Cubic function fields with prescribed ramification

Abstract

This paper describes cubic function fields L/K with prescribed ramification, where K is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure K′/K of L/K is of genus zero, and a description of the twists of L/K up to isomorphism over K. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows Möbius transformations on K. The paper concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point.