If I have a modular curve, how does one in general find a Hauptmodul for this curve?
2026-04-01 14:27:54.1775053674
Hauptmoduls for modular curves
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As Pete mentions in the comments, computing Hauptmoduls for genus zero modular curves has a long history (results go back to work by Klein and Gierster published in 1879). Since those early works, there have been many papers on Hauptmodules. For a nice discussion about Hauptmodules (including references to the aforementioned papers of Klein and Gierster) see On Rationally Parametrized Modular Equations by Robert Maier, which includes a table of Hauptmoduls for $X_0(N)$ on page 11. Though one can construct Hauptmoduls for $X_0(N)$ using eta-functions, this does not work in general. In general one can instead use Siegel functions (or the closely related Klein forms) (see especially this paper by Sutherland and Zywina which explains a general procedure for constructing Hauptmoduls from Siegel functions starting on page 11). (One can also construct Hauptmodules from "generalized eta-functions", which are essentially Siegel functions.)