Purely cubic function fields with short periods

Abstract

A "function field version" of Voronoi's algorithm can be used to compute the fundamental unit of a purely cubic complex congruence function field of characteristic at least 5. This is accomplished by generating a sequence of minima in the maximal order of the field. The number of mimima computed is the period of the field. Generally, the period is very large -it is proportional to the regulator and exponential in the genus of the field -but there are classes of fields with very short periods. For several infinite families of such fields, we explicitly compute the Voronoi continued fraction expansions and the fundamental units. We also investigate the case of period length 1 where the minima in the maximal order are exactly the units of the field. Finally, we explore the connection between regulator and period and other cases of small periods and regulators.