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

2023/01/22 by John Cullinan, Nathan O. Kaplan, Cullinan, John +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2301.09176

openalex publication_date 2023/01/22 · openalex created_date 2023/01/25 · openalex updated_date 2026/07/28

Abstract

Let ℓ be a prime number and let E and E' be ℓ-isogenous elliptic curves defined over a finite field k of characteristic p ≠ ℓ. Suppose the groups E(k) and E'(k) are isomorphic, but E(K) \not ≃ E'(K), where K is an ℓ-power extension of k. In a previous work we have shown that, under mild rationality hypotheses, the case of interest is when ℓ=2 and K is the unique quadratic extension of k. In this paper we study the likelihood of such an occurrence by fixing a pair of 2-isogenous elliptic curves E, E' over Q and asking for the proportion of primes p for which E(F_p) ≃ E'(F_p) and E(F_p²) \not ≃ E'(F_p²).

Related

• The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, II
• On group structures realized by elliptic curves over a fixed finite field
• On the p-isogenies of elliptic curves with multiplicative reduction over quadratic fields
• Quantitative upper bounds related to an isogeny criterion for elliptic curves
• On the number of elliptic curves with prescribed isogeny or torsion group over number fields of prime degree