The Effect of Quadratic Base Change on Torsion of Elliptic Curves
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917820677226496 |
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| author | Balçık, Irmak Newman, Burton |
| author_facet | Balçık, Irmak Newman, Burton |
| contents | Let $K$ be a quadratic number field and let $E$ be an elliptic curve defined over $K$ such that $E[2] \not\subseteq E(K).$ In this paper, we study the effect of quadratic base change on $E(K)_{\text{tor}}.$ Moreover, for a given elliptic curve $E/K$ with prescribed torsion group over $K,$ (no restriction on its $2$-torsion part) we describe a fast algorithm to find all quadratic extensions $L/K$ in which $E(K)_{\text{tor}} \subsetneq E(L)_{\text{tor}}$ and describe $E(L)_{\text{tor}}$ in each such case. In particular, we determine the growth of $E(K)_{\text{tor}}$ upon quadratic base change when $K$ is any quadratic cyclotomic field, which completes the earlier work of the second author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_03709 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Effect of Quadratic Base Change on Torsion of Elliptic Curves Balçık, Irmak Newman, Burton Number Theory Let $K$ be a quadratic number field and let $E$ be an elliptic curve defined over $K$ such that $E[2] \not\subseteq E(K).$ In this paper, we study the effect of quadratic base change on $E(K)_{\text{tor}}.$ Moreover, for a given elliptic curve $E/K$ with prescribed torsion group over $K,$ (no restriction on its $2$-torsion part) we describe a fast algorithm to find all quadratic extensions $L/K$ in which $E(K)_{\text{tor}} \subsetneq E(L)_{\text{tor}}$ and describe $E(L)_{\text{tor}}$ in each such case. In particular, we determine the growth of $E(K)_{\text{tor}}$ upon quadratic base change when $K$ is any quadratic cyclotomic field, which completes the earlier work of the second author. |
| title | The Effect of Quadratic Base Change on Torsion of Elliptic Curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2310.03709 |