Algebraic curves uniformized by congruence subgroups of triangle groups
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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.References
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Additional Information
- Pete L. Clark
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 767639
- 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
- MR Author ID: 727424
- ORCID: 0000-0001-7494-8732
- Email: jvoight@gmail.com
- 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).
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 33-82
- MSC (2010): Primary 11F06
- DOI: https://doi.org/10.1090/tran/7139
- MathSciNet review: 3885137