二次齐次簇上纤维丛的Linnik问题
Linnik's problem in fiber bundles over quadratic homogeneous varieties
摘要：
我们计算统计的$SL_{d}(\mathbb{Z})$矩阵躺在水平集上的一个整数多项式定义的$SL_{d}(\mathbb{R})$，结果是一个著名的定理证明的Linnik关于均匀分布的径向投影积分向量从一个大的球到单位球的变体。利用上述结果，我们推广了Aka，Einsiedler和Shapira的工作在各个方向。例如，我们计算剩余类模$q$和位于水平集$-(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+x_{4}^{2}=N$上的本原积分向量的适当归一化正交格的联合分布为$N\to\infty$，其中归一化正交格位于$\mathbb{R}^{4}$的秩-$3$离散子群的模空间的子流形中。
英文摘要：
We compute the statistics of $SL_{d}(\mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(\mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of radially projected integral vectors from a large sphere into the unit sphere. Using the above result we generalize the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo $q$ and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set $-(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+x_{4}^{2}=N$ as $N\to\infty$, where the normalized orthogonal lattices sit in a submanifold of the moduli space of rank-$3$ discrete subgroups of $\mathbb{R}^{4}$.
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