Torsion points of abelian varieties in abelian extensions

TL;DR

This paper characterizes when an abelian variety over a number field has finitely many torsion points in the maximal abelian extension, linking this to the absence of CM subvarieties, and provides a new proof of Ribet's finiteness result.

Contribution

It establishes a criterion for finiteness of torsion points over abelian extensions based on the presence of CM subvarieties, and offers an alternative proof of Ribet's theorem.

Findings

Finitely many torsion points in abelian extensions iff no CM subvariety exists

Provides a new proof of Ribet's finiteness theorem

Connects torsion point finiteness to complex multiplication properties

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.