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

Abstract

Suppose that [Formula: see text] is an elliptic curve with a rational point [Formula: see text] of order [Formula: see text] and [Formula: see text] is a point of infinite order. We consider the problem of determining the density of primes [Formula: see text] for which [Formula: see text] has odd order. This density is determined by the image of the arboreal Galois representation [Formula: see text]. Assuming that [Formula: see text] is primitive (that is neither [Formula: see text] nor [Formula: see text] is twice a point over [Formula: see text]) and that the image of the ordinary [Formula: see text]-adic Galois representation is as large as possible (subject to [Formula: see text] having a rational point of order [Formula: see text]), we determine that there are [Formula: see text] possibilities for the image of [Formula: see text]. As a consequence, the density of primes [Formula: see text] for which the order of [Formula: see text] is odd is between 1/14 and [Formula: see text].