In , Kodama, Top, Washio studied the maximality of a family of elliptic curves, mostly of genus 3, over a finite field. They used the Jacobians of the curves and differential forms to obtain their results. In this note, in order to prove the maximality of the curves under study, we use analytical tools, namely character and Jacobsthal sums, together with an important result which says that if a curve is the image of a maximal curve under a rational map, then it is itself maximal. Character sums are suitable for counting the number of points on a curve over a finite field, and their use makes the proofs natural and rather elementary. The norm and trace curves are utilized to construct rational maps to the hyperelliptic curves. As an application, the maximality of a certain hyperelliptic curve is used to find the explicit value of a character sum.
"On the Maximality of Certain Hyperellptic Curves with an Application to Character Sums Peter McCalla and Francois Ramaroson." Commun. Math. Anal. 18 (2) 76 - 85, 2015.