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

 

Minimal polynomials of singular moduli


Author: Eric Errthum
Journal: Math. Comp. 83 (2014), 411-420
MSC (2010): Primary 11G18; Secondary 11Y40
Published electronically: May 6, 2013
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Abstract: Given a properly normalized parametrization of a genus-0 modular curve, the complex multiplication points map to algebraic numbers called singular moduli. In both cases there are known algorithms for algebraically computing the rational norms of the singular moduli without relying on the the recognition of a decimal or $ p$-adic expansion as a rational number. We demonstrate a method of extending these norm algorithms to determine the minimal polynomial of the singular moduli below a discriminant threshold. We then use these minimal polynomials to compute the algebraic $ abc$-ratios for the singular moduli.


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Additional Information

Eric Errthum
Affiliation: Department of Mathematics and Statistics, Winona State University, Winona, Minnesota 55987
Email: eerrthum@winona.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2013-02709-3
PII: S 0025-5718(2013)02709-3
Received by editor(s): November 14, 2010
Received by editor(s) in revised form: October 25, 2011, and May 2, 2012
Published electronically: May 6, 2013
Dedicated: This paper is dedicated to my wife Kate.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.