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 explicit procedure to obtain a defining equation when the purely cubic closure $K^{\prime }/K$ 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 article concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point.

24 pages; published version

References in corpus (2)

• Identities for field extensions generalizing the Ohno-Nakagawa relations
• A complete classification of cubic function fields over any finite field