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Conjugacy classes of hyperbolic matrices in $ {\rm Sl}(n,\,{\bf Z})$ and ideal classes in an order


Author: D. I. Wallace
Journal: Trans. Amer. Math. Soc. 283 (1984), 177-184
MSC: Primary 11F06; Secondary 11R80
MathSciNet review: 735415
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Abstract: A bijection is proved between $ \operatorname{Sl} (n,{\mathbf{Z}})$-conjugacy classes of hyperbolic matrices with eigenvalues $ \{ {\lambda _1}, \ldots ,{\lambda _n}\} $ which are units in an $ n$-degree number field, and narrow ideal classes of the ring $ {R_k} = {\mathbf{Z}}[{\lambda _i}]$. A bijection between $ \operatorname{Gl} (n,{\mathbf{Z}})$-conjugacy classes and the wide ideal classes, which had been known, is repeated with a different proof.


References [Enhancements On Off] (What's this?)

  • [1] Carl Friedrich Gauss, Disquisitiones arithmeticae, Translated into English by Arthur A. Clarke, S. J, Yale University Press, New Haven, Conn.-London, 1966. MR 0197380
  • [2] Dennis A. Hejhal, The Selberg trace formula for 𝑃𝑆𝐿(2,𝑅). Vol. I, Lecture Notes in Mathematics, Vol. 548, Springer-Verlag, Berlin-New York, 1976. MR 0439755
  • [3] Claiborne G. Latimer and C. C. MacDuffee, A correspondence between classes of ideals and classes of matrices, Ann. of Math. (2) 34 (1933), no. 2, 313–316. MR 1503108, 10.2307/1968204
  • [4] P. Samuel, Algebraic theory of numbers, Houghton Miflin, Boston, Mass., 1979, p. 24.
  • [5] Peter Sarnak, Prime geodesic theorems, Ph. D. thesis, Stanford University, 1980, pp. 1, 10, 19, 44.
  • [6] P. Sarnak and P. Cohen, Discrete groups and harmonic analysis (to appear).
  • [7] Olga Taussky, Classes of matrices and quadratic fields, Pacific J. Math. 1 (1951), 127–132. MR 0043064
  • [8] Olga Taussky, Composition of binary integral quadratic forms via integral 2×2 matrices and composition of matrix classes, Linear and Multilinear Algebra 10 (1981), no. 4, 309–318. MR 638125, 10.1080/03081088108817421
  • [9] Olga Taussky, On a theorem of Latimer and MacDuffee, Canadian J. Math. 1 (1949), 300–302. MR 0030491
  • [10] Audrey Terras, Harmonic analysis on symmetric spaces and applications, UCSD lecture notes.
  • [11] Audrey Terras, Analysis on positive matrices as it might have occurred to Fourier, Lecture Notes in Math., vol. 899, Springer, Berlin-New York, 1981, pp. 442–478. MR 654544
  • [12] D. I. Wallace, Explicit form of the hyperbolic term in the trace formula for $ \operatorname{SL} (3,{\mathbf{R}})$ and Pell's equation for hyperbolics in $ \operatorname{Sl} (3,{\mathbf{Z}})$ (to appear).

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0735415-0
Article copyright: © Copyright 1984 American Mathematical Society