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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Construction of hyperelliptic function fields of high three-rank


Authors: M. Bauer, M. J. Jacobson Jr., Y. Lee and R. Scheidler
Journal: Math. Comp. 77 (2008), 503-530
MSC (2000): Primary 11R11; Secondary 11R65, 11Y16, 11Y40, 14H05, 14H40
Published electronically: July 26, 2007
MathSciNet review: 2353964
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Abstract | References | Similar Articles | Additional Information

Abstract: We present several explicit constructions of hyperelliptic function fields whose Jacobian or ideal class group has large $ 3$-rank. Our focus is on finding examples for which the genus and the base field are as small as possible. Most of our methods are adapted from analogous techniques used for generating quadratic number fields whose ideal class groups have high $ 3$-rank, but one method, applicable to finding large $ l$-ranks for odd primes $ l \geq 3,$ is new and unique to function fields. Algorithms, examples, and numerical data are included.


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Additional Information

M. Bauer
Affiliation: Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
Email: mbauer@math.ucalgary.ca

M. J. Jacobson Jr.
Affiliation: Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
Email: jacobs@cpsc.ucalgary.ca

Y. Lee
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada, V5A 1S6
Email: yoonjinl@sfu.ca

R. Scheidler
Affiliation: Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
Email: rscheidl@math.ucalgary.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-07-02001-7
PII: S 0025-5718(07)02001-7
Keywords: Hyperelliptic function field, ideal class group, Jacobian, 3-rank
Received by editor(s): July 26, 2005
Received by editor(s) in revised form: November 8, 2006
Published electronically: July 26, 2007
Additional Notes: The first, second, and fourth authors were supported by NSERC of Canada
The third author was supported by an AWM-NSF Mentoring Grant
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.