On the Visibility category of the Shafarevich--Tate group

TL;DR

This work develops a categorical framework for visualizing elements of the Shafarevich--Tate group via the Visibility category, addressing Mazur's questions about minimality and dimensional variation. It proves that multiple minimal visualizations of a fixed σ can exist in different dimensions, and provides explicit, computable constructions for n=2 and n=3: an explicit genus-2 Cremona–Mazur-type visualization for order-2 and a genus-4 Cremona–Mazur curve for order-3, with overwhelming computational evidence for minimality. It also confirms the Agashe–Stein construction yields minimal visualizations for orders 2 and 3 and shows Cremona–Mazur curves may fail to be minimal in some cases, illustrating a genuine difference between 𝒱(E, σ) and 𝒱(E, ord σ). Collectively, the paper advances understanding of Sha finiteness through visibility, provides explicit minimal objects, and outlines concrete directions for extending computations and explicit constructions. The results have implications for computational arithmetic geometry and the explicit realization of Sha elements in ambient abelian varieties.

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.

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10
• Remark 1.11
• Remark 1.12
• Theorem 1.13