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Title:Infrastructure, Arithmetic, and Class Number Computations in Purely Cubic Function Fields of Characteristic at Least 5
Author(s):Landquist, Eric
Doctoral Committee Chair(s):Zaharescu, Alexandru
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
Abstract: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.
Issue Date:2009
Type:Text
Language:English
Description:194 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2009.
URI:http://hdl.handle.net/2142/86922
Other Identifier(s):(MiAaPQ)AAI3363008
Date Available in IDEALS:2015-09-28
Date Deposited:2009


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