论文标题
一属属的比例超过$ \ mathbb {q} $，由二进制四分之一定义
The proportion of genus one curves over $\mathbb{Q}$ defined by a binary quartic that everywhere locally have a point
论文作者
论文摘要
我们考虑了$ z^2 = f（x，y）$的$ \ mathbb {q} $属属的比例，其中$ f（x，y）\ in \ mathbb {z，y）[x，y] $是二元四分之一的Quartic form（或更一般的表格$ z^2+z^2+h（x，x，x，x，x，x，x，x，x，x，x，x，x，x，x，x，x，x，x，y） $ h（x，y）\ in \ mathbb {z} [x，y] $是二进制二次形式），在本地无处不在。我们表明，这些本地溶解的这些曲线的比例大约为75.96％。我们证明，所有奇数$ p $ $ p $ $ 9 $的理性函数给出了Prime $ p $的本地密度（对于所有奇数$ p $）（对于广义方程式，相同的理性函数在每个元素下都可以使局部密度）。进行其他分析以严格估算真实位置的局部密度。
We consider the proportion of genus one curves over $\mathbb{Q}$ of the form $z^2=f(x,y)$ where $f(x,y)\in\mathbb{Z}[x,y]$ is a binary quartic form (or more generally of the form $z^2+h(x,y)z=f(x,y)$ where also $h(x,y)\in\mathbb{Z}[x,y]$ 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 generalised 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.
