Curves with many points over finite fields: the class field theory approach

The problem of constructing curves with many points over finite fields has received considerable attention in the recent years. Using the class field theory approach, we construct new examples of curves ameliorating some of the known bounds. More precisely, we improve the lower bounds on the maximal number of points $N_q(g)$ for many values of the genus $g$ and of the cardinality $q$ of the finite field $\mathbb F_q$, by looking at coverings of all genus three smooth projective curves over $\mathbb F_q$, for $q$ is an odd prime less than 19.