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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.

MSC:
03C60 Model-theoretic algebra
14G05 Rational points
11G10 Abelian varieties of dimension \(> 1\)
03C45 Classification theory, stability, and related concepts in model theory
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