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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2210.16977 |
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| _version_ | 1866914889983852544 |
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| author | Genao, Tyler |
| author_facet | Genao, Tyler |
| contents | Given a number field $F_0$ that contains no Hilbert class field of any imaginary quadratic field, we show that under GRH there exists an effectively computable constant $B:=B(F_0)\in\mathbb{Z}^+$ for which the following holds: for any finite extension $L/F_0$ whose degree $[L:F_0]$ is coprime to $B$, one has for all elliptic curves $E_{/F_0}$ that the $L$-rational torsion subgroup $E(L)[\textrm{tors}]=E(F_0)[\textrm{tors}]$. This generalizes a previous result of González-Jiménez and Najman over $F_0=\mathbb{Q}$.
Towards showing this, we also prove a result on relative uniform divisibility of the index of a mod-$\ell$ Galois representation of an elliptic curve over $F_0$. Additionally, we show that the main result's conclusion fails when we allow $F_0$ to have rationally defined CM, due to the existence of $F_0$-rational isogenies of arbitrarily large prime degrees satisfying certain congruency conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_16977 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Growth of torsion groups of elliptic curves upon base change from number fields Genao, Tyler Number Theory 11G05, 11G15 Given a number field $F_0$ that contains no Hilbert class field of any imaginary quadratic field, we show that under GRH there exists an effectively computable constant $B:=B(F_0)\in\mathbb{Z}^+$ for which the following holds: for any finite extension $L/F_0$ whose degree $[L:F_0]$ is coprime to $B$, one has for all elliptic curves $E_{/F_0}$ that the $L$-rational torsion subgroup $E(L)[\textrm{tors}]=E(F_0)[\textrm{tors}]$. This generalizes a previous result of González-Jiménez and Najman over $F_0=\mathbb{Q}$. Towards showing this, we also prove a result on relative uniform divisibility of the index of a mod-$\ell$ Galois representation of an elliptic curve over $F_0$. Additionally, we show that the main result's conclusion fails when we allow $F_0$ to have rationally defined CM, due to the existence of $F_0$-rational isogenies of arbitrarily large prime degrees satisfying certain congruency conditions. |
| title | Growth of torsion groups of elliptic curves upon base change from number fields |
| topic | Number Theory 11G05, 11G15 |
| url | https://arxiv.org/abs/2210.16977 |