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

2024/01/11 by John Cullinan, Shanna Dobson, Cullinan, John +13
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic

paper · pdf · doi:10.48550/arxiv.2401.06250

Abstract

Let E and E' be 2-isogenous elliptic curves over \Q. Following \citeck, we call a good prime p anomalous if E(\F_p) ≃ E'(\F_p) but E(\F_p²) \not ≃ E'(\F_p²). 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.

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