The Eisenstein ideal at prime-square level

In his famous Eisenstein-ideal paper, Barry Mazur studied congruences modulo p between Eisenstein series and cuspforms of prime level N. Among other things, he proved that such congruences exist only if N is congruent to 1 modulo p. The trickiest part of this result is to prove that such congruences do not exist if N is -1 modulo p. I'll talk about recent work with Jackie Lang in which we prove that, for such primes N, Eisenstein congruences do exist at level N², that these congruences can be used to construct non-trivial elements of certain class groups, and that the structure of the congruences is much more rigid than in prime-level case.