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

TL;DR

This paper investigates the density of primes for which a point of infinite order on an elliptic curve with a rational 2-torsion point has odd order modulo p, using Galois representations and classifying possible images.

Contribution

It classifies the possible images of the arboreal Galois representation for such elliptic curves and determines bounds on the density of primes with odd order points.

Findings

Identifies 63 possible images of the Galois representation.

Establishes that the density of primes with odd order points lies between 1/14 and 89/168.

Provides a framework for understanding the distribution of prime orders in elliptic curve points.

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 $\tau_{E,2^{k}} : {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to {\rm AGL}_{2}(\mathbb{Z}/2^{k}\mathbb{Z})$. 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 $89/168$.