Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Half-step modular equations

Author: Harvey Cohn
Journal: Math. Comp. 64 (1995), 1267-1285
MSC: Primary 11F03; Secondary 11F11
MathSciNet review: 1284665
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11F03, 11F11

Retrieve articles in all journals with MSC: 11F03, 11F11

Additional Information

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