On the Visibility category of the Shafarevich--Tate group

Given an elliptic curve $E$ over $\text{Q}$ and a nontrivial element $\sigma$ of its Shafarevich–Tate group $\text{Sha}(E)$, we introduce the $\mathbf{\text{Visualization}}\mathbf{\text{category}}$ $\text{V}(E;\sigma )$ of abelian varieties that ``visualize’’ $\sigma$ 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. We revisit two constructions of visualizing abelian varieties: restriction of scalars (as in the work of Agashe and Stein), and a construction due to de Jong (as in the work of Cremona and Mazur). We show that restriction of scalars typically produces minimal visualizations. When $\sigma$ has order $2$ or $3$, we build upon the de Jong construction and make it totally explicit. While the de Jong construction can produce non-minimal objects, an appropriate choice in the construction for order $2$ elements $\sigma$ 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 absence of a $3$-isogeny, the de Jong construction yields a minimal visualization.