论文标题
集体订单组中的平均$ 2 $ torsion元素的平均数量
The mean number of $2$-torsion elements in the class groups of cubic orders
论文作者
论文摘要
当判别列出此类秩序时，我们确定了立方顺序的班级组中的2个扭转元素的平均数量。 Specifically, we prove that when isomorphism classes of totally real (resp., complex) cubic orders are enumerated by discriminant, the average $2$-torsion in the class group is $1 + \frac{1}{4} \times \frac{ζ(2)}{ζ(4)}$ (resp., $1 + \frac{1}{2} \ times \ frac {ζ（2）} {ζ（4）} $）。特别是，我们发现，班级组中的平均$ 2 $ torsion在一个立方领域的所有订单上均增加，而不是限制到立方领域整数的子范围，在该阶段的平均$ 2 $ 2 $ 2 $ 2 $ 2 $ \ frac ucct of Bhargava的平均$ \ frac compoct中是$ \ frac copt的。 $ \ frac {3} {2} $）。 通过bhargava-varma的工作，证明了这一结果等于在“可还原” $ \ perperatorName {sl} _3（\ m athbb {z}）$ orbits上获得“可还原” $ \ permatatorName的渐近计数\ operatatorName {sym}^2 \ mathbb {z}^3 $ of $ 3 \ times 3 $对称整数矩阵具有界限不变性并满足局部条件。在本文中，我们解决了这个轨道计数问题的概括，其中尺寸$ 3 $被任何固定的奇数整数$ n \ geq 3 $取代。更准确地说，我们确定可简化$ \ permatorName {sl} _n（\ Mathbb {z}）$ - orbits的渐近公式无限的一致条件集。
We determine the mean number of 2-torsion elements in class groups of cubic orders, when such orders are enumerated by discriminant. Specifically, we prove that when isomorphism classes of totally real (resp., complex) cubic orders are enumerated by discriminant, the average $2$-torsion in the class group is $1 + \frac{1}{4} \times \frac{ζ(2)}{ζ(4)}$ (resp., $1 + \frac{1}{2} \times \frac{ζ(2)}{ζ(4)}$). In particular, we find that the average $2$-torsion in the class group increases when one ranges over all orders in cubic fields instead of restricting to the subfamily of rings of integers of cubic fields, where the average $2$-torsion in the class group was first determined in work of Bhargava to be $\frac{5}{4}$ (resp., $\frac{3}{2}$). By work of Bhargava--Varma, proving this result amounts to obtaining an asymptotic count of the number of "reducible" $\operatorname{SL}_3(\mathbb{Z})$-orbits on the space $\mathbb{Z}^2 \otimes_{\mathbb{Z}} \operatorname{Sym}^2 \mathbb{Z}^3$ of $3 \times 3$ symmetric integer matrices having bounded invariants and satisfying local conditions. In this paper, we resolve the generalization of this orbit-counting problem where the dimension $3$ is replaced by any fixed odd integer $N \geq 3$. More precisely, we determine asymptotic formulas for the number of reducible $\operatorname{SL}_N(\mathbb{Z})$-orbits on $\mathbb{Z}^2 \otimes_{\mathbb{Z}} \operatorname{Sym}^2 \mathbb{Z}^N$ satisfying general infinite sets of congruence conditions.
