Higher Eisenstein Congruences
Heilbronn Number Theory Seminar
28th February 2018, 4:00 pm – 5:00 pm
Howard House, 4th Floor Seminar Room
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.