The proportion of genus one curves over ℚ defined by a binary quartic that everywhere locally have a point

Abstract

We consider the proportion of genus one curves over [Formula: see text] of the form [Formula: see text] where [Formula: see text] is a binary quartic form (or more generally of the form [Formula: see text] where also [Formula: see text] is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime [Formula: see text] is given by a fixed degree-[Formula: see text] rational function of [Formula: see text] for all odd [Formula: see text] (and for the generalized equation, the same rational function gives the local density at every prime). An additional analysis is carried out to estimate rigorously the local density at the real place.