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.

中文翻译：

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由二元四次方程定义的在 ℚ 上的属一曲线的比例，该二元四次方程在任何地方都有一个点

我们考虑属一曲线的比例ℚ形式的z2 = F(X,是的)在哪里F(X,是的) ∈ ℤ[X,是的]是二元四次形式（或更一般的形式z2 + H(X,是的)z = F(X,是的)哪里还有H(X,是的) ∈ ℤ[X,是的]是二元二次形式），在局部各处都有点。我们表明，这些曲线在局部可溶的比例（计算为局部密度的乘积）约为 75.96%。我们证明了局部密度在素数p由一个固定的度数给出-9的有理函数p对于所有奇怪的p（对于广义方程，相同的有理函数给出每个素数的局部密度）。进行了额外的分析以严格估计真实地点的局部密度。