The modular curve of order N, relating j(z) and j(z/N), can be simplified by a system of involutions first due to Fricke. A computation is presented for finding the genus of the simplified curves by using the method of isometric circles due to Poincaré. This is an internal construction (in memory) for the fundamental domain. A survey of the results for N ≤ 310 confirms the 64 values of N (largest 119) for which the genus reduces to 0 and 65 values of N (largest 288) for which it reduces to 1. It also reveals 55 values of N (largest 390) for which the genus reduces to 2. While other surveys have been made, (e.g., by A. Ogg and by A.O.L. Atkin), the present one has the advantage of using minimal computational tools and theoretical techniques. The elliptic cases are interesting because in some instances e. g., N = 40,48,63,65,75, the involution process does not produce a genus reduction for the torus, and Poincaré’s method shows how this works geometrically. Also, in many cases the elliptic curves can be parametrised analytically using discriminant functions in a manner originally introduced by Fricke.
Elliptic Curve Fundamental Domain Modular Function Modular Curve Modular Equation
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