varieties over function fields of characteristic zero, Math. Ann. 297 (1993), no. 2, 303 – 307. · Zbl 0789.14017
[5] Françoise Delon, Idéaux et types sur les corps séparablement clos, Mém. Soc. Math. France (N.S.) 33 (1988), 76 (French, with English summary). · Zbl 0678.03016
[6] Ehud Hrushovski, Unidimensional theories are superstable, Ann. Pure Appl. Logic 50 (1990), no. 2, 117 – 138. · Zbl 0713.03015
[7] U. Hrushovski and A. Pillay, Weakly normal groups, Logic colloquium ’85 (Orsay, 1985) Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987, pp. 233 – 244.
[8] E. Hrushovski and Z. Sokolovic, Minimal subsets of differentially closed fields, Trans. Amer. Math. Soc. (to appear).
[9] E. Hrushovski and B. Zil’ber, Zariski geometries, J. Amer. Math. Soc. 9 (1996), 1–56. CMP 95:06
[10] Ehud Hrushovski and Boris Zilber, Zariski geometries, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 315 – 323. · Zbl 0781.03023
[11] Serge Lang, Division points on curves, Ann. Mat. Pura Appl. (4) 70 (1965), 229 – 234. · Zbl 0151.27401
[12] Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag, Berlin, 1991. Diophantine geometry. · Zbl 0744.14012
[13] Daniel Lascar, Ranks and definability in superstable theories, Israel J. Math. 23 (1976), no. 1, 53 – 87. · Zbl 0326.02038
[14] Ju. I. Manin, Algebraic curves over fields with differentiation, Izv. Akad. Nauk SSSR. Ser. Mat. 22 (1958), 737 – 756 (Russian). · Zbl 0151.27503
[15] Ju. I. Manin, Rational points on algebraic curves over function fields, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1395 – 1440 (Russian). · Zbl 0166.16901
[16] M. Messmer, Groups and fields interpretable in separably closed fields, preprint. · Zbl 0803.03023
[17] A. Pillay, Model theory, stability theory, and stable groups, The Model Theory of Groups , Notre Dame Math. Lectures, no. 11, Univ. Notre Dame Press, Notre Dame, IN, 1989, pp. 1–22.CMP 21:09 · Zbl 0792.03019
[18] A. Robinson and P. Roquette, On the finiteness theorem of Siegel and Mahler concerning Diophantine equations, J. Number Theory 7 (1975), 121 – 176. · Zbl 0299.12107
[19] Gerald E. Sacks, Saturated model theory, W. A. Benjamin, Inc., Reading, Mass., 1972. Mathematics Lecture Note Series. · Zbl 0242.02054
[20] Z. Sokolovic, Model theory of differential fields, Ph.D. Thesis, Notre Dame, July, 1992.
[21] A. Weil, Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948. · Zbl 0037.16202
[22] Carol Wood, Notes on the stability of separably closed fields, J. Symbolic Logic 44 (1979), no. 3, 412 – 416. · Zbl 0424.03014
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