$\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$-action on level $\Gamma(N)$ modular forms

Question

Let $k\geq 0$, $N\geq 2$ be integers. I suspect that there is an action of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$ on the finite-dimensional $\mathbb{Q}$-vector space of holomorphic modular forms of level $\Gamma(N)$ and weight $k$ such that the invariant subspace is the space of holomorphic modular forms of level $\Gamma(1)$ and weight $k$ (considered as level $\Gamma(N)$ modular forms).

I think so because in the moduli interpretation of the modular curve $X(N)$ the group $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$ acts on the set of pairs of generators of the $N$-torsion group on a generalized elliptic curve (but for $N=2$ the curve is stacky so I am not sure).

First question: does such an action indeed exist and where can I read more about it? Second question: can I explicitly write down what this action does to the Fourier coefficients of modular forms?

Answer 1

There is an action of $\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$. Let $f$ be a modular form of weight $k$, invariant for $\Gamma(N)$. Suppose $\gamma \in \text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ and lift $\gamma$ to a matrix $\widetilde{\gamma} \in \text{SL}_2(\mathbb{Z})$ (this is the hard part of the strong approximation theorem).

Then $f|_k \widetilde{\gamma}$ is clearly independent of the choice of lift (another choice differs on the left by something in $\Gamma(N)$), and it's invariant for the action of $\Gamma(N)$ because $\Gamma(N)$ is normal in $\text{SL}_2(\mathbb{Z})$. I'm not sure if there's a nice formula for the Fourier coefficients -- probably there is!

Your argument with the moduli interpretation suggests this extends to an action of $\mathbb{GL}_2(\mathbb{Z}/N\mathbb{Z})$, but this is misleading because the fine moduli space for this curve is geometrically disconnected and its complex points are isomorphic to $\phi(N)$ copies of $\mathcal{H}/\Gamma(N)$. This is where you get an action of $GL_2$. See more at The modular curve X(N).

Answer 2

The Riemann surface $X(N)$ can be obtained from $Y(N)$ plus its few cusps $$Y(N)= \{ \Bbb{C}^*(\Bbb{Z}+z\Bbb{Z}), \Bbb{C}^*(p+\Bbb{Z}+z\Bbb{Z}) ,\Bbb{C}^*(q+\Bbb{Z}+z\Bbb{Z})\}$$ where $p, q \in \Bbb{C}/(\Bbb{Z}+z\Bbb{Z})$ are two points that generate the $N$ torsion of the complex torus $\Bbb{C}/(\Bbb{Z}+z\Bbb{Z})$ and $\Bbb{C}^*(\Bbb{Z}+z\Bbb{Z})$ is the set of all lattices (=complex tori) isomorphic to $\Bbb{Z}+z\Bbb{Z}$.

If $ \pmatrix{a & b \\ c & d} \in GL_2(\Bbb{Z}/N\Bbb{Z})$ you might want to try the map $$( z,p,q) \mapsto ( z, ap+bq, cp+dq)$$ which does permute the points of $Y(N)$.

If this map is holomorphic then it is continuous, and since locally $z,p,q$ are complex numbers, the map is given by some $a,b,c,d \in \Bbb{Z}$, which means it extends holomorphically to the map $$\Bbb{Z}+z\Bbb{Z} \mapsto (a+bz)\Bbb{Z}+(c+dz) \Bbb{Z} $$ and that map is well-defined on $\Bbb{C}^* (\Bbb{Z}+z\Bbb{Z})$ iff $ \pmatrix{a & b \\ c & d} \in GL_2(\Bbb{Z})= \pm SL_2(\Bbb{Z})$.

Whence the group of holomorphic automorphisms of $Y(N)$ is $$\pm SL_2(\Bbb{Z}) / \ker(\pm SL_2(\Bbb{Z}) \to GL_2(\Bbb{Z}/N\Bbb{Z})) = \pm SL_2(\Bbb{Z}/N\Bbb{Z})$$