Let p be a prime and Fq be the finite field of q elements of characteristic p. Given a, b ∈ Fq, and positive integer n ≥ 2, let Nn(a, b) = #{α ∈ Fqn | T raceFqn /Fq (α) = a, NormFqn /Fq (α) = b}. Motivated by various applications, many authors have investigated sharp estimates for the number Nn(a, b). In 2010, after associating Nn(a, b) with the number of rational points on certain toric Calabi-Yau hypersurface, Moisio and Wan [1] proved that the following: Theorem 0.1. | Nn(a, b) − q n − 1 q − 1 |≤ (n − 1)q n−2 2 . In this talk, we will discuss how to associate Nn(a, b) with the number of rational points on certain algebraic curves and point out improvements on Moisio-Wan’s bound for a certain range of q. References [1] M. Moisio, D. Wan, On Katz’s bound for the number of elements with given trace and norm, J. Reine Angew. Math. 638 (2010), 69–7
