二元型定义的环中理想类的新参数化及其应用
A new parametrization for ideal classes in rings defined by binary forms, and applications
AI总结：
本文通过共正则表示的轨道为二元型定义的环中理想类平方根建立新参数化，并将其应用于界定数域族中2-类群平均大小等若干此前难以解决的分布问题。
摘要：
我们利用一个共正则表示的轨道，给出了由二元型定义的环的逆差分的理想类的平方根的一个参数化。这个参数化可以被解释为 Bhargava 发现并由 Wood 推广的“高次合成律”的一个新的整模型，它是解决一系列此前难以处理的关于类群、Selmer 群及相关对象分布的公开问题所缺少的关键要素。例如，在本文中，我们应用该参数化来界定由二元 n 次型定义的数域族中 2-类群的平均大小，其中 n ≥ 3 是任意整数，无论奇偶；在论文 [41] 中，我们应用它证明了大多数整系数奇次二元型不能本原地表示一个平方数；在论文 [11] 中，我们与 Bhargava 和 Shankar 合作，应用它来界定椭圆曲线的 2-Selmer 群大小的二阶矩。
英文摘要：
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.
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