Zeta functions of trinomial curves and maximal curves

TL;DR

This paper derives explicit formulas for the zeta functions and genus of trinomial curves over finite fields, and investigates conditions under which these curves are maximal, contributing to the understanding of their arithmetic properties.

Contribution

It provides explicit formulas for the zeta functions and genus of trinomial curves, and characterizes when they are maximal over finite fields.

Findings

Explicit formulas for zeta functions in terms of Gauss and Jacobi sums.

A formula for the genus of trinomial curves over finite fields.

Conditions for trinomial curves to be maximal over finite fields.

Abstract

We determine the zeta functions of trinomial curves in terms of Gauss sums and Jacobi sums, and we obtain an explicit formula of the genus of a trinomial curve over a finite field, then we study the conditions for a trinomial curve to be a maximal curve over a finite field.