Abstract. We present a method for tabulating all cubic function fields over F q (t) whose discriminant D has either odd degree or even degree and the leading coefficient of −3D is a non-square in F * q , up to a given bound B on deg(D). Our method is based on a generalization of Belabas' method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms. The algorithm requires O(B 4 q B ) field operations as B → ∞. The algorithm, examples and numerical data for q = 5, 7, 11, 13 are included.

Abstract. We give a general method for tabulating all cubic function fields over Fq(t) whose discriminant D has odd degree, or even degree such that the leading coefficient of −3D is a non-square in F * q , up to a given bound on |D| = q deg(D) . The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields. We present numerical data for cubic function fields over F5 and over F7 with deg(D) ≤ 7 and deg(D) odd in both cases.

Abstract. 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.

This paper presents an algorithm for generating all imaginary and unusual discriminants up to a fixed degree bound that define a quadratic function field of positive 3-rank. Our method makes use of function field adaptations of a method due to Belabas for finding quadratic number fields of high 3-rank and of a refined function field version of a theorem due to Hasse. We provide numerical data for discriminant degree up to 11 over the finite fields F5, F7, F11 and F13. A special feature of our technique is that it produces quadratic function fields of minimal genus for any given 3-rank. Taking advantage of certain Fq(t)-automorphisms in conjunction with Horner's rule for evaluating polynomials significantly speeds up our algorithm in the imaginary case; this improvement is unique to function fields and does not apply to number field tabulation. These automorphisms also account for certain divisibility properties in the number of fields found with positive 3-rank. Our numerical data mostly agrees with the predicted heuristics of Friedman-Washington and partial results on the distribution of such values due to Ellenberg-Venkatesh-Westerland for quadratic function fields over the finite field Fq where q ≡ −1 (mod 3). The corresponding data for q ≡ 1 (mod 3) does not agree closely with the previously mentioned heuristics and results, but does agree more closely 1 arXiv:1003.1287v3 [math.NT] 3 May 2012 with some recent number field conjectures of Malle and some work in progress on proving such conjectures for function fields due to Garton.

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