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.

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# Higher Eisenstein Congruences

Cathy Hsu, University of Oregon

Tuesday, December 5, 2017 - 11:00am to 12:00pm

PDL C-401