Research-On the Visibility category of the Shafarevich-Tate group

Research

Number Theory Seminar

On the Visibility category of the Shafarevich-Tate group

Abstract

Given an elliptic curve over $\mathbb{Q}$ and a nontrivial element $\sigma$ of its Shafarevich--Tate group $\Sha(E)$, we introduce the Visualization category of abelian varieties that ``visualize'' $\sigma$, in the sense of Cremona--Mazur, and we study minimal objects in this category. In particular, we show that there can be several minimal visualizing abelian varieties of different dimensions, answering a question of Mazur. This is joint work in progress with Jerson Caro (Boston University) and Shiva Chidambaram (Wisconsin-Madison).