On the growth of Tate-Shafarevich groups of $p$-supersingular abelian varieties of ${\rm GL}_2$-type over $\mathbb{Z}_p$-extensions of number fields
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918159597961216 |
|---|---|
| author | Isik, Erman Lei, Antonio |
| author_facet | Isik, Erman Lei, Antonio |
| contents | We study the boundedness of the Mordell-Weil rank and the growth of the $v$-primary part of the Tate-Shafarevich group of $p$-supersingular abelian varieties of ${\rm GL}_2$-type with real multiplication over $\mathbb{Z}_p$-extensions of number fields, where $v$ is a prime lying above $p$. Building on the work of Iovita-Pollack in the case of elliptic curves, under precise ramification and splitting conditions on $p$, we construct explicit systems of local points using the theory of Lubin-Tate formal groups. We then define signed Coleman maps, which in turn allow us to formulate and analyse signed Selmer groups. Assuming these Selmer groups are cotorsion, we prove that the Mordell-Weil groups are bounded over any subextensions of the $\mathbb{Z}_p$-extension and provide an asymptotic formula for the growth of the $v$-primary part of the Tate-Shafarevich groups. Our results extend those of Kobayashi, Pollack, and Sprung on $p$-supersingular elliptic curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the growth of Tate-Shafarevich groups of $p$-supersingular abelian varieties of ${\rm GL}_2$-type over $\mathbb{Z}_p$-extensions of number fields Isik, Erman Lei, Antonio Number Theory 11R23 (primary), 11G05, 11R20 (secondary) We study the boundedness of the Mordell-Weil rank and the growth of the $v$-primary part of the Tate-Shafarevich group of $p$-supersingular abelian varieties of ${\rm GL}_2$-type with real multiplication over $\mathbb{Z}_p$-extensions of number fields, where $v$ is a prime lying above $p$. Building on the work of Iovita-Pollack in the case of elliptic curves, under precise ramification and splitting conditions on $p$, we construct explicit systems of local points using the theory of Lubin-Tate formal groups. We then define signed Coleman maps, which in turn allow us to formulate and analyse signed Selmer groups. Assuming these Selmer groups are cotorsion, we prove that the Mordell-Weil groups are bounded over any subextensions of the $\mathbb{Z}_p$-extension and provide an asymptotic formula for the growth of the $v$-primary part of the Tate-Shafarevich groups. Our results extend those of Kobayashi, Pollack, and Sprung on $p$-supersingular elliptic curves. |
| title | On the growth of Tate-Shafarevich groups of $p$-supersingular abelian varieties of ${\rm GL}_2$-type over $\mathbb{Z}_p$-extensions of number fields |
| topic | Number Theory 11R23 (primary), 11G05, 11R20 (secondary) |
| url | https://arxiv.org/abs/2510.11511 |