Eischen, EllenHsu, Catherine2018-09-062018-09-062018-09-06https://hdl.handle.net/1794/23742In 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 dissertation, 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 N. Specifically, we use a commutative algebra result of Berger, Klosin, and Kramer to bound the depth of mod p Eisenstein congruences (from below) by the p-adic valuation of φ(N). We then show how this depth of congruence controls the local principality of the associated Eisenstein ideal.en-USAll Rights Reserved.Algebraic number theoryCongruencesModular formsElectronic Thesis or Dissertation