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Mathematics of Computation

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Half-step modular equations

Author: Harvey Cohn
Journal: Math. Comp. 64 (1995), 1267-1285
MSC: Primary 11F03; Secondary 11F11
MathSciNet review: 1284665
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Abstract: The classical modular equations relating Klein-Weber's $ j(\tau )$ to $ j(b\tau )$ can be computed as the composition of two "half-step" equations relating $ {j_m}(\tau )$ and $ {j_m}(\tau \sqrt b )$, where $ {j_m}$ is an extended modular function (corresponding to $ \tau \to \tau + \sqrt m ,\tau \to - 1/\tau $, et al.). The half-step equations are easily constructed and manipulated in computer algebra. The cases computed here are b prime, $ m = a$ (or ab), $ \gcd (a,b) = 1,ab\vert 30$. This includes many cases where the property of "normal parametrization" occurs, which is of interest in class field theory. Extended modular functions have found recent application in group character theory but they arose in the present context as traces at $ \infty $ of Hilbert modular equations.

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Keywords: Klein and Hecke modular functions, Atkin-Lehner involutions, modular equations
Article copyright: © Copyright 1995 American Mathematical Society

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