Cubic function fields with prescribed ramification

Abstract

This article describes cubic function fields L/K with prescribed ramification, where K is a rational function field. We give general equations for such extensions, an algorithm to obtain an explicit defining equation when the purely cubic closure K ′ /K of L/K has 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 article concludes by studying covers of elliptic and hyperelliptic curves that are ramified above exactly one point.