Roughly 30 years ago, I proved: Suppose that A is an abelian variety over a number field K. Then A has only a finite number of torsion points defined over the maximal cyclotomic extension of K. After explaining the ingredients of the proof, I will highlight some questions suggested by this theorem. One natural project is to compute the group of cyclotomic torsion points in some specific examples. If A is J_0(N), where N is a prime number, then the group of torsion points on A over the maximal cyclotomic extension of Q is the kernel of the Eisenstein ideal on A.