On Eisenstein ideals and the cuspidal group of J (0)(N)

摘要

Let C (N) be the cuspidal subgroup of the Jacobian J (0)(N) for a square-free integer N > 6. For any Eisenstein maximal ideal m of the Hecke ring of level N, we show that C (N) [m] not equal 0. To prove this, we calculate the index of an Eisenstein ideal I contained in m by computing the order of the cuspidal divisor annihilated by I.