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

2004/11/12 by Tetsushi Ito, Ito, Tetsushi · 1 citation
Mathematics · #14G15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #Primary: 14K12 #Secondary: 11G10 #math.AG #math.NT #msc:11G10 #msc:14G15 #msc:14K12

paper · pdf · doi:10.48550/arxiv.math/0411291

4 pages

arxiv created 2004/11/12 · openalex publication_date 2004/11/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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