Stability of torsion subgroups of elliptic curves over non-Galois extensions of odd prime degree
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| Format: | Preprint |
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2025
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| _version_ | 1866914188745506816 |
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| author | Im, Bo-Hae Kim, Hansol |
| author_facet | Im, Bo-Hae Kim, Hansol |
| contents | Let $K$ be a field of characteristic $0$ and $E/K$ an elliptic curve over $K$. For a finite extension $L/K$ and a prime~$\ell$, we provide Galois-theoretic sufficient conditions on $L/K$ under which $E\left(L\right)\left[\ell^{\infty}\right] = E\left(K\right)\left[\ell^{\infty}\right]$. For a non-Galois extension $L/K$ of prime degree, we relate the growth of the $\ell^{\infty}$-torsion subgroup of $E$ under the base change $L/K$ to the image of the mod-$\ell$ cyclotomic character. In particular, In particular, we refine Gonz{á}lez-Jim{é}nez's result by ruling out certain torsion structures for quintic non-Galois extensions $L/\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18194 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability of torsion subgroups of elliptic curves over non-Galois extensions of odd prime degree Im, Bo-Hae Kim, Hansol Number Theory Primary: 11G05, Secondary: 14H52 Let $K$ be a field of characteristic $0$ and $E/K$ an elliptic curve over $K$. For a finite extension $L/K$ and a prime~$\ell$, we provide Galois-theoretic sufficient conditions on $L/K$ under which $E\left(L\right)\left[\ell^{\infty}\right] = E\left(K\right)\left[\ell^{\infty}\right]$. For a non-Galois extension $L/K$ of prime degree, we relate the growth of the $\ell^{\infty}$-torsion subgroup of $E$ under the base change $L/K$ to the image of the mod-$\ell$ cyclotomic character. In particular, In particular, we refine Gonz{á}lez-Jim{é}nez's result by ruling out certain torsion structures for quintic non-Galois extensions $L/\mathbb{Q}$. |
| title | Stability of torsion subgroups of elliptic curves over non-Galois extensions of odd prime degree |
| topic | Number Theory Primary: 11G05, Secondary: 14H52 |
| url | https://arxiv.org/abs/2510.18194 |