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On the Manin-Mumford and Mordell-Lang conjectures in positive characteristic. (English) Zbl 1295.14024

The authors prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture in the situation where the ambient space is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is finitely generated. As a consequence a proof of the Mordell-Lang conjecture that does not depend on model theory methods but rather on algebraic-geometric methods is provided. In sectin 2 of the paper, the authors developed the necessary machinery to work with relative jet schemes via the reduction of scalars functor (represented by \(W\)). With the use of jet spaces, some natural “critical schemes” are constructed in section 3 to “catch rational points”. Using Galois-theory methods, the proof of proposition 4.1 in section 4 allows to show the sparsity of points over finite fields that are liftable to highly \(p\)-divisible unramified points. Exceptional sets were also introduced by A. Buium [Ann. Math. (2) 136, No. 3, 557–567 (1992; Zbl 0817.14021)], where the structure of Exc is studied via differential equations. However, the use of Galois theory to study the critical sets is essentially new to the approach of the authors in this paper. The fundamental result relevant to the logical connection between the Manin-Mumford and Mordell-Lang conjectures is the following.
Theorem. Let \(K_0\) be the function field of a smooth curve over \(\bar{F}_p\). Let \(A\) be an abelian variety over \(K_0\), and let \(X \hookrightarrow A\) be a closed integral scheme. Also let \(\Gamma \subset A(K_0)\) be a finitely generated subgroup. Suppose that for any field extension \(L_0|K_0\) and every \(Q \in A(L_0)\), the set \(X_{L_0}^{+,Q} \cap \mathrm{Tor}(A(L_0))\) is not Zariski dense in \(X_{L_0}^{+,Q}\), then \(X \cap \Gamma\) is not Zariski dense in \(X\).
In the statement of the theorem, \(X_{L_0}^{+,Q}\) is denoting the set theoretical image of \(X_{L_0}\) under translation by \(Q\) in \(A_{L_0}\).
In section 2B of the article, the authors present the theory of jet schemes of smooth commutative groups, with the definition of the maps \[ \lambda_n^W : W(U) \rightarrow J^n(W/U)(U), \] while working with \(U\)-schemes and associated jets \(J^n(W/U)\). Hints for a discussion of cases when the Manin-Mumford could be considered special case of Mordell-Lang or Mordell-Lang conjecture reduce to Manin-Mumford are included in remark 4.9 and remark (important) in the introduction.

MSC:

14G05 Rational points
14K12 Subvarieties of abelian varieties
14G17 Positive characteristic ground fields in algebraic geometry
14K99 Abelian varieties and schemes
11G10 Abelian varieties of dimension \(> 1\)

Citations:

Zbl 0817.14021
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