Endomorphisms and torsion of Abelian varieties (Q1095214)

It is proved the following result which is a final solution of the torsion problem for abelian varieties A over the maximal abelian extension \(K^{ab}\) of a given number field K. Let A be K-isogenous to the product \(X_ 1\times...\times X_ n\) of simple abelian varieties \(X_ i\) over K. Then the torsion subgroup \(A(K^{ab})_{tors}\) is finite iff all \(X_ i\) are not of CM-type over K (1\(\leq i\leq h)\). For CM-type A we have \(A(K^{ab})_{tors}=A(\bar K)_{tors}\) where \(\bar K\) is the algebraic closure of K.