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The Mordell-Lang theorem for Drinfeld modules. (English) Zbl 1158.11030
The 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.

##### MSC:
 11G09 Drinfel’d modules; higher-dimensional motives, etc. 14L15 Group schemes
##### Keywords:
Drinfeld modules; Mordell-Lang conjecture
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