论文标题
类型中央简单代数中本地瓷砖订单的数量
Type numbers of locally tiled orders in central simple algebras
论文作者
论文摘要
让$ a $是一个中央简单的代数，上面是数字字段$ k $，带有整数$ \ MATHCAL {O} _K $，以便既要么代数$ n \ ge 3 $或$ n = 2 $和$ a $的程度，而且不是完全确定的quaternion algebra。然后，强近似值$ a $，这使我们能够描述$ \ Mathcal {o} _k $ - 订单$γ\ subset a $ a $的属。我们考虑在每个有限位置$ n $ k $的订单$γ$，并使用$ sl_n（k_ν）$的bruhat-titt building the bruhat-titt building for $γ$的本地正常化器提供几何描述。我们还提供明确的公式和算法来计算$γ$的类型数。我们的结果将Vignéras的工作推广到高级中央简单代数中的订单。
Let $A$ be a central simple algebra over a number field $K$ with ring of integers $\mathcal{O}_K$, such that either the degree of the algebra $n \ge 3$, or $n=2$ and $A$ is not a totally definite quaternion algebra. Then strong approximation holds in $A$, which allows us to describe the genus of an $\mathcal{O}_K$-order $Γ\subset A$ in terms of idelic quotients of the field $K$. We consider orders $Γ$ that are tiled at every finite place $ν$ of $K$ and use the Bruhat-Tits building for $SL_n(K_ν)$ to give a geometric description for the local normalizers of $Γ$. We also give explicit formulas and algorithms to compute the type number of $Γ$. Our results generalize work of Vignéras for orders in higher degree central simple algebras.
