Divisibility of orders of reductions of elliptic curves

2023/01/02 by Antigona Pajaziti, Pajaziti, Antigona, Mohammad Sadek +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2301.00711

openalex publication_date 2023/01/02 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28

Abstract

Let E be an elliptic curve defined over \mathbb Q and \widetildeE_p denote the reduction of E modulo a prime p of good reduction for E. The divisibility of |\widetildeE_p(\mathbbF_p)| by an integer m≥ 2 for a set of primes p of density 1 is determined by the torsion subgroups of elliptic curves that are \mathbb Q-isogenous to E. In this work, we give explicit families of elliptic curves E over \mathbb Q together with integers m_E such that the congruence class of |\widetildeE_p(\mathbbF_p)| modulo m_E can be computed explicitly. In addition, we can estimate the density of primes p for which each congruence class occurs. These include elliptic curves over \mathbb Q whose torsion grows over a quadratic field K where m_E is determined by the K-torsion subgroups in the \mathbb Q-isogeny class of E. We also exhibit elliptic curves over \mathbb Q(t) for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than 1 are divisible by given small integers.

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