2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73216We 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.4 pagesNumber TheoryAlgebraic GeometryPrimary: 14K12; Secondary: 11G10, 14G15text