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arXiv · 2305.10261

On the Manin-Mumford Theorem for Algebraic Groups

Harry Schmidt·Immanuel van Santen
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Abstract

We describe the Zariski-closure of sets of torsion points in connected algebraic groups. This is a generalization of the Manin-Mumford conjecture for commutative algebraic groups proved by Hindry. He proved that every subset with Zariski-dense torsion points is the finite union of torsion-translates of algebraic subgroups. We formulate and prove an analogous theorem for arbitrary connected algebraic groups. We also define a canonical height on connected algebraic groups that coincides with a N\'eron-Tate height if $G$ is a (semi-) abelian variety. This motivates a generalization of the Bogomolov conjecture to arbitrary connected algebraic groups defined over a number field. We prove such a generalization as well.

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Harry Schmidt, Immanuel van Santen. 2023-05-17. On the Manin-Mumford Theorem for Algebraic Groups. https://arxiv.org/abs/2305.10261

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