Mathematische Annalen

, Volume 371, Issue 1–2, pp 41–126 | Cite as

Moduli interpretations for noncongruence modular curves

  • William Yun ChenEmail author


We consider the moduli of elliptic curves with G-structures, where G is a finite 2-generated group. When G is abelian, a G-structure is the same as a classical congruence level structure. There is a natural action of \(\text {SL}_2(\mathbb {Z})\) on these level structures. If \(\Gamma \) is a stabilizer of this action, then the quotient of the upper half plane by \(\Gamma \) parametrizes isomorphism classes of elliptic curves equipped with G-structures. When G is sufficiently nonabelian, the stabilizers \(\Gamma \) are noncongruence. Using this, we obtain arithmetic models of noncongruence modular curves as moduli spaces of elliptic curves equipped with nonabelian G-structures. As applications we describe a link to the Inverse Galois Problem, and show how our moduli interpretations explains the bad primes for the Unbounded Denominators Conjecture, and allows us to translate the conjecture into the language of geometry and Galois theory.



The author is grateful to Pierre Deligne, Winnie Li, John Voight, Jordan Ellenberg, Ching-Li Chai, and Christelle Vincent for their generous comments and helpful suggestions while revising this paper. He would also like to thank Hilaf Hasson and Jeff Yelton for many enlightening discussions where much confusion was both generated and dispersed. The paper was partially written during the author’s stay at the National Center for Theoretical Sciences (NCTS) in Taiwan, and revised during the author’s visit at ICERM, and postdoc at the IAS. He would like to thank NCTS, ICERM, and the IAS for their support and hospitality.


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Authors and Affiliations

  1. 1.Institute for Advanced StudyPrincetonUSA

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