Mordell exceptional locus for subvarieties of the additive group
Let \(X\) be an irreducible closed subvariety of a semiabelian variety \(G\) (an extension of an abelian variety by a torus) over an algebraically closed field \(K\). The Mordell exceptional locus \(Z(X) \subset X(K)\) is defined as the union of all translates of positive-dimensional algebraic subgroups of \(G\) that lie inside \(X\). \textit{D. Abramovich} [Compos. Math. 90, No. 1, 37--52 (1994; Zbl 0814.14041)] has shown that \(Z(X)\) is Zariski closed. In the present paper the author shows the analogous statement for subvarieties of a product of Drinfeld modules of generic characteristic. More precisely, let \(K\) be an algebraically closed field containing the polynomial ring \(A = \mathbb{F}_p[t] \) and \(\phi_1, \ldots, \phi_n : A \to \mathrm{End}(\mathbb{G}_{a,K}) \) a collection of Drinfeld modules of generic characteristic. Then \(\phi_1 \times \cdots \times \phi_n\) defines an \(A\)-module structure on \( E = \mathbb{G}_{a,K}^n \). Let \(V\) be a closed subvariety of \(E\). Define \( Z(V) \subset V\) to be the union of all translates of positive-dimensional algebraic sub-\(A\)-modules of \(E\) that lie inside \(V\), then it is shown that \(Z(V)\) is closed.
scientific article; zbMATH DE number 5302012
scientific article