Conjugacy classes of hyperbolic matrices in and ideal classes in an order
Author:
D. I. Wallace
Journal:
Trans. Amer. Math. Soc. 283 (1984), 177184
MSC:
Primary 11F06; Secondary 11R80
MathSciNet review:
735415
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Abstract: A bijection is proved between conjugacy classes of hyperbolic matrices with eigenvalues which are units in an degree number field, and narrow ideal classes of the ring . A bijection between conjugacy classes and the wide ideal classes, which had been known, is repeated with a different proof.
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 C. F. Gauss, Disquisitiones arithmeticae, Chelsea, New York, 1889, 1965, p. 18. (German) MR 0197380 (33:5545)
 [2]
 Dennis Hejhal, The Selberg trace formula for , Vol. I, Lecture Notes in Math., SpringerVerlag, New York, 1976, p. 5. MR 0439755 (55:12641)
 [3]
 C. G. Latimer and C. C. MacDuffee, A correspondence between classes of ideals and classes of matrices, Ann. of Math. (2) 34 (1933), 313316. MR 1503108
 [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), 127132. MR 0043064 (13:201b)
 [8]
 , Composition of binary integral quadratic forms via integral matrices and composition of matrix classes, Linear and Multilinear Algebra 10 (1981), 309318. MR 638125 (82m:15030)
 [9]
 , On a theorem of Latimer and MacDuffee, Reprinted from Canad. J. Math. 1 (1949), 300302. MR 0030491 (11:3k)
 [10]
 Audrey Terras, Harmonic analysis on symmetric spaces and applications, UCSD lecture notes.
 [11]
 , Analysis on positive matrices as it might have occurred to Fourier, Lecture Notes in Math., vol. 899, SpringerVerlag, Berlin and New York, 1981, pp. 442478. MR 654544 (84b:22025)
 [12]
 D. I. Wallace, Explicit form of the hyperbolic term in the trace formula for and Pell's equation for hyperbolics in (to appear).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198407354150
PII:
S 00029947(1984)07354150
Article copyright:
© Copyright 1984
American Mathematical Society
