The elliptic curves $x^3+y^3=N$ and the conjecture of Birch and Swinnerton-Dyer

Zoom Meeting ID: 919 6497 4060
For password see the email or contact Pieter Moree (moree@mpim...).

In this talk, I will report on some recent progress on the conjecture of Birch and Swinnerton-Dyer for elliptic curves E of the form x^{^3}+y^{^3}=N for cube-free positive integers N. They are cubic twists of the Fermat elliptic curve x^{^3}+y^{^3}=1, and admit complex multiplication by the ring of integers of $\mathbb{Q}(\sqrt{-3})$. First, I will explain the Tamagawa number divisibility satisfied by the central L-values, and exhibit a curious relation between the 3-part of the Tate-Shafarevich group of E and the number of prime divisors of N which are inert in $\mathbb{Q}(\sqrt{-3})$. I will then explain my joint work with Yongxiong Li, studying in more detail the cases when N=2p or 2p^{^2} for an odd prime number p congruent to 2 or 5 modulo 9. For these curves, we establish the 3-part of the Birch-Swinnerton-Dyer conjecture and a relation between the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ and the 2-Selmer group of E, which can be used to study non-triviality of the 2-part of their Tate-Shafarevich group.