由矩阵分解为初等矩阵定义的簇上的整点
Integral points on varieties defined by matrix factorization into elementary matrices
摘要：
设${\mathcal O}$是数域$K$中的$S$ -整数环.对于$A\in\rm{SL}_{2}(\mathcal{O})$和$k\geq 1$，我们定义了${\mathcal O}$上的矩阵分解簇$V_k(A)$，它将$A$参数化为$k$初等矩阵的乘积;定义$V_k(A)$的方程用欧拉连续多项式来表示。我们发现，$V_k(A)$是合理的$(k-3)$ -折叠与诱导纤维化结构。我们结合这种几何结构和算术结果来研究$V_k(A)$的${\mathcal O}$ -点的Zapriki闭包。证明了对$k\geq 4$，若单位群${\mathcal O}^{\times}$为无穷大，则当$V_{k}(A)({\mathcal O})\neq\emptyset$为无穷大时，$k\geq 4$上的${\mathcal O}$ -点是Zerkki稠密的.这表明，如果$A$可以写成$k\geq 4$初等矩阵的乘积，那么这可以在最强意义上以无限多种方式完成。然后，这可以与分解成${\rm SL}_{2}({\mathcal O})$的初等矩阵的结果相结合。一个结果是，对于$k\geq 9$，如果${\mathcal O}^{\times}$是无限的，则$V_{k}(A)$上的${\mathcal O}$ -点是Zeromki稠密的.
英文摘要：
Let ${\mathcal O}$ be the ring of $S$-integers in a number field $K$. For $A\in\rm{SL}_{2}(\mathcal{O})$ and $k\geq 1$, we define matrix-factorization varieties $V_k(A)$ over ${\mathcal O}$ which parametrize factoring $A$ into a product of $k$ elementary matrices; the equations defining $V_k(A)$ are written in terms of Euler's continuant polynomials. We show that the $V_k(A)$ are rational $(k-3)$-folds with an inductive fibration structure. We combine this geometric structure with arithmetic results to study the Zariski closure of the ${\mathcal O}$-points of $V_k(A)$. We prove that for $k\geq 4$ the ${\mathcal O}$-points on $V_k(A)$ are Zariski dense if $V_{k}(A)({\mathcal O})\neq\emptyset$ assuming the group of units ${\mathcal O}^{\times}$ is infinite. This shows that if $A$ can be written as a product of $k\geq 4$ elementary matrices, then this can be done in infinitely many ways in the strongest sense possible. This can then be combined with results on factoring into elementary matrices for ${\rm SL}_{2}({\mathcal O})$. One result is that for $k\geq 9$ the ${\mathcal O}$-points on $V_{k}(A)$ are Zariski dense if ${\mathcal O}^{\times}$ is infinite.
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