  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.
