Reducibility and rational torsion in modular abelian varieties

Abstract

Let N be a square-free positive integer and let f be a newform of weight 2 on \Gamma_0(N). Let A denote the abelian subvariety of J_0(N) associated to f and let m be a maximal ideal of the Hecke algebra T that contains Ann_T(f) and has residue characteristic r such that r does not divide 6N. We show that if either A[m] or the canonical representation \rho_m over T/m associated to m is reducible, then r divides the order of the cuspidal subgroup of J_0(N) and A[m] has a nontrivial rational point. We mention some applications of this result, including an application to the second part of the Birch and Swinnerton-Dyer conjecture for A.