论文标题
由二进制形式定义的环中理想类的新参数化和应用程序
A new parametrization for ideal classes in rings defined by binary forms, and applications
论文作者
论文摘要
我们给出了由二进制形式定义的逆环的理想等级的平方根的参数化，该环的轨道形式根据核心表示的轨道定义。该参数化可以解释为Bhargava发现的``更高构图定律''的新积分模型，并由木材概括，是解决一系列有关课堂，selmer组和相关对象的一系列先前棘手的开放问题所需的缺失成分。例如，在本文中，我们将参数化应用于由二进制$ n $ -ic表单定义的数字字段中$ 2 $ class组的平均大小，其中$ n \ geq 3 $是任意的整数，奇数，甚至是奇数；在论文[41]中，我们将其应用于证明大多数积分奇数二进制形式无法原始代表正方形。在与Bhargava和Shankar的关节的论文[11]中，我们将其应用于$ 2 $ -Sellmer组的椭圆形曲线的第二刻。
We give a parametrization of square roots of the ideal class of the inverse different of rings defined by binary forms in terms of the orbits of a coregular representation. This parametrization, which can be construed as a new integral model of a ``higher composition law'' discovered by Bhargava and generalized by Wood, was the missing ingredient needed to solve a range of previously intractable open problems concerning distributions of class groups, Selmer groups, and related objects. For instance, in this paper, we apply the parametrization to bound the average size of the $2$-class group in families of number fields defined by binary $n$-ic forms, where $n \geq 3$ is an arbitrary integer, odd or even; in the paper [41], we applied it to prove that most integral odd-degree binary forms fail to primitively represent a square; and in the paper [11], joint with Bhargava and Shankar, we applied it to bound the second moment of the size of the $2$-Selmer group of elliptic curves.
