Moduli Interpretations for Noncongruence Modular Curves

TL;DR

This paper introduces a new moduli interpretation of noncongruence modular curves via nonabelian G-structures on elliptic curves, linking them to Galois theory and the Inverse Galois Problem.

Contribution

It defines G-structures for elliptic curves, generalizing classical level structures, and shows how noncongruence modular curves can be realized as moduli spaces of such structures.

Findings

Noncongruence modular curves are moduli spaces of elliptic curves with nonabelian G-structures.

Connections established between these moduli spaces, the Inverse Galois Problem, and the Unbounded Denominators Conjecture.

The framework explains bad primes and translates conjectures into geometric and Galois-theoretic language.

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.