Explicit formulas and exact values for the number of rational points on singular curves over finite fields

We provide new explicit formulas for bounding the number of rational points on singular curves over finite fields. This enables us to obtain exact values of N q (g, $\pi$) which is defined as the maximum number of rational points over F q on a curve of geometric genus g and arithmetic genus $\pi$. We also give special attention to the case g = 2 in order to extend the work of Aubry and Iezzi on N q (0, $\pi$) and N q (1, $\pi$).