The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II

The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II

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...

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Veröffentlicht in:arXiv.org 2024-01
Hauptverfasser: Cullinan, John, Dobson, Shanna, Frey, Linda, Hamakiotes, Asimina, Hernandez, Roberto, Kaplan, Nathan, Mello, Jorge, Scullard, Gabrielle
Format: Artikel
Sprache:eng
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Curves
Fields (mathematics)
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Zusammenfassung: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.
ISSN:2331-8422