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Fricke’s two-valued modular equations

Author: Harvey Cohn
Journal: Math. Comp. 51 (1988), 787-807
MSC: Primary 11F03; Secondary 11F27
MathSciNet review: 935079
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Abstract: The modular equation of order b is a polynomial relation between $j(z)$ and $j(z/b)$, which has astronomically large coefficients even for small values of b. Fricke showed that a two-valued relation exists for 37 small values of b. This relation would have much smaller coefficients and would also be convenient for finding singular moduli. Although Fricke produced no two-valued relations explicitly (no doubt because of the tedious amount of algebraic manipulation), they are now found by use of MACSYMA. For 31 cases ranging from $b = 2$ to 49, Fricke provided the equations necessary to generate the relations (with two corrections required). The remaining six cases (of order 39, 41, 47, 50, 59, 71) require an extension of Fricke’s methods, using the discriminant function, theta functions, and power series approximations.

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Keywords: Modular equation, singular moduli, theta-functions
Article copyright: © Copyright 1988 American Mathematical Society