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Computing quadratic function fields with high 3-rank via cubic field tabulation. (English) Zbl 1350.11095

This paper is continuation of previous work of the authors [Math. Comput. 81, No. 280, 2335–2359 (2012; Zbl 1290.11154)] and of the PhD Thesis of the first author [Fast tabulation of cubic function fields. Calgary: University of Calgary (PhD Thesis) (2009)]. Let \(D\) be a square-free polynomial in \({\mathbb F}_q [t]\) of positive degree. A polynomial \(F\in {\mathbb F}_q[t]\) is called imaginary if \(\deg (F)\) is odd, unusual if \(\deg(F)\) is even and \(\text{sgn}(F)\) (the leading coefficient of \(F\)) is a non-square in \({\mathbb F}^{\ast}_q\) and real if \(\deg(F)\) is even and \(\text{sgn}(F)\) is a square in \({\mathbb F}^{\ast}_q\). The main purpose of this paper is to obtain extensive numerical data on quadratic function fields of discriminant \(D\) with bounded degree, where \(-3D\) is imaginary or unusual, and non-zero \(3\)-rank. The algorithm is given in Section 3 (Algorithm 3.1). The rapid tabulation of all cubic function fields with bounded discriminant degree is key to this approach. In Section 4, the improved algorithm for quadratic function fields where \(-3D\) is imaginary is presented (Algorithm 4.4). Its complexity is studied in Section 5.
The Friedman-Washington heuristics [E. Friedman and L. C. Washington, in: Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 227–239 (1989; Zbl 0693.12013)] attempt to explain statistical observations about divisor class groups of quadratic function fields. The data produced in this paper yields evidence for the Friedman-Washington heuristics for \(q=5,11\) but for \(q=7,13\) the data does not agree closely with the Friedman-Washington heuristics. This is studied in the last three sections.

MSC:

11R11 Quadratic extensions
11R58 Arithmetic theory of algebraic function fields
11R65 Class groups and Picard groups of orders
11Y40 Algebraic number theory computations

Software:

NTL
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References:

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