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 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.
| Identifer | oai:union.ndltd.org:uoregon.edu/oai:scholarsbank.uoregon.edu:1794/23742 |
| Date | 06 September 2018 |
| Creators | Hsu, Catherine |
| Contributors | Eischen, Ellen |
| Publisher | University of Oregon |
| Source Sets | University of Oregon |
| Language | en_US |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Rights | All Rights Reserved. |
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