The Mordell-Lang theorem for Drinfeld modules.

*(English)*Zbl 1158.11030The Mordell-Lang conjecture (now a theorem by G. Faltings [Perspect. Math. 15, 175–182 (1994; Zbl 0823.14009)]) asserts that if \( A / K \) is an abelian variety over an algebraically closed field of characteristic zero, \( \Gamma \subset A(K) \) a finitely generated subgroup, and \( X \subset A(K) \) a Zarisky closed subset then the intersection \( X \cap \Gamma \) is a finite union of translates of subgroups of \( \Gamma \). This is a generalization of Falting’s theorem stating that the number of rational points on a curve \( C \) of genus at least two is finite (embed \( C \) in its Jacobian, and use that the group of rational points on \( J \) is finitely generated.)

In the present paper similar theorems are shown for powers of a Drinfeld module (so the underlying variety is \(\mathbb{G}_a^n \) and the module structure is by a diagonal Drinfeld module action), both in the finite characteristic and the infinite characteristic cases. The statements are roughly of the type: “if \(X\) is a Zariski closed subset and \(\Gamma\) a finitely generated sub-module then \( X \cap \Gamma \) is a finite union of translates of submodules of \( \Gamma \)”, but there are new conditions that do not occur in the classical Mordell-Lang conjecture. The author does give examples to show that these conditions can not be dropped.

In the present paper similar theorems are shown for powers of a Drinfeld module (so the underlying variety is \(\mathbb{G}_a^n \) and the module structure is by a diagonal Drinfeld module action), both in the finite characteristic and the infinite characteristic cases. The statements are roughly of the type: “if \(X\) is a Zariski closed subset and \(\Gamma\) a finitely generated sub-module then \( X \cap \Gamma \) is a finite union of translates of submodules of \( \Gamma \)”, but there are new conditions that do not occur in the classical Mordell-Lang conjecture. The author does give examples to show that these conditions can not be dropped.

Reviewer: Lenny Taelman (Leiden)