Tamagawa number divisibility of central L-values
Yukako Kezuka (MPI (Bonn))
Abstract: 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.
number theory
Audience: researchers in the topic
Dublin Algebra and Number Theory Seminar
Series comments: Passcode: The 3-digit prime numerator of Riemann zeta at -11
Organizers: | Kazim Buyukboduk*, Robert Osburn |
*contact for this listing |