Algebraic curves uniformized by congruence subgroups of triangle groups

Clark, Pete; Voight, John

doi:10.1090/tran/7139

Algebraic curves uniformized by congruence subgroups of triangle groups
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by Pete L. Clark and John Voight PDF

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Abstract:

We construct certain subgroups of hyperbolic triangle groups which we call “congruence” subgroups. These groups include the classical congruence subgroups of $\mathrm {SL}_2(\mathbb {Z})$, Hecke triangle groups, and $19$ families of arithmetic triangle groups associated to Shimura curves. We determine the field of moduli of the curves associated to these groups and thereby realize the groups $\mathrm {PSL}_2(\mathbb {F}_q)$ and $\mathrm {PGL}_2(\mathbb {F}_q)$ regularly as Galois groups.

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