We consider the proportion of genus one curves over Q of the form z2 = f(x, y) where f(x, y) ∈ Z[x, y] is a binary quartic form (or more generally of the form z2 + h(x, y)z = f(x, y) where also h(x, y) ∈ Z[x, y] 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 p is given by a fixed degree-9 rational function of p for all odd p (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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Genus one curves
- Local solubililty
- Random diophantine equations