 
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, , ..., n1
} 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, , ..., n1
]T
(therefore, it has eigenvalues = 1, 2, ..., n,
