Rocky Mountain Journal of Mathematics

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

P. Rozenhart, M.J. Jacobson, Jr, and R. Scheidler

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Abstract

In this paper, we present extensive numerical data on quadratic function fields with non-zero 3-rank. We use a function field adaptation of a method due to Belabas for finding quadratic number fields of high 3-rank. Our algorithm relies on previous work for tabulating cubic function fields of bounded discriminant \cite {Pieter3} but includes a significant novel improvement when the discriminants are imaginary. We provide numerical data for discriminant degree up to 11 over the finite fields $\mathbb{F}_5, \mathbb{F}_7, \mathbb{F}_11$ and $\mathbb{F}_13$ and $\mathbb{F}_13$. In addition to presenting new examples of fields of minimal discriminant degree with a given 3-rank, we compare our data with a variety of heuristics on the density of such fields with a given 3-rank, which in most cases supports their validity.

Article information

Source
Rocky Mountain J. Math., Volume 45, Number 6 (2015), 1985-2022.

Dates
First available in Project Euclid: 14 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1457960344

Digital Object Identifier
doi:10.1216/RMJ-2015-45-6-1985

Mathematical Reviews number (MathSciNet)
MR3473164

Zentralblatt MATH identifier
1350.11095

Subjects
Primary: 11R11: Quadratic extensions 11R58: Arithmetic theory of algebraic function fields [See also 14-XX] 11R65: Class groups and Picard groups of orders 11Y40: Algebraic number theory computations

Keywords
Quadratic function field ideal class group three rank

Citation

Rozenhart, P.; Jr, M.J. Jacobson,; Scheidler, R. Computing quadratic function fields with high 3-rank via cubic field tabulation. Rocky Mountain J. Math. 45 (2015), no. 6, 1985--2022. doi:10.1216/RMJ-2015-45-6-1985. https://projecteuclid.org/euclid.rmjm/1457960344


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