Zeta functions of curves over finite fields with many rational points

Currently, the best known bounds on the number of rational points on an absolutely irreducible, smooth, projective curve defined over a finite field in most cases come from the optimization of the explicit formulae method, which is detailed in [7] and [9]. In [4], we gave an example of a case where the best known bound could not be met. The purpose of this paper is to give another new example of this phenomenon, and to explain the method used to obtain these results. This method gives rise to lists of all possible zeta functions, even in cases where we cannot conclude that no curve exists to meet the bound.