On the non-triviality of the 2-part of the Tate-Shafarevich group

Abstract: The conjecture of Birch and Swinnerton-Dyer concerns a deep connection between the arithmetic of elliptic curves and the behaviour of their associated complex $L$-functions at $s=1$. The conjecture was formulated in the early 60's, and much of it remains mysterious today. Indeed, the exact Birch-Swinnerton-Dyer formula remains unknown even for the classical family of elliptic curves $E$ of the form $x^3+y^3=N$, where $N$ is a positive integer.

In this talk, I will study the "$p$-part" of the conjecture for these curves at small primes $p$. These cases are often eschewed, but they seem to make up a most significant part of the full conjecture.

First, I will study the $3$-adic valuation of the algebraic part of their central $L$-values, and use it to show that the "analytic" order of the Tate-Shafarevich group of $E$ is a perfect square for some $N$. In the second part of the talk, I will explain how we can obtain the $3$-part of the Birch-Swinnerton-Dyer conjecture in certain special cases of $N$ where the rank of $E$ is known to be equal to $0$ or $1$. For the $2$-part of the conjecture, I will explain a relation between the ideal class group of a corresponding cubic field extension and the $2$-Selmer group of $E$. This can be used to study non-triviality of the $2$-part of the Tate-Shafarevich group of $E$, even when $E$ has rank $1$.

The second part of this talk is joint work with Yongxiong Li.

algebraic geometrynumber theory

Audience: researchers in the topic

( paper | slides )

---

Rendez-vous on special values and periods

Series comments: The main objective of this conference is to gather together young researchers interested in special values of L-functions and periods. These objects are at the crossroads of many recent important developments in arithmetic geometry, such as Euler systems or the theory of motives. The different talks will portray the variety of viewpoints with which L-functions and periods are studied at present.

Registration is free and mandatory, to get access to the livestream and recording of the talks.

---