中心单代数的序嵌入
Embedding Orders Into Central Simple Algebras
摘要：
将域嵌入数域$K$上的中心单代数$B$的问题是类域理论的领域。将包含在这样一个代数的极大子域$L$的整数环中的阶嵌入到该代数中的阶的主题更加微妙。第一个这样的结果沿着这些路线是一个优雅的结果Chevalley \cite{Chevalley-book}说，与$B = M_n(K)$的比例同构类的最大订单在$B$到环的整数$L$可以嵌入（总数目的类）是$[L \cap \widetilde K : K]^{-1}$其中$\widetilde K$是希尔伯特类领域的$K$。Chinburg和Friedman（\cite{Chinburg-Friedman}）考虑了满足Eichler条件的四元数代数中的任意二次阶，而Arenas-Carmona \cite{Arenas-Carmona}考虑了整数环嵌入到一类更高秩中心单代数中的最大阶。本文考虑了维数为$p^2$，$p$为奇素数的中心单代数，证明了$K$的一次$p$扩张中的任意交换阶嵌入到最大阶的$p$同构类中的零个、全部或恰好一个.那些在这个意义上是选择性的交换序被明确地刻画了;类域起着关键的作用。钦伯格和弗里德曼的论点的一个重要组成部分是$SL_2$在局部域上的最大阶树的结构。在这项工作中，我们推广Chinburg和弗里德曼的结果取代树的Bruhat-Tits建设$SL_p$。
英文摘要：
The question of embedding fields into central simple algebras $B$ over a number field $K$ was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields $L$ of such an algebra into orders in that algebra is more nuanced. The first such result along those lines is an elegant result of Chevalley \cite{Chevalley-book} which says that with $B = M_n(K)$ the ratio of the number of isomorphism classes of maximal orders in $B$ into which the ring of integers of $L$ can be embedded (to the total number of classes) is $[L \cap \widetilde K : K]^{-1}$ where $\widetilde K$ is the Hilbert class field of $K$. Chinburg and Friedman (\cite{Chinburg-Friedman}) consider arbitrary quadratic orders in quaternion algebras satisfying the Eichler condition, and Arenas-Carmona \cite{Arenas-Carmona} considers embeddings of the ring of integers into maximal orders in a broad class of higher rank central simple algebras. In this paper, we consider central simple algebras of dimension $p^2$, $p$ an odd prime, and we show that arbitrary commutative orders in a degree $p$ extension of $K$, embed into none, all or exactly one out of $p$ isomorphism classes of maximal orders. Those commutative orders which are selective in this sense are explicitly characterized; class fields play a pivotal role. A crucial ingredient of Chinberg and Friedman's argument was the structure of the tree of maximal orders for $SL_2$ over a local field. In this work, we generalize Chinburg and Friedman's results replacing the tree by the Bruhat-Tits building for $SL_p$.
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