---
date: 2022-05-03 22:29
modification date: Tuesday 3rd May 2022 22:29:33
title: "Mordell-Lang conjecture"
aliases: [Mordell-Lang conjecture]
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# Mordell-Lang conjecture

![](attachments/Pasted%20image%2020220503222936.png)

The Mordell-Lang #open/conjectures 
Let $X$ be a closed geometrically integral subvariety of a semiabelian variety $A$ defined over a field $K$ of characteristic 0 . Let $\Gamma$ be a finitely generated subgroup of $A(\bar{K})$ and $\Gamma^{\prime}$ a subgroup of the divisible hull of $\Gamma$ (i.e. for each $x \in \Gamma^{\prime}$ there exists a non-zero integer $n$ such that $\left.n x \in \Gamma\right)$. If $X$ is not a translate of a semi-abelian subvariety of $A$, then $X(\bar{K}) \cap \Gamma^{\prime}$ is not Zariski dense in $X .$
![](attachments/Pasted%20image%2020220503223132.png)
![](attachments/Pasted%20image%2020220503223146.png)