On the number of conjugacy classes in SL(2, Z)

Abstract

Let H(t) be the number of conjugacy classes of elements in SL(2, L ) with trace t, and let h(n) be the number of equivalence classes of binary quadratic forms with discriminant n. Then for t≠±2, H(t)= h(t 2−4) . For all real θ > 0 there is a T( θ) such that whenever | t|> T( θ), H(t)>|t| 1−θ . There is a c>0 such that for those t such that t 2−4 is squarefree, H(t)≤c|t| .