The K(π,1)-conjecture implies the center conjecture for Artin groups

Jankiewicz, Kasia; Schreve, Kevin

doi:10.1016/j.jalgebra.2022.10.024

We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of

{SL}_{2} <#comment/> (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])

, and a close description of the Fuchsian uniformization

D (0, 1) / {Γ<#comment/>}_{N}

of the Riemann surface

C ∖<#comment/> {μ<#comment/>}_{N}

.

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