On the conjectures of Mordell and Lang in positive characteristics (Q1814488)

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.