Conjugacy classes of hyperbolic matrices in 𝑆𝑙(𝑛,𝑍) and ideal classes in an order

Abstract

A bijection is proved between Sl ⁡ ( n , Z ) \operatorname {Sl} (n,{\mathbf {Z}}) -conjugacy classes of hyperbolic matrices with eigenvalues { λ 1 , … , λ n } \{ {\lambda _1}, \ldots ,{\lambda _n}\} which are units in an n n -degree number field, and narrow ideal classes of the ring R k = Z [ λ i ] {R_k} = {\mathbf {Z}}[{\lambda _i}] . A bijection between Gl ⁡ ( n , Z ) \operatorname {Gl} (n,{\mathbf {Z}}) -conjugacy classes and the wide ideal classes, which had been known, is repeated with a different proof.