On rational Eisenstein primes and the rational cuspidal groups of modular Jacobian varieties

Abstract

Let N be a non-squarefree positive integer and let l be an odd prime such that l(2) does not divide N. Consider the Hecke ring T(N) of weight 2 for Gamma(0)(N) and its rational Eisenstein primes of T(N) containing l. If m is such a rational Eisenstein prime, then we prove that m is of the form (l, I-M,N(D)), where we also define the ideal I-M,N(D) of T(N). Furthermore, we prove that C(N)[m] not equal 0, where C(N) is the rational cuspidal group of J(0)(N). To do this, we compute the precise order of the cuspidal divisor C-M,N(D) and the index of I-M,N(D) in T(N) circle times Z(l).