TORSION OF RATIONAL ELLIPTIC CURVES OVER THE MAXIMAL ABELIAN EXTENSION OF Q

Let E be an elliptic curve defined over Q, and let Q(ab) be the maximal abelian extension of Q. In this article we classify the groups that can arise as E(Q(ab))(tors) up to isomorphism. The method illustrates techniques for finding explicit models of modular curves of mixed level structure. Moreover, we provide an explicit algorithm to compute E(Q(ab))(tors) for any elliptic curve E/Q.