On the Visibility category of the Shafarevich--Tate group

TL;DR

This paper introduces the visualization category for elements of the Shafarevich--Tate group of elliptic curves, studies minimal objects within it, and provides explicit constructions and computational evidence for minimal visualizations.

Contribution

It defines the visualization category for $ ext{Sha}(E)$ elements, analyzes minimal objects, and explicitly constructs minimal visualizations using restriction of scalars and de Jong's methods.

Findings

Restriction of scalars often yields minimal visualizations.

Explicit genus 2 curves are constructed for order 2 elements.

Computational evidence suggests de Jong's construction is minimal without 3-isogeny.

Abstract

Given an elliptic curve $E$ over $\Q$ and a nontrivial element $\sigma$ of its Shafarevich--Tate group $\Sha(E)$, we introduce the \textbf{Visualization category} $\V(E; \sigma)$ 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.