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Transactions of the American Mathematical Society

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Algebraic curves uniformized by congruence subgroups of triangle groups


Authors: Pete L. Clark and John Voight
Journal: Trans. Amer. Math. Soc. 371 (2019), 33-82
MSC (2010): Primary 11F06
DOI: https://doi.org/10.1090/tran/7139
Published electronically: July 20, 2018
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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 $ \textup {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 $ \textup {PSL}_2(\mathbb{F}_q)$ and $ \textup {PGL}_2(\mathbb{F}_q)$ regularly as Galois groups.


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Additional Information

Pete L. Clark
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: pete@math.uga.edu

John Voight
Affiliation: Department of Mathematics and Statistics, University of Vermont, 16 Colchester Avenue, Burlington, Vermont 05401 – and – Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, New Hampshire 03755
Email: jvoight@gmail.com

DOI: https://doi.org/10.1090/tran/7139
Received by editor(s): September 12, 2016
Received by editor(s) in revised form: November 22, 2016
Published electronically: July 20, 2018
Additional Notes: The second author was supported by an NSF CAREER Award (DMS-1151047).
Article copyright: © Copyright 2018 American Mathematical Society