$p$-twisted Selmer near-companion curves
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918107961884672 |
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| author | Kim, Minseok |
| author_facet | Kim, Minseok |
| contents | Let $E_1$ and $E_2$ be elliptic curves over a number field $K$. In \cite{scc}, Mazur and Rubin define the concept of $n$-Selmer near-companions and conjecture that if $E_1$ and $E_2$ are $n$-Selmer near-companions over $K$, then $E_1[n]$ is $G_K$-isomorphic to $E_2[n]$. Yu proves the conjecture on $n$-Selmer near-companion curves in the case $n=2$. We we introduce the notion of $p$-twisted Selmer near-companions ($p$-TSNC) over $K$ and prove that if $E_1$ and $E_2$ are $p$-TSNC over $K$, then $K(E_1[p])=K(E_2[p])$ under certain conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10896 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $p$-twisted Selmer near-companion curves Kim, Minseok Number Theory Let $E_1$ and $E_2$ be elliptic curves over a number field $K$. In \cite{scc}, Mazur and Rubin define the concept of $n$-Selmer near-companions and conjecture that if $E_1$ and $E_2$ are $n$-Selmer near-companions over $K$, then $E_1[n]$ is $G_K$-isomorphic to $E_2[n]$. Yu proves the conjecture on $n$-Selmer near-companion curves in the case $n=2$. We we introduce the notion of $p$-twisted Selmer near-companions ($p$-TSNC) over $K$ and prove that if $E_1$ and $E_2$ are $p$-TSNC over $K$, then $K(E_1[p])=K(E_2[p])$ under certain conditions. |
| title | $p$-twisted Selmer near-companion curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.10896 |