Selmer stability for elliptic curves in Galois $\ell$-extensions
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_ | 1866914312899002368 |
|---|---|
| author | Pathak, Siddhi Ray, Anwesh |
| author_facet | Pathak, Siddhi Ray, Anwesh |
| contents | We study the behavior of Selmer groups of an elliptic curve $E/\mathbb{Q}$ in finite Galois extensions with prescribed Galois group. Fix a prime $\ell \geq 5$, a finite group $G$ with $\#G = \ell^n$, and an elliptic curve $E/\mathbb{Q}$ with $Sel_\ell(E/\mathbb{Q}) = 0$ and surjective mod-$\ell$ Galois representation. We show that there exist infinitely many Galois extensions $F/\mathbb{Q}$ with Galois group $Gal(F/\mathbb{Q}) \simeq G$ for which the $\ell$-Selmer group $Sel_\ell(E/F)$ also vanishes. We obtain an asymptotic lower bound for the number $M(G, E; X)$ of such fields $F$ with absolute discriminant $|Δ_F|\leq X$, proving that there is an explicit constant $δ>0$ such that $M(G, E; X) \gg X^{\frac{1}{\ell^{n-1}(\ell - 1)}} (\log X)^{δ- 1}$. The asymptotic for $M(G, E; X)$ matches the conjectural count for all $G$-extensions $F/\mathbb{Q}$ for which $|Δ_F|\leq X$, up to a power of $\log X$. This demonstrates that Selmer stability is not a rare phenomenon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15945 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Selmer stability for elliptic curves in Galois $\ell$-extensions Pathak, Siddhi Ray, Anwesh Number Theory 11G05, 11R45 We study the behavior of Selmer groups of an elliptic curve $E/\mathbb{Q}$ in finite Galois extensions with prescribed Galois group. Fix a prime $\ell \geq 5$, a finite group $G$ with $\#G = \ell^n$, and an elliptic curve $E/\mathbb{Q}$ with $Sel_\ell(E/\mathbb{Q}) = 0$ and surjective mod-$\ell$ Galois representation. We show that there exist infinitely many Galois extensions $F/\mathbb{Q}$ with Galois group $Gal(F/\mathbb{Q}) \simeq G$ for which the $\ell$-Selmer group $Sel_\ell(E/F)$ also vanishes. We obtain an asymptotic lower bound for the number $M(G, E; X)$ of such fields $F$ with absolute discriminant $|Δ_F|\leq X$, proving that there is an explicit constant $δ>0$ such that $M(G, E; X) \gg X^{\frac{1}{\ell^{n-1}(\ell - 1)}} (\log X)^{δ- 1}$. The asymptotic for $M(G, E; X)$ matches the conjectural count for all $G$-extensions $F/\mathbb{Q}$ for which $|Δ_F|\leq X$, up to a power of $\log X$. This demonstrates that Selmer stability is not a rare phenomenon. |
| title | Selmer stability for elliptic curves in Galois $\ell$-extensions |
| topic | Number Theory 11G05, 11R45 |
| url | https://arxiv.org/abs/2504.15945 |