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Journal Articles Rocky Mountain Journal of Mathematics Year : 2015

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

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Abstract

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 $ \geq 1$, of imaginary or unusual discriminant $D$, for a fixed $|D| = q^{\deg(D)}$. We present numerical data for quadratic function fields over $\mathbb{F}_{5}, \mathbb{F}_{7}, \mathbb{F}_{11}$ and $\mathbb{F}_{13}$ with $\deg(D) \leq 11$. Our algorithm produces quadratic function fields of minimal genus for any given 3-rank. Our numerical data mostly agrees with the Friedman-Washington heuristics for quadratic function fields over the finite field $\mathbb{F}_{q}$ where $q \equiv -1 \pmod{3}$. The data for quadratic function fields over the finite field $\mathbb{F}_{q}$ where $q \equiv 1 \pmod{3}$ does not agree closely with Friedman-Washington, but does agree more closely with some recent conjectures of Malle.

Dates and versions

inria-00462008 , version 1 (08-03-2010)

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Pieter Rozenhart, Michael Jacobson Jr., Renate Scheidler. Computing quadratic function fields with high 3-rank via cubic field tabulation. Rocky Mountain Journal of Mathematics, 2015, 45 (6), pp.1985-2022. ⟨10.1216/RMJ-2015-45-6-1985⟩. ⟨inria-00462008⟩
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