Mathematics – Number Theory
Scientific paper
2010-03-05
Mathematics
Number Theory
19 pages. Typos fixed, references added, better algorithm timings included in this version. Submitted for publication
Scientific paper
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.
Michael Jacobson Jr.
Rozenhart Pieter
Scheidler Renate
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