RESEARCH BLOG 3/10/03 In blog 2/27/03 Train Tracks on train tracks, I mentioned Lee Summary: RESEARCH BLOG 3/10/03 In blog 2/27/03 Train Tracks on train tracks, I mentioned Lee Mosher had said that the number of conjugacy classes of elements in SL(2, Z) with the same trace is the class number of the field Q adjoin the eigenvalue with that trace. It turns out that the formula is a class number, but not just of the field. I'll describe this in some greater generality. Suppose that we have a matrix A SL(n, Z), such that its char- acteristic polynomial pA(x) is irreducible. Let be an eigenvalue of A. Consider the field Q(), and let O Q() be the ring of integers, then O is a unit, since det(A) = 1. Then Z[] O is an order, that is a subring with 1 of O of rank n. If we take {1, , ..., n-1 } as a basis for Q(), identifying it with Qn , then multiplication by acts as an element R SL(n, Z) GL(n, Q), and this is the usual rational canonical form for matrices in GL(n, Q) with eigenvalue and eigenvec- tor = [1, , ..., n-1 ]T (therefore, it has eigenvalues = 1, 2, ..., n, Collections: Mathematics