# Embedding orders into central simple algebras

Benjamin Linowitz^{[1]}; Thomas R. Shemanske^{[1]}

- [1] Department of Mathematics 6188 Kemeny Hall Dartmouth College Hanover, NH 03755

Journal de Théorie des Nombres de Bordeaux (2012)

- Volume: 24, Issue: 2, page 405-424
- ISSN: 1246-7405

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topLinowitz, Benjamin, and Shemanske, Thomas R.. "Embedding orders into central simple algebras." Journal de Théorie des Nombres de Bordeaux 24.2 (2012): 405-424. <http://eudml.org/doc/251051>.

@article{Linowitz2012,

abstract = {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 $B = M_n(K)$ 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 ([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 $SL_2$ over a local field. In this work, we generalize Chinburg and Friedman’s results replacing the tree by the Bruhat-Tits building for $SL_p$.},

affiliation = {Department of Mathematics 6188 Kemeny Hall Dartmouth College Hanover, NH 03755; Department of Mathematics 6188 Kemeny Hall Dartmouth College Hanover, NH 03755},

author = {Linowitz, Benjamin, Shemanske, Thomas R.},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {Order; central simple algebra; affine building; embedding; order},

language = {eng},

month = {6},

number = {2},

pages = {405-424},

publisher = {Société Arithmétique de Bordeaux},

title = {Embedding orders into central simple algebras},

url = {http://eudml.org/doc/251051},

volume = {24},

year = {2012},

}

TY - JOUR

AU - Linowitz, Benjamin

AU - Shemanske, Thomas R.

TI - Embedding orders into central simple algebras

JO - Journal de Théorie des Nombres de Bordeaux

DA - 2012/6//

PB - Société Arithmétique de Bordeaux

VL - 24

IS - 2

SP - 405

EP - 424

AB - 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 $B = M_n(K)$ 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 ([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 $SL_2$ over a local field. In this work, we generalize Chinburg and Friedman’s results replacing the tree by the Bruhat-Tits building for $SL_p$.

LA - eng

KW - Order; central simple algebra; affine building; embedding; order

UR - http://eudml.org/doc/251051

ER -

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