CONJUGACY CLASSES OF HYPERBOLIC MATRICES IN Sl(n, Z) AND IDEAL CLASSES IN AN ORDER

A bijection is proved between Sl( n, Z)-conjugacy classes of hyperbolic matrices with eigenvalues {A,,_A,,} which are units in an n-degree number field, and narrow ideal classes of the ring Rk = Z[A,]. A bijection between Gl(«,Z)-conjugacy classes and the wide ideal classes, which had been known, is repeated with a different proof. In 1980, Peter Sarnak was able to obtain an estimate of the growth of the class number of real quadratic number fields using the Selberg Trace Formula and a bijection between hyperbolic elements of Sl(2, Z) and quadratic forms. The "class number" counted was the number of congruence classes of quadratic forms as studied by Gauss [1]. In this paper we will translate this bijection into modern number-theoretic terms by counting ideal classes in a ring of integers associated to a given field. In this way a bijection is proved between conjugacy classes of hyperbolic matrices in Sl(2, Z) with a given set of eigenvalues and ideal classes in a certain order (i.e. subring of dimension n over Z) associated to the ring of integers 0K in a real n th degree number field K. This more direct method is necessary for generalizing the bijection to higher dimensional cases because Sarnak's result depends upon quadratic forms, Pell's equation and other things which are well understood only in the case of S1(2,Z). We must mention the work of Latimer and MacDuffee [3] who first proved Theorem 2 in a slightly different fashion. Important also is the extensive work of Taussky [7-9], who simplified the results of Latimer and MacDuffee and extended them in certain directions, as well as doing much work on the Sl(2, Z) case. It follows from a brief examination of the characteristic polynomial for a matrix A in Sl(n,Z) that the eigenvalues of A are conjugate units in an extension of Q. We shall insist in the remainder of this paper that A be "hyperbolic" with irreducible characteristic polynomial, that is, A will have distinct real eigenvalues A(,), each of which is of degree n over Q. Proposition I. IfX is an eigenvalue for a matrix A e SL( n, Z), then for any field K containing X there exists an eigenvector co, Tu = (w,,..., wn), with w, e 0K. Received by the editors March 30, 1983. 1980 Mathematics Subject Classification. Primary 10-02, 15-02; Secondary 15A18, 15A36, 10C07. ©1984 American Mathematical Society 0025-5726/84 $1.00 + $.25 per page 177 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use