Voronoi's algorithm in purely cubic congruence function fields of unit rank 1

Abstract

The first part of this paper classifies all purely cubic function fields over a finite field of characteristic not equal to 3. In the remainder, we describe a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1 and characteristic at least 5. The technique is based on Voronoi's algorithm for generating a chain of successive minima in a multiplicative cubic lattice, which is used for calculating the fundamental unit and regulator of a purely cubic number field.

Algorithmic Aspects of Cubic Function Fields

“…Using the signature description for purely cubic function fields given in Theorem 2.1 of [12] and the well-known characterization of hyperelliptic function fields (see for example Proposition 14.6 on p. 248 of [9]), we can reformulate and summarize Theorem 4.2 as follows. …”

“…For the rest, we can reason as in the proof of Theorem 2.1 of [12]: in the cases where there is only one (inert or totally ramified) place at infinity in K, S 0 is trivial and hence R = 1. For signature (1, 1, 1, 1, 1, 1), the formula for R follows from the fact that the map that permutes the three roots of f (Y ) also permutes the three places at infinity.…”

“…To unify the notation for both unit rank scenarios, we set |θ| 0 = |θ|, |θ| 1 = |θ| 2 = |θ θ | 1/2 for all θ ∈ K in the unit rank 1 case. The regulator and fundamental unit(s) of K/k(x) can be computed exactly as described in [12] for the unit rank 1 case and [6] for the case of unit rank 2, so we only give the minimal necessary background here and recall the algorithm for completeness. We focus our discussion on fractional ideals of O, i.e.…”

“…Since general-purpose methods tend to be computationally inefficient, the discussion of fast algorithms has so far predominantly focused on elliptic and hyperelliptic curves. In addition, the arithmetic of purely cubic curves y 3 = D(x) has been investigated in considerable detail [12,10,11,2]; Picard curves represent a special case thereof. Several other particular classes of of curves have also been studied from an algorithmic point of view, such as superelliptic curves (curves of the form y n = D(x)) [5] and C ab curves [1].…”

“…We also investigate efficient arithmetic of reduced fractional ideals in the maximal order of the field and show how to use this arithmetic to find the fundamental unit(s) and the regulator of the extension. Our method is based on a procedure that was originally developed by Voronoi for cubic number fields [14] and was recently adapted to purely cubic function fields [12,6].…”