Infrastructure, Arithmetic, and Class Number Computations in Purely Cubic Function Fields of Characteristic at Least 5

Finally, we describe methods to compute the divisor class number, h, of K, and in the case that O has unit rank 1 or 2, the regulator and ideal class number of O as well. A method of Scheidler and Stein [SS07, SS08] determines sharper upper and lower bounds on h, for a given cubic function field, than those given by the Hasse-Weil Theorem. We then employ Shanks' Baby Step-Giant Step algorithm [Sha71] and Pollard's Kangaroo method [Pol78], to search this interval and compute the desired invariants for purely cubic function fields of unit rank 0 and 1. The total complexity of the method to compute these invariants is O (q(2 g-1)/5+epsilon(g )) ideal operations as q → infinity, where 0 ≤ epsilon( g) ≤ 1/5. With this approach, we computed the 28 decimal digit divisor class numbers of two purely cubic function fields of genus 3: one of unit rank 0 and one of unit rank 1. We also computed the 25 decimal digit divisor class numbers of two purely cubic function fields of genus 4: one of unit rank 0 and one of unit rank 1. In the unit rank 1 examples, we factored the divisor class numbers into the ideal class numbers and the respective 26 and 24 decimal digit S-regulators. We believe that these are the largest divisor class numbers ever computed for a cubic function field of genus at least 4 and the largest regulators ever computed for any cubic function field, respectively.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Landquist, Eric

Contributors dc:contributor

• Zaharescu, Alexandru

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3363008

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86922