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The following question was suggested by Sasha's answer to the following question : Is the derivative of a modular function a modular function .

Question. What are the modular functions with respect to $\Gamma(n)$ of weight zero?

I know that the $j$-invariant is of weight zero with respect to $\Gamma(1)$.

Is the modular function $\lambda$ of weight zero with respect to $\Gamma(2)$?

More generally, is a Hauptmodul for $\Gamma(n)$ of weight zero?

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For any $n$, the space $Y(n) = \Gamma(n) \backslash \mathcal{H}$ is a Riemann surface, and if you add in points for each cusp, you get a compact Riemann surface, the principal modular curve $X(n)$. The weight 0 modular functions of level $\Gamma(n)$ are then the meromorphic functions on this Riemann surface. This is a field, and since $X(n)$ is compact, it's finitely generated over $\mathbb{C}$. Nonetheless it is still rather large, e.g. it is not finite-dimensional over $\mathbb{C}$ or anything like that; rather, it'll be a finite Galois extension of the rational function field $\mathbb{C}(X)$, which is what you get for level $1$.

(This field is called the "modular function field of level $\Gamma(n)$". See the Wikipedia entry Modular curve for more information.)

For certain small $n$, this field is generated by one element over $\mathbb{C}$, which corresponds to the modular curve $X(n)$ having genus 0. That is what a Hauptmodul for $\Gamma(n)$ is: a modular function that generates the function field of $\Gamma(n)$. However these only exist for a few small $n$. So yes, a Hauptmodul is always of weight 0.

In particular, the $\lambda$ function, which is a Hauptmodul of level $\Gamma(2)$, has weight 0.

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Dear @David Loeffler,could you please suggest me some research papers or books where I can find more on Hauptmodul for $\Gamma(n)$, $\Gamma_0^{1}{n}$, j-invariant and Hauptmodul, j-invariant in terms of Dedekind eta function. Any explanation would be of great help. Thanks in advance –  Sushma May 9 at 8:04
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