Cubic function fields with prescribed ramification

This paper describes cubic function fields [Formula: see text] with prescribed ramification, where [Formula: see text] 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 [Formula: see text] of [Formula: see text] is of genus zero, and a description of the twists of [Formula: see text] up to isomorphism over [Formula: see text]. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows Möbius transformations on [Formula: see text]. The paper concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point.

Construction of all cubic function fields of a given square-free discriminant

For any square-free polynomial D over a finite field of characteristic at least 5, we present an algorithm for generating all cubic function fields of discriminant D. We also provide a count of all these fields according to their splitting at infinity. When D′ = D/(-3) has even degree and a leading coefficient that is a square, i.e. D′ is the discriminant of a real quadratic function field, this method makes use of the infrastructures of this field. This infrastructure method was first proposed by Shanks for cubic number fields in an unpublished manuscript from the late 1980s. While the mathematical ingredients of our construction are largely classical, our algorithm has the major computational advantage of finding very small minimal polynomials for the fields in question.

An Explicit Treatment of Cubic Function Fields with Applications

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

EFFECTIVE DETERMINATION OF PRIME DECOMPOSITIONS OF CUBIC FUNCTION FIELDS

In this paper, we determine completely the prime decomposition of cubic function fields by effective and explicit methods.