Half-step modular equations

Cohn, Harvey

doi:10.1090/S0025-5718-1995-1284665-7

Half-step modular equations
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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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