Rational torsion in elliptic curves and the cuspidal subgroup

Abstract

Let A be an elliptic curve over ℚ of square free conductor N that has a rational torsion point of prime order r such that r does not divide 6N. We show that then r divides the order of the cuspidal subgroup C of J 0 (N). If A is optimal, then viewing A as an abelian subvariety of J 0 (N), our proof shows more precisely that r divides the order of A∩C. Also, under the hypotheses above minus the hypothesis that r does not divide N, we show that for some prime p that divides N, the eigenvalue of the Atkin–Lehner involution W p acting on the newform associated to A is -1.