The density of odd order reductions for elliptic curves with a rational point of order 2

Abstract

Suppose that $E/\mathbb{Q}$ is an elliptic curve with a rational point $T$ of order $2$ and $\alpha \in E(\mathbb{Q})$ is a point of infinite order. We consider the problem of determining the density of primes $p$ for which $\alpha \in E({\mathbb{F}}_p)$ has odd order. This density is determined by the image of the arboreal Galois representation ${\tau }_{E,2^k}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\mathrm{AGL}}_2(\mathbb{Z}/2^k\mathbb{Z})$. Assuming that $\alpha$ is primitive (that is, neither $\alpha$ nor $\alpha +T$ is twice a point over $\mathbb{Q}$) and that the image of the ordinary mod $2^k$ Galois representation is as large as possible (subject to $E$ having a rational point of order $2$), we determine that there are $63$ possibilities for the image of ${\tau }_{E,2^k}$. As a consequence, the density of primes $p$ for which the order of $\alpha$ is odd is between $1/14$ and $89/168$.

15 pages