Catherine Hsu – University of Oregon

Name: Catherine Hsu – University of Oregon
Start: 2017-09-18T16:30:00-04:00
End: 2017-09-18T17:30:00-04:00
Location: Martin M-102

September 18, 2017 @ 4:30 pm - 5:30 pm EDT

Higher Eisenstein Congruences

In this talk, we examine the relationship between the “depth” of certain Eisenstein congruences and the local structure of the Eisenstein ideal. Specifically, let $p\geq 3$ be prime. For squarefree level $N>6$ and weight $k=2,$ we use a commutative algebra result of Berger, Klosin, and Kramer to bound the depth of Eisenstein congruences modulo $p$ (from below) by the $p$ -adic valuation of the numerator of $\frac{\varphi(N)}{24}$ . We then show that if $N$ has at least three prime factors and some prime $p\geq 5$ divides $\varphi(N),$ the Eisenstein ideal is not locally principal. Lastly, we illustrate these results with explicit computations and discuss generalizations to higher weights.