On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction

Details On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction

2004-11-12

arxiv.org/abs/math/0411291v1

Mathematics

Number Theory

4 pages

Scientific paper

We give a short proof of the "prime-to-$p$ version" of the Manin-Mumford conjecture for an abelian variety over a number field, when it has supersingular reduction at a prime dividing $p$, by combining the methods of Bogomolov, Hrushovski, and Pink-Roessler. Our proof here is quite simple and short, and neither $p$-adic Hodge theory nor model theory is used. The observation is that a power of a lift of the Frobenius element at a supersingular prime acts on the prime-to-$p$ torsion points via nontrivial homothety.

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