Tabulation of cubic function fields via polynomial binary cubic forms (Q2840015)

Cubic function fields over finite fields are interesting objects in arithmetic geometry. Let \(\mathbb{F}_q\) be a fixed finite field with \(q=p^s\) elements, where \(s\) is a positive integer and \(p>3\). Furthermore, let \(\mathbb{F}_q[t]\) be the ring of polynomials in \(t\) over \(\mathbb{F}_q\) and \(\mathbb{F}_q(t)\) its field of fractions. A cubic function field is a finite (separable) extension of \(\mathbb{F}_q(t)\) of degree \(3\). In this article, the authors describe an algorithm for tabulating all \(\mathbb{F}_q(t)\)-isomorphism classes of cubic function fields over \(\mathbb{F}_q\) of discriminant \(D\) with degree bounded above and with certain restrictions on \(D\). These restrictions on \(D\) are that \(D\) has either odd degree or even degree and the leading coefficient of \(-3D\) is a non-square in \(\mathbb{F}_q^*\).NEWLINENEWLINE The main theoretical result is a generalization of a theorem of \textit{H. Davenport} and \textit{H. Heilbronn} [Proc. R. Soc. Lond., Ser. A 322, 405--420 (1971; Zbl 0212.08101)] for cubic number fields to cubic function fields. The authors thus obtain a bijection between the \(\mathbb{F}_q(t)\)-isomorphism classes of the special cubic function fields of discriminant \(D\in \mathbb{F}_q[t]\) and certain equivalence classes of primitive irreducible binary cubic forms over \(\mathbb{F}_q[t]\) of discriminant \(D\). The latter classes can be easily enumerated. The new tabulation algorithm generalizes the tabulation algorithm for cubic number fields by \textit{K. Belabas} [Math. Comput. 66, No. 219, 1213--1237 (1997; Zbl 0882.11070)] to cubic function fields. The authors also list the corresponding tabulation algorithm together with a detailed complexity analysis and various tabulation results.