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The problem of constructing curves with many points over finite fields hasreceived considerable attention in the recent years. Using the class fieldtheory approach, we construct new examples of curves ameliorating some of theknown bounds. More precisely, we improve the lower bounds on the maximal numberof 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 threesmooth projective curves over $\mathbb F_q$, for $q$ is an odd prime less than19.

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