The unbounded denominators conjecture

Calegari, Frank; Dimitrov, Vesselin; Tang, Yunqing

doi:10.1090/jams/1053

The unbounded denominators conjecture
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by Frank Calegari, Vesselin Dimitrov and Yunqing Tang;

J. Amer. Math. Soc. 38 (2025), 627-702

DOI: https://doi.org/10.1090/jams/1053

Published electronically: February 6, 2025

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Abstract:

We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of $\operatorname {SL}_2(\mathbf {Z})$. 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 second main theorem, the congruence subgroup property of $\operatorname {SL}_2(\mathbf {Z}[1/p])$, and a close description of the Fuchsian uniformization $D(0,1)/\Gamma _N$ of the Riemann surface $\mathbf {C} \smallsetminus \mu _N$.

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The unbounded denominators conjecture
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Abstract: