Embedding orders into central simple algebras
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 Ω 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 Ω dans R ? La première réponse à cette question est un résultat élégant de Chevalley [6]. Avec B=M n (K), [L:K]=n et Ω l’anneau des entiers de L, Chevalley démontre que la proportion des ordres maximaux de B qui admettent un plongement de Ω est [LK ˜:K] -1 , où 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 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 Ω 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 SL 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 SL 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 (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 [LK ˜:K] -1 where 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 .

DOI : 10.5802/jtnb.803
Classification : 11R54, 11S45, 20E42
Mots clés : Order, central simple algebra, affine building, embedding
Linowitz, Benjamin 1 ; Shemanske, Thomas R. 1

1 Department of Mathematics 6188 Kemeny Hall Dartmouth College Hanover, NH 03755
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Linowitz, Benjamin; Shemanske, Thomas R. Embedding orders into central simple algebras. Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 2, pp. 405-424. doi : 10.5802/jtnb.803. http://archive.numdam.org/articles/10.5802/jtnb.803/

[1] Peter Abramenko and Kenneth S. Brown, Buildings. Graduate Texts in Mathematics, vol. 248, Springer, New York, 2008, Theory and applications. MR MR2439729 | MR | Zbl

[2] Luis Arenas-Carmona, Applications of spinor class fields: embeddings of orders and quaternionic lattices. Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7, 2021–2038. MR MR2044166 (2005b:11044) | Numdam | MR

[3] Kenneth S. Brown, Buildings. Springer-Verlag, New York, 1989. MR MR969123 (90e:20001) | MR | Zbl

[4] J. Brzezinski, On two classical theorems in the theory of orders. J. Number Theory 34 (1990), no. 1, 21–32. MR MR1039764 (91d:11142) | MR | Zbl

[5] Wai Kiu Chan and Fei Xu, On representations of spinor genera. Compos. Math. 140 (2004), no. 2, 287–300. MR MR2027190 (2004j:11035) | MR

[6] C. Chevalley, Algebraic number fields. L’arithmétique dans les algèbres de matrices, Herman, Paris, 1936. | Zbl

[7] Ted Chinburg and Eduardo Friedman, An embedding theorem for quaternion algebras. J. London Math. Soc. (2) 60 (1999), no. 1, 33–44. MR MR1721813 (2000j:11173) | MR | Zbl

[8] A. Fröhlich, Locally free modules over arithmetic orders. J. Reine Angew. Math. 274/275 (1975), 112–124, Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, III. MR MR0376619 (51 #12794) | MR | Zbl

[9] Paul Garrett, Buildings and classical groups. Chapman & Hall, London, 1997. MR 98k:20081 | MR | Zbl

[10] Xuejun Guo and Hourong Qin, An embedding theorem for Eichler orders. J. Number Theory 107 (2004), no. 2, 207–214. MR MR2072384 (2005c:11141) | MR

[11] P. J. Higgins, Introduction to topological groups. Cambridge University Press, London, 1974, London Mathematical Society Lecture Note Series, No. 15. MR MR0360908 (50 #13355) | MR | Zbl

[12] Serge Lang, Algebraic number theory, second ed. Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York, 1994. MR MR1282723 (95f:11085) | MR | Zbl

[13] B. Linowitz, Selectivity in quaternion algebras. J. of Number Theory 132 (2012), 1425–1437.

[14] C. Maclachlan, Optimal embeddings in quaternion algebras. J. Number Theory 128 (2008), 2852–2860. | MR

[15] Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, second ed. Springer-Verlag, Berlin, 1990. MR MR1055830 (91h:11107) | MR | Zbl

[16] Jürgen Neukirch, Algebraic number theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999, Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder. MR MR1697859 (2000m:11104) | MR | Zbl

[17] Richard S. Pierce, Associative algebras. Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York, 1982, , Studies in the History of Modern Science, 9. MR MR674652 (84c:16001) | MR | Zbl

[18] I. Reiner, Maximal orders. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1975, London Mathematical Society Monographs, No. 5. MR MR0393100 (52 #13910) | MR | Zbl

[19] Mark Ronan, Lectures on buildings. Academic Press Inc., Boston, MA, 1989. MR 90j:20001 | MR | Zbl

[20] Thomas R. Shemanske, Split orders and convex polytopes in buildings. J. Number Theory 130 (2010), no. 1, 101–115. MR MR2569844 | MR

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