Equations over finite fields: Zeta function and Weil conjectures

Abstract

This work is a review of the congruent zeta function and the Weil conjectures for non-singular curves. We derive an equation to obtain the number of solutions of equations over finite fields using Jacobi sums in order to compute the Zeta function for specific equations. Also, we introduce the necessary algebraic concepts to prove the rationality and functionality of the zeta function.