Divisibility of orders of reductions of elliptic curves

Abstract

Let $E$ be an elliptic curve defined over $Q$ and ${\tilde{E}}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $\left|{\tilde{E}}_p\left(F_p\right)\right|$ by an integer $m\ge 2$ 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 $m_E$ such that the congruence class of $\left|{\tilde{E}}_p\left(F_p\right)\right|$ 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 $Q$ whose torsion grows over a quadratic field $K$ where $m_E$ is determined by the $K$-torsion subgroups in the $Q$-isogeny class of $E$. We also exhibit elliptic curves over $Q\left(t\right)$ 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.