Let $G$ be a finite 2-generated group. In this paper I study Teichmuller structures of level $G$ (or just $G$-structures) on elliptic curves $E$, which roughly correspond to a $G$-torsor on $E$, etale away from the origin. The corresponding moduli stacks $\mathcal{M}(G)$ are defined over $\mathbb{Q}$ (or more generally any $\mathbb{Z}[1/|G|]$-scheme), and over $\mathbb{C}$, $\mathcal{M}(G)_{\mathbb{C}}$ is a disjoint union of "modular curves" $\mathcal{H}/\Gamma$, where $\Gamma\le\text{SL}_2(\mathbb{Z})$ is finite index. If $G$ is abelian, then $\Gamma$ is a congruence subgroup, and one recovers the standard congruence modular curves. If $\Gamma$ is sufficiently nonabelian, then $\mathcal{M}(G)_{\mathbb{C}}$ is a union of noncongruence modular curves. By a result of Asada, for every finite index $\Gamma$, $\mathcal{H}/\Gamma$ can be realized as a component of such a moduli space, for a suitable $G$. As applications we outline a connection to the inverse Galois problem, and show that noncongruence modular forms for $\Gamma$ have bounded denominators at primes not dividing $|G|$.


If $G$ is a 2-generated abelian group, then it is easy to see that $G$ structures correspond to classical congruence level structures. When $G$ is nonabelian, it is often difficult to tell if the components of $\mathcal{M}(G)$ will be congruence or noncongruence. In this paper, we show that if $G$ is metabelian, then $G$-structures are again equivalent to classical congruence level structures. We do this by studying the outer automorphism group $\text{Out}(\widehat{M})$ of the rank 2 free profinite metabelian group $\widehat{M}$, which can be identified with the maximal pro-metabelian quotient of the fundamental group of a punctured elliptic curve over $\overline{\mathbb{Q}}$, and show that the natural image of $\pi_1(\mathcal{M}(1)_{\mathbb{Q}})$ in $\text{Out}(\pi_1(E^\circ_{\overline{\mathbb{Q}}})^{metabelian})\cong\text{Out}(\widehat{M})$ is isomorphic to its image in $\text{Out}(\pi_1(E^\circ_{\overline{\mathbb{Q}}})^{ab}) = \text{GL}_2(\widehat{\mathbb{Z}})$, where $\mathcal{M}(1)$ is the moduli stack of elliptic curves. The result is perhaps best expressed as follows: If $G$ is a finite 2-generated metabelian group of exponent $e$, then there is a sandwich: $$\mathcal{M}((\mathbb{Z}/e^2)^2)_{\mathbb{Q}}\rightarrow\mathcal{M}(G)_\mathbb{Q}\rightarrow\mathcal{M}(G^{ab})_\mathbb{Q}$$ where the second map is surjective, and the first map is surjective onto a connected component of $\mathcal{M}(G)_\mathbb{Q}$. Moreover, we show that all the components of $\mathcal{M}(G)_\mathbb{Q}$ are isomorphic, and so this sandwich gives a "complete" picture of $\mathcal{M}(G)_\mathbb{Q}$.


In these notes I give a detailed expository account of Katz's definition of modular forms and the $q$-expansion principle, in a way which makes sense for noncongruence subgroups of $SL(2,\mathbb{Z})$, or more precisely, for general stacks finite etale over the moduli stack of elliptic curves.