Representation fields for orders of small rank

• Arenas Carmona, Luis ^[1]
  1. [1] Universidad de Chile

     Universidad de Chile

     Santiago, Chile

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 36, Nº. 1, 2017, págs. 131-148
• Idioma: inglés
• DOI: 10.4067/S0716-09172017000100008
• Enlaces
  • Texto completo
• Resumen

  • A representation field for a non-maximal order H in a central simple algebra is a subfield of the spinor class field of maximal orders which determines the set of spinor genera of maximal orders representing H. In our previous work we have proved the existence of the representation field for several important families of suborders, like commutative orders, while we have also found examples where the representation field fails to exist. To be precise, we have found full-rank orders, in central simple algebras of dimension 9 or larger over a suitable field, whose representation field is undefined. In this article, we prove that the representation field is defined for any order H of rank r < 7. This is done by defining representation fields for arbitrary representations of orders into central simple algebra and showing that the computation of these generalized representation fields can be reduced to the case of irreducible representations. The same technique yields the existence of representation fields for any order in an algebra whose semi-simple reduction is commutative. We also construct a rank-8 order, in a 16-dimensional matrix algebra, whose representation field is not defined.

• Referencias bibliográficas
  • Citas [1] L. Arenas-Carmona, Applications of spinor class fields: embeddings of orders and quaternionic lattices, Ann. Inst. Fourier (Grenoble)...
  • [2] L. Arenas-Carmona. Relative spinor class fields: A counterexample, Archiv. Math. 91, pp. 486-491, (2008).
  • [3] L. Arenas-Carmona, Spinor class fields for commutative orders, Ann. Inst. Fourier (Grenoble) 62, pp. 807-819, (2012).
  • [4] L. Arenas-Carmona, Spinor class fields for cyclic orders, Acta Arith. 156, pp. 143-156, (2012).
  • [5] L. Arenas-Carmona, Computing quaternion quotient graphs via representations of orders, J. Algebra. 402, pp. 258-279, (2014).
  • [6] L. Arenas-Carmona, Selectivity in division algebras, Archiv Math. 103, pp. 139-146, (2014).
  • [7] L. Arenas-Carmona, Roots of unity in definite quaternion orders, Acta Arith. 170, pp. 381-393, (2015).
  • [8] J. Brzezinski, Riemann-Roch Theorem for locally principal orders, Math. Ann., 276, pp. 529-536, (1987).
  • [9] C. Chevalley, L’arithm´ etique sur les alg` ebres de matrices, Herman: Paris (1936).
  • [10] T. Chinburg and E. Friedman, An embedding theorem for quaternion algebras, J. London Math. Soc. 60.2 (1999), 33-44.
  • [11] J.S. Hsia, Y. Y. Shao, and F. Xu, Representations of indefinite quadratic forms, J. Reine Angew. Math. 494, pp. 129-140, (1998).
  • [12] B. Linowitz and T. Shemanske, Embedding orders in central simple algebras, J. Théor. Nombres Bordeaux 24, pp. 405-424, (2012).
  • [13] M. Papikian, Local Diophantine properties of modular curves of Delliptic sheaves, J. Reine Angew. Math. 664, pp. 115-140, (2012).