Torsion points of abelian varieties in abelian extensions

1998/03/13 by Wolfgang M. Ruppert, Ruppert, Wolfgang M.
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT

paper · pdf · doi:10.48550/arxiv.math/9803169

arxiv created 1998/03/13 · openalex publication_date 1998/03/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be an abelian variety defined over a number field K and let Kab be the maximal abelian extension of K. We show that there only finitely many torsion points of A which are defined over Kab iff A has no abelian subvariety with complex multiplication over K. We use this to give another proof of Ribet's result that A has only finitely many torsion points which are defined over the cyclotomic extension of K.

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