On the Visibility category of the Shafarevich--Tate group

Abstract: Given an elliptic curve over $\mathbb{Q}$ and a nontrivial element $σ$ of its Shafarevich--Tate group, we introduce the Visualization category $\mathcal{V}(E; σ)$ of abelian varieties that ``visualize'' $σ$ in the sense of 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. In the case that $σ$ has order $2$ or $3$, we revisit two constructions of visualizing abelian varieties, due to Agashe and Stein, and Cremona and Mazur. We show that the Agashe--Stein construction always yields minimal visualizations for these orders. We also build upon the Cremona--Mazur construction and show how it can be made totally explicit. While the Cremona--Mazur construction can produce non-minimal objects, an appropriate choice in the construction for order $2$ elements $σ$ yields an explicit genus $2$ curve whose Jacobian is a minimal visualization. For order $3$ elements we apply our algorithmic construction to Fisher's database of such elements, and obtain computational evidence that, in the majority of cases, the Cremona--Mazur construction yields a minimal visualization.