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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)

##### MSC:
 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
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