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The Unbounded Denominators Conjecture

Frank Calegari, Vesselin Dimitrov, Yunqing Tang

TL;DR

This work resolves the Unbounded Denominators Conjecture for all noncongruence subgroups of $\mathrm{SL}_2(\mathbf{Z})$ by establishing a quantitative holonomy bound for integral power-series solutions tied to modular lambda, Nevanlinna theory, and Wohlfahrt-level growth. The authors develop an arithmetic holonomicity framework, with multiple proofs, and exploit an integrated mean-growth bound for universal covering maps $F_N: D(0,1) \to \mathbf{C} \setminus \mu_N$, together with a global equidistribution argument, to force a contradiction unless a given form is congruence. They extend the result to vector-valued modular forms, proving Mason’s conjecture in that setting, and connect the theory to rational conformal field theory via congruence properties. The approach blends Diophantine approximation, harmonic analysis on moduli, and the arithmetic of Fuchsian uniformizations, yielding a robust path from arithmetic holonomicity to modular-congruence classification with potential implications for related Galois and π1 questions in arithmetic geometry.

Abstract

We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of SL_2(Z). Our result includes also Mason's generalization of the original conjecture to the setting of vector-valued modular forms, thereby supplying a new path to the congruence property in rational conformal field theory. The proof involves a new arithmetic holonomicity bound of a potential-theoretic flavor, together with Nevanlinna's second main theorem, the congruence subgroup property of SL_2(Z[1/p]), and a close description of the Fuchsian uniformization D(0,1)/Γ_N of the Riemann surface C \setminus μ_N.

The Unbounded Denominators Conjecture

TL;DR

This work resolves the Unbounded Denominators Conjecture for all noncongruence subgroups of by establishing a quantitative holonomy bound for integral power-series solutions tied to modular lambda, Nevanlinna theory, and Wohlfahrt-level growth. The authors develop an arithmetic holonomicity framework, with multiple proofs, and exploit an integrated mean-growth bound for universal covering maps , together with a global equidistribution argument, to force a contradiction unless a given form is congruence. They extend the result to vector-valued modular forms, proving Mason’s conjecture in that setting, and connect the theory to rational conformal field theory via congruence properties. The approach blends Diophantine approximation, harmonic analysis on moduli, and the arithmetic of Fuchsian uniformizations, yielding a robust path from arithmetic holonomicity to modular-congruence classification with potential implications for related Galois and π1 questions in arithmetic geometry.

Abstract

We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of SL_2(Z). Our result includes also Mason's generalization of the original conjecture to the setting of vector-valued modular forms, thereby supplying a new path to the congruence property in rational conformal field theory. The proof involves a new arithmetic holonomicity bound of a potential-theoretic flavor, together with Nevanlinna's second main theorem, the congruence subgroup property of SL_2(Z[1/p]), and a close description of the Fuchsian uniformization D(0,1)/Γ_N of the Riemann surface C \setminus μ_N.

Paper Structure

This paper contains 54 sections, 53 theorems, 263 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A sketch of the main ideas
  3. The Diophantine principle
  4. Modularity and simultaneous uniformizations of $f$ and $\lambda$
  5. A finite local monodromy leads to an overconvergence
  6. The dimension bound can be leveraged with growing level $N$
  7. Nevanlinna theory for Fuchsian groups
  8. The arithmetic holonomicity theorem
  9. The auxiliary construction
  10. Extrapolation and proof of Lemma \ref{['meromorphic form']}
  11. Canonical factorization and proof of Theorem \ref{['abstract form']}
  12. A first alternative proof
  13. Lemma on the lexicographically lowest coefficient
  14. Extrapolation and first alternative proof of Theorem \ref{['abstract form']}
  15. A second alternative proof
  16. ...and 39 more sections

Key Result

Theorem 1.0.1

Let $N$ be any positive integer, and let $f(\tau) \in \mathbf{Z} \llbracket q^{1/N} \rrbracket$ for $q = \exp(\pi i \tau)$ be a holomorphic function on the upper half plane $\mathbf{H}$. Suppose there exists an integer $k$ and a finite index subgroup $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ such and suppose that $f(\tau)$ is meromorphic at the cusps, that is, locally extends to a meromorphic f

Figures (1)

  • Figure 5.2.5: A (partial) tiling of $D(0,1)$ by a fundamental domain for $\Phi_3$, together with a density plot of $|F^3_3 - 1|$ which vanishes at $z=1$.

Theorems & Definitions (131)

  • Theorem 1.0.1: Unbounded Denominators Conjecture
  • Lemma 1.1.2
  • Remark 1.1.10: Big $O$ and small $o$ notation, $\mathbf{N}$ and $\mathbf{N}_{> 0}$
  • Theorem 2.0.1
  • Definition 2.0.4
  • Corollary 2.0.5
  • proof
  • Lemma 2.0.8
  • Remark 2.0.10
  • Lemma 2.1.2
  • ...and 121 more