Abstract
We find plane models for all \(X_0(N)\), \(N\ge 2\). We observe a map from the modular curve \(X_0(N)\) to the projective plane constructed using modular forms of weight 12 for the group \(\Gamma _0(N)\); the Ramanujan function \(\Delta \), \(\Delta (N\cdot )\) and the third power of Eisestein series of weight 4, \(E_4^3\), and prove that this map is birational equivalence for every \(N\ge 2\). The equation of the model is the minimal polynomial of \(\Delta (N\cdot )/\Delta \) over \({\mathbb {C}}(j)\).
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Communicated by A. Constantin.
The author acknowledges Croatian Science Foundation Grant No. 9364.
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Kodrnja, I. On a simple model of \(X_0(N)\). Monatsh Math 186, 653–661 (2018). https://doi.org/10.1007/s00605-018-1161-3
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DOI: https://doi.org/10.1007/s00605-018-1161-3