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arXiv · 2109.09040

The Unbounded Denominators Conjecture

Frank Calegari·Vesselin Dimitrov·Yunqing Tang
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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)/\Gamma_N of the Riemann surface C \setminus \mu_N.

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Frank Calegari, Vesselin Dimitrov, Yunqing Tang. 2021-09-19. The Unbounded Denominators Conjecture. https://arxiv.org/abs/2109.09040

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