On the Visibility category of the Shafarevich--Tate group
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913074649235456 |
|---|---|
| author | Banwait, Barinder S. Caro, Jerson Chidambaram, Shiva |
| author_facet | Banwait, Barinder S. Caro, Jerson Chidambaram, Shiva |
| contents | Given an elliptic curve $E$ over $\Q$ and a nontrivial element $σ$ of its Shafarevich--Tate group $\Sha(E)$, we introduce the \textbf{Visualization category} $\V(E; σ)$ 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. 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 $σ$ 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 $σ$ 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 absence of a $3$-isogeny, the de Jong construction yields a minimal visualization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21519 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Visibility category of the Shafarevich--Tate group Banwait, Barinder S. Caro, Jerson Chidambaram, Shiva Number Theory 11G05 (primary), 11G10, 11Y16 (secondary) Given an elliptic curve $E$ over $\Q$ and a nontrivial element $σ$ of its Shafarevich--Tate group $\Sha(E)$, we introduce the \textbf{Visualization category} $\V(E; σ)$ 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. 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 $σ$ 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 $σ$ 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 absence of a $3$-isogeny, the de Jong construction yields a minimal visualization. |
| title | On the Visibility category of the Shafarevich--Tate group |
| topic | Number Theory 11G05 (primary), 11G10, 11Y16 (secondary) |
| url | https://arxiv.org/abs/2601.21519 |