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

 

Explicit bounds and heuristics on class numbers in hyperelliptic function fields


Authors: Andreas Stein and Edlyn Teske
Journal: Math. Comp. 71 (2002), 837-861
MSC (2000): Primary 11Y16, 11Y40, 11R29, 11R58; Secondary 11M38, 11R65
Published electronically: October 4, 2001
MathSciNet review: 1885633
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Abstract: In this paper, we provide tight estimates for the divisor class number of hyperelliptic function fields. We extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to $g$ with the help of new results. We thus obtain a faster method of computing regulators and class numbers. Furthermore, we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number. These heuristics are based on recent results by Katz and Sarnak. Our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest.


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Additional Information

Andreas Stein
Affiliation: University of Illinois at Urbana-Champaign, Department of Mathematics, 1409 West Green Street. Urbana, Illinois 61801
Email: andreas@math.uiuc.edu

Edlyn Teske
Affiliation: University of Waterloo, Department of Combinatorics and Optimization, Waterloo, Ontario, Canada N2L 3G1
Email: eteske@math.uwaterloo.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-01-01385-0
PII: S 0025-5718(01)01385-0
Keywords: Hyperelliptic function field, class numbers, regulator, truncated Euler products
Received by editor(s): July 27, 1999
Received by editor(s) in revised form: August 2, 2000
Published electronically: October 4, 2001
Article copyright: © Copyright 2001 American Mathematical Society