Catherine Hsu, University of Oregon

Higher Eisenstein Congruences

Abstract: In his seminal work on modular curves and the Eisenstein
ideal, Mazur studied the existence of congruences between certain
Eisenstein series and newforms, proving that Eisenstein ideals
associated to weight 2 cusp forms of prime level are locally principal.
In this talk, we re-examine Eisenstein congruences, incorporating a
notion of "depth of congruence," in order to understand the local
structure of Eisenstein ideals associated to weight 2 cusp forms of
squarefree level. Specifically, we use a commutative algebra result of
Berger, Klosin, and Kramer to strictly bound the depth of mod p
Eisenstein congruences (from below) by the p-adic valuation of the
numerator of \varphi(N)/24. We then show how this depth of congruence
controls the local principality of the associated Eisenstein ideal.