THE UNBOUNDED DENOMINATORS CONJECTURE

• 成果类型: Article
• 署名作者: Calegari, Frank; Dimitrov, Vesselin; Tang, Yunqing
• 署名单位: University of Chicago; California Institute of Technology; University of California System; University of California Berkeley
• 刊物名称: JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
• ISSN/ISSBN: 0894-0347
• DOI: 10.1090/jams/1053
• 发表日期: 2025
• 页码: 627-702
• 关键词: generalized modular-forms; fourier coefficients; noncongruence subgroups; Invariance; series

摘要

We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of SL2(Z). Our result includes also Masons 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 SL2(Z[1/p]), and a close description of the Fuchsian uniformization of the Riemann surface C\mu N.