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

Ito, Tetsushi

doi:10.1090/S0002-9939-06-08518-2

On the Manin-Mumford conjecture for abelian varieties with a prime of supersingular reduction
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Proc. Amer. Math. Soc. 134 (2006), 2857-2860 Request permission

Abstract:

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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