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Speaker:
Yukako Kezuka
Zugehörigkeit:
MPIM
Datum:
Mit, 17/03/2021 - 14:30 - 15:30
Parent event:
Number theory lunch seminar 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.© MPI f. Mathematik, Bonn | Impressum & Datenschutz |