Moduli Interpretations for Noncongruence Modular Curves

Abstract

We define the notion of a $G$-structure for elliptic curves, where $G$ is a finite 2-generated group. When $G$ is abelian, a $G$-structure is the same as a classical congruence level structure. There is a natural action of $\text{SL}_2(\mathbb{Z})$ on these level structures. If $\Gamma$ is a stabilizer of this action, then the quotient of the upper half plane by $\Gamma$ parametrizes isomorphism classes of elliptic curves equipped with $G$-structures. When $G$ is "sufficiently" nonabelian, the stabilizers $\Gamma$ are noncongruence. As a result we realize noncongruence modular curves as moduli spaces of elliptic curves equipped with nonabelian $G$-structures. As applications we describe links to the Inverse Galois Problem, and show how our moduli interpretations explains the bad primes for the Unbounded Denominators Conjecture, and allows us to translate the conjecture into the language of geometry and Galois theory.