Event Type:

Number Theory Seminar

Date/Time:

Tuesday, January 30, 2018 - 16:00 to 17:00

Location:

BEXL 323

Guest Speaker:

Institution:

University of Oregon

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.