In his influential paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and the Eisenstein series of weight 2 and prime level N. In particular, he defined the Eisenstein ideal in the relevant Hecke algebra, and showed that it is
locally principal. We'll discuss the analogous situation for certain squarefree levels N, and show that, while the Eisenstein ideal may not be locally principal, we can count the minimal number of generators and explain the arithmetic significance of this number. This is joint work with Carl Wang-Erickson.