Curves over Finite Fields with Many Rational Points Obtained by Ray Class Field Extensions

A general type of ray class fields of global function fields is investigated. The computation of their genera is reduced to the determination of the degrees of these extensions, which turns out to be the main difficulty. While in two special situations explicit formulas for the degrees are known, the general problem is solved algorithmically. The systematic application of the methods described yields several new examples of algebraic curves over \(\mathbb{F}_{2}, \mathbb{F}_{3}, \mathbb{F}_{4}, \mathbb{F}_{5}\) and \( \mathbb{F}_{7}\) with comparatively many rational points.

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