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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



How branching properties determine modular equations

Author: Harvey Cohn
Journal: Math. Comp. 61 (1993), 155-170
MSC: Primary 11F11; Secondary 11Y16
MathSciNet review: 1195433
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Abstract: If a prime p is decomposed as ${x^2} + 4{y^2}$, the power ${2^m}||y$ can be determined by an algorithm of polynomial efficiency based on use of singular moduli from the modular equation of order 2. The properties of the modular functions required in this algorithm are simple branching and parametrization properties, which in turn define the modular functions and equations (essentially uniquely). The well-known equations of "Klein’s Icosahedron" and their Hecke analogues come into play here, and to some extent they can be uniquely characterized in this fashion. The extraneous cases which arise are in some sense interesting analogues of modular equations.

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Keywords: Klein and Hecke modular functions, modular equations
Article copyright: © Copyright 1993 American Mathematical Society