Divisibility of orders of reductions of elliptic curves

Official URL: https://dx.doi.org/10.1016/j.exmath.2025.125679

Let E be an elliptic curve defined over Q and E˜p denote the reduction of E modulo a prime p of good reduction for E. The divisibility of |E˜p(Fp)| 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 Q-isogenous to E. In this work, we give explicit families of elliptic curves E over Q together with integers mE such that the congruence class of |E˜p(Fp)| modulo mE 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 Q whose torsion grows over a quadratic field K where mE is determined by the K-torsion subgroups in the Q-isogeny class of E. We also exhibit elliptic curves over 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.