The Unbounded Denominators Conjecture
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866929500967665664 |
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| author | Calegari, Frank Dimitrov, Vesselin Tang, Yunqing |
| author_facet | Calegari, Frank Dimitrov, Vesselin Tang, Yunqing |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_09040 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Unbounded Denominators Conjecture Calegari, Frank Dimitrov, Vesselin Tang, Yunqing Number Theory 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. |
| title | The Unbounded Denominators Conjecture |
| topic | Number Theory |
| url | https://arxiv.org/abs/2109.09040 |