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Award ID contains: 2231958

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  1. ; ; (, Journal of the American Mathematical Society)
    We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of SL 2 ⁡<#comment/> ( Z ) \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 SL 2 ⁡<#comment/> ( Z [ 1 / p ] ) \operatorname {SL}_2(\mathbf {Z}[1/p]) , and a close description of the Fuchsian uniformization D ( 0 , 1 ) / Γ<#comment/> N D(0,1)/\Gamma _N of the Riemann surface C ∖<#comment/> μ<#comment/> N \mathbf {C} \smallsetminus \mu _N . 
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    Free, publicly-accessible full text available February 6, 2026
  2. ; (, Journal of Number Theory)
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