Counting conjugacy classes in Sylow pp-subgroups of Chevalley groups

Let pp be a prime and let qq be a power of pp. Let U(q)U(q) be a Sylow pp-subgroup of a finite Chevalley group G(q)G(q) and let B(q)B(q) be the normalizer of U(q)U(q) in G(q)G(q). In this paper we prove rationality of the zeta functions associated to: the number of conjugacy classes of U(q)U(q); the number of B(q)B(q)-conjugacy classes in U(q)U(q); and the number of conjugacy classes of B(q)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.