Elliptic curves with rank one and nontrivial 2-part of Tate Shafarevich groups over the $\mathbb{Z}_2$-extension of $\mathbb{Q}$
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| Format: | Preprint |
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2026
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| _version_ | 1866908887567826944 |
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| author | Deng, Li-Tong Li, Yong-Xiong |
| author_facet | Deng, Li-Tong Li, Yong-Xiong |
| contents | Let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_2$-extension over $\mathbb{Q}$. For each integer $n\geq1$, let $\mathbb{Q}_n$ denote the unique subfield in $\mathbb{Q}_\infty$ such that $[\mathbb{Q}_\infty:\mathbb{Q}]=2^n$. Denote by $\mathbb{Z}_2[{\rm Gal}(\mathbb{Q}_n/\mathbb{Q})]$ the group ring of ${\rm Gal}(\mathbb{Q}_\infty/\mathbb{Q})$. For any elliptic curve defined over $\mathbb{Q}$ with odd conductor, the Mazur-Tate modular element associated with the curve is an element of $\mathbb{Z}_2[{\rm Gal}(\mathbb{Q}_n/\mathbb{Q})]$. In this paper, for each $n$, we study the $2$-adic properties of Mazur-Tate modular elements associated with quadratic twists of elliptic curves, under specializations by finite order characters of ${\rm Gal}(\mathbb{Q}_n/\mathbb{Q})$. Using the congruence properties of Heegner points and an equivariant version of the Coates-Wiles theorem, we construct an elliptic curve $E/\mathbb{Q}$ and a family of quadratic twists $E^{(m)}$ of $E$ such that each $E^{(m)}$ has both analytic and algebraic rank one over $\mathbb{Q}_\infty$, and whose Tate-Shafarevich group is infinite over $\mathbb{Q}_\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14234 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Elliptic curves with rank one and nontrivial 2-part of Tate Shafarevich groups over the $\mathbb{Z}_2$-extension of $\mathbb{Q}$ Deng, Li-Tong Li, Yong-Xiong Number Theory 11G40 (primary), 11G05, 11R23 (secondary) Let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_2$-extension over $\mathbb{Q}$. For each integer $n\geq1$, let $\mathbb{Q}_n$ denote the unique subfield in $\mathbb{Q}_\infty$ such that $[\mathbb{Q}_\infty:\mathbb{Q}]=2^n$. Denote by $\mathbb{Z}_2[{\rm Gal}(\mathbb{Q}_n/\mathbb{Q})]$ the group ring of ${\rm Gal}(\mathbb{Q}_\infty/\mathbb{Q})$. For any elliptic curve defined over $\mathbb{Q}$ with odd conductor, the Mazur-Tate modular element associated with the curve is an element of $\mathbb{Z}_2[{\rm Gal}(\mathbb{Q}_n/\mathbb{Q})]$. In this paper, for each $n$, we study the $2$-adic properties of Mazur-Tate modular elements associated with quadratic twists of elliptic curves, under specializations by finite order characters of ${\rm Gal}(\mathbb{Q}_n/\mathbb{Q})$. Using the congruence properties of Heegner points and an equivariant version of the Coates-Wiles theorem, we construct an elliptic curve $E/\mathbb{Q}$ and a family of quadratic twists $E^{(m)}$ of $E$ such that each $E^{(m)}$ has both analytic and algebraic rank one over $\mathbb{Q}_\infty$, and whose Tate-Shafarevich group is infinite over $\mathbb{Q}_\infty$. |
| title | Elliptic curves with rank one and nontrivial 2-part of Tate Shafarevich groups over the $\mathbb{Z}_2$-extension of $\mathbb{Q}$ |
| topic | Number Theory 11G40 (primary), 11G05, 11R23 (secondary) |
| url | https://arxiv.org/abs/2603.14234 |