Counting conjugacy classes in Sylow p-subgroups of Chevalley groups

Let p be a prime and let q be a power of p. Let U(q) be a Sylow p-subgroup of a finite Chevalley group G(q) and let B(q) be the normalizer of U(q) in G(q). In this paper we prove rationality of the zeta functions associated to: the number of conjugacy classes of U(q); the number of B(q)-conjugacy classes in U(q); and the number of conjugacy classes of B(q). Our proof is constructive and provides a parametrization of the conjugacy classes; it also leads to a method to calculate the zeta functions.