Tamagawa number divisibility of central $L$-values

Abstract: In this talk, I will study 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 study the $p$-adic valuation of the algebraic part of their 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 $\mathbb{Q}(\sqrt{-3})$. In the second part of the talk, I will study 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. The second part of this talk is joint work with Yongxiong Li.

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic