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On the conjectures of Mordell and Lang in positive characteristics. (English) Zbl 0735.14019
Let \(k\) be a number field embedded in a fixed algebraic closure \(\overline \mathbb{Q}\) of \(\mathbb{Q}\). Let \(A\) be an abelian variety defined over \(k\) and let \(V\subset A\) be a sub-variety. Let \(\Gamma\) be a finitely generated subgroup of \(A(\overline\mathbb{Q})\) and set \(\overline\Gamma=\{x\in A(\overline\mathbb{Q})\mid\exists m\in\mathbb{N} \hbox{ with } mx\in\Gamma\}\). Then one has the following (conjecture of Lang-Manin-Mumford) theorem (Faltings, Hindry): The variety \(V\) contains a finite number of translates \(\gamma_ i+B_ i\) of abelian subvarieties of \(A\) such that \(V(\overline\mathbb{Q})\cap\overline\Gamma=\bigcup_{1\leq i\leq t}(\gamma_ i+B_ i\cap\overline\Gamma)\).
The case \(\Gamma=\{0\}\) of the above theorem is due to Raynaud and Hindry. — The author considers a variant of this result in characteristic \(p\) and where \(V\) is a curve. Thus, let \(\mathcal C\) be a curve of genus \(g\) over an algebraically closed field \(\mathcal K\) of finite characteristic \(p\). We assume that \(\mathcal C\) is ordinary in that its Jacobian has \(p^ g\) points of \(p\)-torsion. Moreover, \(\mathcal C\) can be defined over a field \(K\subseteq{\mathcal K}\) which is finitely generated over the prime field; thus \(K\) is the function field of a variety \(W\). The curve \(\mathcal C\) is then classified by a morphism from \(W\) to the moduli space of curves of genus \(g\); we also assume that \(\mathcal C\) is non- isotrivial in that we require this map to be non-constant. Let \(J\) be the Jacobian of \(\mathcal C\) and \(\Gamma\subset J({\mathcal K})\) a finitely generated subgroup. Let \(\overline\Gamma=\{x\in J({\mathcal K})\mid\exists m\in\mathbb{N} \hbox{ with } (m,p)=1 \hbox{ and }mx\in\Gamma\}\). The author then proves the following theorem: Embed \(\mathcal C\) into its Jacobian via the Albanese map. Then \({\mathcal C}({\mathcal K})\cap\overline\Gamma\) is finite.
Reviewer: D.Goss (Columbus)

14G05 Rational points
14G27 Other nonalgebraically closed ground fields in algebraic geometry
14H40 Jacobians, Prym varieties
11R58 Arithmetic theory of algebraic function fields
14H99 Curves in algebraic geometry
Full Text: DOI EuDML
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