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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Half-step modular equations
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by Harvey Cohn PDF
Math. Comp. 64 (1995), 1267-1285 Request permission

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), $\gcd (a,b) = 1,ab|30$. 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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Math. Comp. 64 (1995), 1267-1285
  • MSC: Primary 11F03; Secondary 11F11
  • DOI: https://doi.org/10.1090/S0025-5718-1995-1284665-7
  • MathSciNet review: 1284665