Minimal polynomials of singular moduli
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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.References
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Additional Information
- Eric Errthum
- Affiliation: Department of Mathematics and Statistics, Winona State University, Winona, Minnesota 55987
- Email: eerrthum@winona.edu
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 83 (2014), 411-420
- MSC (2010): Primary 11G18; Secondary 11Y40
- DOI: https://doi.org/10.1090/S0025-5718-2013-02709-3
- MathSciNet review: 3120596
Dedicated: This paper is dedicated to my wife Kate.