Shiva.Chidambaram

Affiliation: University of Wisconsin-Madison

Title Of Talk: Minimal objects in the visibility category of Shafarevich-Tate groups

URLS:
https://arxiv.org/abs/2601.21519

Abstract: For an elliptic curve $E$ and an element $\sigma$ of its Shafarevich–Tate group, the visualization category consists of abelian varieties that contain $E$ and visualize the torsor representing $\sigma$. We answer several fundamental questions about minimal objects in this category, including Mazur's question about how different two minimal objects can be. We revisit two constructions for proving visibility, due to Cremona-Mazur, and Agashe-Stein. When $\xi$ has order $2$, by making the Cremona-Mazur construction explicit, we produce interesting examples, and a totally explicit proof of visibility of such elements in abelian surfaces (originally due to Bruin, and Klenke). Conditions for minimal visualization in the Agashe-Stein construction, naturally translate to interesting arithmetic statistics questions about Galois groups of number fields.

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