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On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction. (English) Zbl 1100.14036
The Manin-Mumford conjecture asserts that an irreducible closed subvariety $$X$$ of abelian variety $$A$$ defined over a number field such that the torsion subgroup of $$A$$ intersects $$X$$ in a Zariski dense subset of $$X$$ is necessarily a translate of an abelian subvariety of $$A$$. This conjecture has been proved by Raynaud and many proofs are now available. The present paper gives a very short proof for the special case that $$A$$ has supersingular reduction at a prime dividing $$p$$ and the prime-to-$$p$$ part of the torsion subgroup of $$A$$ has Zariski-dense intersection with $$X$$. It combines methods of Bogomolov, Hrushovski and Pink and Roessler. The idea is that at the supersingular reduction a power of Frobenius acts as a power $$q^m$$ of the cardinality $$q$$ of the residue field on the prime-to-$$p$$ part of the torsion, hence using the fact that the prime-to-$$p$$ torsion of $$A$$ is isomorphic to that of the supersingular fibre one concludes that a lift of Frobenius acts on the prime-to-$$p$$ part of the torsion of $$A$$ as $$q^m$$, hence $$X$$ is stable under multiplication by $$q^m$$. This implies that $$X$$ is a translate of an abelian subvariety. The paper also discusses the relation with the other proofs.

##### MSC:
 14K12 Subvarieties of abelian varieties 11G10 Abelian varieties of dimension $$> 1$$ 14G15 Finite ground fields in algebraic geometry
##### Keywords:
Manin-Mumford conjecture
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