The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917565329047552 |
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| author | Cullinan, John Dobson, Shanna Frey, Linda Hamakiotes, Asimina Hernandez, Roberto Kaplan, Nathan Mello, Jorge Scullard, Gabrielle |
| author_facet | Cullinan, John Dobson, Shanna Frey, Linda Hamakiotes, Asimina Hernandez, Roberto Kaplan, Nathan Mello, Jorge Scullard, Gabrielle |
| contents | Let $E$ and $E'$ be 2-isogenous elliptic curves over $\Q$. Following \cite{ck}, we call a good prime $p$ \emph{anomalous} if $E(\F_p) \simeq E'(\F_p)$ but $E(\F_{p^2}) \not \simeq E'(\F_{p^2})$. Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II Cullinan, John Dobson, Shanna Frey, Linda Hamakiotes, Asimina Hernandez, Roberto Kaplan, Nathan Mello, Jorge Scullard, Gabrielle Number Theory Let $E$ and $E'$ be 2-isogenous elliptic curves over $\Q$. Following \cite{ck}, we call a good prime $p$ \emph{anomalous} if $E(\F_p) \simeq E'(\F_p)$ but $E(\F_{p^2}) \not \simeq E'(\F_{p^2})$. Our main result is an explicit formula for the proportion of anomalous primes for any such pair of elliptic curves. We consider both the CM case and the non-CM case. |
| title | The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.06250 |