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Noncongruence modular curves as Hurwitz spaces

William Y. Chen

TL;DR

The paper develops a unifying framework in which noncongruence modular curves are realized as Hurwitz moduli spaces for elliptic curves with nonabelian level structures, using stacks of G-structures and admissible G-covers. It establishes universality results (Asada, Ellenberg–McReynolds) that connect components of these stacks to classical modular curves, and it analyzes the arithmetic and geometric monodromy actions (mapping class group and Galois group) that shape their fields of definition and component structure. Through explicit examples (dihedral, metabelian, PSL2-type, and Sz(8)), the work highlights when noncongruence occurs, when components are congruence, and how these moduli spaces encode inverse Galois data, Higman invariants, and Burau representations. The framework also clarifies questions about compactifications, integral models, and potential Hecke-like structures in the noncongruence setting, while drawing deep connections to anabelian geometry and the arithmetic of nonabelian covers of elliptic curves. Overall, it provides a comprehensive, stack-theoretic approach to understanding noncongruence phenomena and their rich interplay with topology, algebraic geometry, and number theory.

Abstract

In this survey article we give an overview of how noncongruence modular curves can be viewed as Hurwitz moduli spaces of covers of elliptic curves at most branched above the origin. We describe some natural questions that arise, and applications of these ideas to the Inverse Galois Problem, Markoff triples and the arithmetic of Fourier coefficients for noncongruence modular forms.

Noncongruence modular curves as Hurwitz spaces

TL;DR

The paper develops a unifying framework in which noncongruence modular curves are realized as Hurwitz moduli spaces for elliptic curves with nonabelian level structures, using stacks of G-structures and admissible G-covers. It establishes universality results (Asada, Ellenberg–McReynolds) that connect components of these stacks to classical modular curves, and it analyzes the arithmetic and geometric monodromy actions (mapping class group and Galois group) that shape their fields of definition and component structure. Through explicit examples (dihedral, metabelian, PSL2-type, and Sz(8)), the work highlights when noncongruence occurs, when components are congruence, and how these moduli spaces encode inverse Galois data, Higman invariants, and Burau representations. The framework also clarifies questions about compactifications, integral models, and potential Hecke-like structures in the noncongruence setting, while drawing deep connections to anabelian geometry and the arithmetic of nonabelian covers of elliptic curves. Overall, it provides a comprehensive, stack-theoretic approach to understanding noncongruence phenomena and their rich interplay with topology, algebraic geometry, and number theory.

Abstract

In this survey article we give an overview of how noncongruence modular curves can be viewed as Hurwitz moduli spaces of covers of elliptic curves at most branched above the origin. We describe some natural questions that arise, and applications of these ideas to the Inverse Galois Problem, Markoff triples and the arithmetic of Fourier coefficients for noncongruence modular forms.

Paper Structure

This paper contains 49 sections, 57 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Noncongruence modular forms and the work of Atkin and Swinnerton-Dyer
  3. Noncongruence modular curves and nonabelian level structures
  4. Some fundamental questions
  5. Structure of the paper
  6. Conventions and notation
  7. The moduli of elliptic curves
  8. Topological properties of elliptic curves and punctured elliptic curves
  9. Fine moduli, coarse moduli, and the necessity of stacks
  10. The moduli stack of elliptic curves
  11. Moduli of framed elliptic curves
  12. Remarks on mirror-image modular curves
  13. Moduli of elliptic curves with Teichmüller markings
  14. Moduli stack of elliptic curves as a quotient stack
  15. Subgroups as coverings: the Galois correspondence and the Riemann existence theorem
  16. ...and 34 more sections

Key Result

Theorem 1.1

Every noncongruence modular curve appears as a quotient of the moduli space of an appropriately chosen $\pi$.The appropriate cover for a given modular curve can be found explicitly by the results of BER11. On the other hand, it can be remarkably difficult to recognize when a cover $\pi$ will result

Theorems & Definitions (134)

  • Theorem 1.1: Asada, Ellenberg-McReynolds, see §\ref{['ss_universality']}
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5: See Ols16
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 124 more