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.