= Compactified modular curve
{title2=$X(\Gamma)$}
For a finite-index subgroup of the <modular group>, the quotient of the <complex upper half-plane> becomes a compact <Riemann surface> after adjoining its cusp classes. At an elliptic fixed point use the quotient coordinate for the finite stabilizer; at a cusp use its exponential local parameter. Weight-zero <modular forms> descend to <holomorphic functions> on this compact surface, and hence are constant by the <maximum modulus principle>.
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