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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.

14K12 Subvarieties of abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
14G15 Finite ground fields in algebraic geometry
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