Computing quadratic function fields with high 3-rank via cubic field tabulation

We present recent results on the computation of quadratic function fields with high 3-rank. Using a generalization of a method of Belabas on cubic field tabulation and a theorem of Hasse, we compute quadratic function fields with 3-rank ≥ 1, of imaginary or unusual discriminant D, for a fixed |D| = q. We present numerical data for quadratic function fields over F5, F7, F11 and F13 with deg(D) ≤ 11. Our algorithm produces quadratic function fields of minimal genus for any given 3-rank. Our numerical data mostly agrees with the FriedmanWashington heuristics for quadratic function fields over the finite field Fq where q ≡ −1 (mod 3). The data for quadratic function fields over the finite field Fq where q ≡ 1 (mod 3) does not agree closely with FriedmanWashington, but does agree more closely with some recent conjectures of Malle.