The objective of this project is the development of an algorithm for computing the regulator and the ideal class number of a purely cubic congruence function field of unit rank 1. Recently, the continued fraction baby step giant step algorithm for finding the regulator of a real quadratic number field has been adapted to work in real quadratic congruence function fields. This method has also been applied to work in real quadratic congruence function fields. This method has also been applied to a variety of areas that are not immediately related to the theory of algebraic congruence function fields, such as elliptic curves and public-key cryptography. An Euler product technique for determining the class number of a real quadratic number field has also been successfully modified to compute the ideal class number of a real quadratic congruence function field. In a complex cubic number field, Voronoi's continued fraction algorithm for obtaining relative minima in cubic lattices is the base for computing the regulator of the field. A giant step optimization has been developed for this method as well. As in the quadratic case, Dirichlet series and Euler products represent a tool for finding the class number of the field, once the regulator is known. Considering the success of modifying number field techniques to work in function fields in the degree 2 situation, a logical approach is the adaptation of the regulator and class number algorithms in purely cubic number fields to the corresponding function field setting, that is, a purely cubic congruence function field of unit rank 1. Such a field takes the form K = k(x)(cube root of D) where k is a finite field whose order is congruent to -1 modulo 3, x is a fixed element that is transcendental over k, and D is a cube-free polynomial in k x whose highest coefficient is a cube in k. Questions to be addressed during the planning period of this project include embedding a purely cubic unit rank one congruence fu nction field K into the field of power series in 1/x over k in order to obtain a theory of continued fractions in K, the development of a reduction theory for ideals in K, and a first design of a regulator algorithm, initially without giant steps. The giant step version as well as the class number computation will be the subject of research following the planning stage.
