Embedding orders into central simple algebras

Linowitz, Benjamin; Shemanske, Thomas R.

Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 2, pp. 405-424.

Soit $B$ une algèbre centrale simple sur un corps de nombres $K$ et $L/K$ une extension finie de corps. Dans la théorie du corps de classes, on étudie les conditions sous lesquelles il existe un plongement de $L$ dans $B$. Considérons un raffinement subtil de ce problème : soit $\Omega$ un ordre d’indice fini dans l’anneau des entiers de $L$, et $R$ un ordre de rang maximal dans $B$. Sous quelles conditions existe-t-il un plongement de $\Omega$ dans $R$ ? La première réponse à cette question est un résultat élégant de Chevalley [6]. Avec $B=M_n\left(K\right)$, $[L:K]=n$ et $\Omega$ l’anneau des entiers de $L$, Chevalley démontre que la proportion des ordres maximaux de $B$ qui admettent un plongement de $\Omega$ est ${[L\cap \tilde{K}:K]}^{-1}$, où $\tilde{K}$ est le corps de classe de Hilbert de $K$. Chinburg et Friedman ([7] ) étudient les plongements d’ordres arbitraires d’un corps de nombres quadratique dans des algèbres de quaternions satisfaisant la condition d’Eichler, et Arenas-Carmona [2] considère les plongements de l’anneau des entiers dans les ordres maximaux d’une grande classe d’algèbres centrales simples. Dans cet article, nous considérons les algèbres centrales simples de dimension $p^2$ où $p$ est un nombre premier impair. Nous démontrons que soit aucune, toutes, ou exactement une des $p$ classes d’isomorphisme des ordres maximaux dans $B$ admettent un plongement d’un ordre $\Omega$ commutatif arbitraire d’une extension de $K$ de degré $p$. Nous caractérisons de manière explicite ces ordres, dits sélectifs. Les corps de classes jouent un rôle central dans cette caractérisation. Un ingrédient important de l’argument de Chinburg et Friedman est la structure de l’arbre des ordres maximaux de $S{L}_2$ sur un corps local. Nous généralisons les résultats de Chinburg et Friedman en remplaçant l’arbre des ordres maximaux par l’immeuble affine de Bruhat-Tits pour $S{L}_p$.

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 [6] which says that with $B=M_n\left(K\right)$ 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 ${[L\cap \tilde{K}:K]}^{-1}$ where $\tilde{K}$ is the Hilbert class field of $K$. Chinburg and Friedman ([7]) consider arbitrary quadratic orders in quaternion algebras satisfying the Eichler condition, and Arenas-Carmona [2] 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 Chinburg and Friedman’s argument was the structure of the tree of maximal orders for $S{L}_2$ over a local field. In this work, we generalize Chinburg and Friedman’s results replacing the tree by the Bruhat-Tits building for $S{L}_p$.

Classification : 11R54, 11S45, 20E42
Mots clés : Order, central simple algebra, affine building, embedding

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