Jaclyn Lang: A new perspective on Eisenstein congruences

Abstract: In Mazur’s celebrated Eisenstein ideal paper, he studies congruences between prime-level cusp forms and the unique weight-2 Eisenstein series of the same level. He shows that (if p is at least 5) such mod-p congruences exist if and only if the level N is congruent to 1 modulo p. In this talk, Jaclyn Lang considers Eisenstein–cuspidal congruences in weight 2 and level N2, still under the condition that N = 1 mod p. In this case, recent work with Pollack and Wake shows that the relevant level-N2 Hecke algebra is a free module over an appropriate inertia-at-N pseudodeformation ring. This structure turns out to be surprisingly powerful. One can recover Mazur’s existence theorem that there exists a mod-p Eisenstein–cuspidal congruence in weight 2 and prime level N when N = 1 mod p. It also allows one to deduce the relevant R=T theorem (for an appropriate pseudodeformation ring R) from the corresponding theorem in prime level, which is due to Wake and Wang–Erickson. Finally, one can recover the results of Merel and Lecouturier that characterize the rank of Mazur’s Hecke algebra in terms of the order of vanishing of a certain zeta element in cases when that rank is at most 3, and Lang explains some of the ideas that go into the aforementioned results. In addition to the joint work with Pollack and Wake, some of these results are joint with Palvannan and Müller.