Week 27.06.2022 – 03.07.2022

The equation E: x^3+y^3=N defines a classical family of elliptic curves as N varies over cube-free positive integers. They admit complex multiplication, which allows us to tackle the conjecture of Birch and Swinnerton-Dyer for E effectively. Indeed, using Iwasawa theory, Rubin was able to show the p-part of the conjecture for E for all primes p, except for the primes 2 and 3. The theory becomes much more complex at these small primes, but at the same time we can observe some interesting phenomena. I will explain a method to study the p-adic valuation of the algebraic part of the central L-value of E, and I will establish the 3-part of the conjecture for E in special cases. I will then explain a relation between the 2-part of a certain ideal class group and the Tate-Shafarevich group of E. Part of this talk is based on joint work with Yongxiong Li.

Schedule:

14:00: Beth Romano -- TBA
14:30: Vaidehee Thatte -- The Defect is bad, deal with it!

Abstracts:

The Defect is bad, deal with it!
The 'defect' (or ramification deficiency) is one of the main obstacles to nice results, such as obtaining resolution of singularities in positive residue characteristic. In this talk, I will discuss why this is a serious obstruction, and how generalized ramification theory helps us understand and treat it. We will take this journey through some examples rather than theory.