Counting cusps of subgroups of $\mathrm {PSL}_2(\mathcal {O}_K)$
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- by Kathleen L. Petersen
- Proc. Amer. Math. Soc. 136 (2008), 2387-2393
- DOI: https://doi.org/10.1090/S0002-9939-08-09262-9
- Published electronically: March 14, 2008
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Abstract:
Let $K$ be a number field with $r$ real places and $s$ complex places, and let $\mathcal {O}_K$ be the ring of integers of $K$. The quotient $[\mathbb {H}^2]^r\times [\mathbb {H}^3]^s/\mathrm {PSL}_2(\mathcal {O}_K)$ has $h_K$ cusps, where $h_K$ is the class number of $K$. We show that under the assumption of the generalized Riemann hypothesis that if $K$ is not $\mathbb {Q}$ or an imaginary quadratic field and if $i \not \in K$, then $\mathrm {PSL}_2(\mathcal {O}_K)$ has infinitely many maximal subgroups with $h_K$ cusps. A key element in the proof is a connection to Artin’s Primitive Root Conjecture.References
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Bibliographic Information
- Kathleen L. Petersen
- Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
- MR Author ID: 811372
- Email: petersen@mast.queensu.ca
- Received by editor(s): June 5, 2006
- Received by editor(s) in revised form: July 12, 2006, November 28, 2006, and June 11, 2007
- Published electronically: March 14, 2008
- Communicated by: Ted Chinburg
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2387-2393
- MSC (2000): Primary 11F23, 22E40, 11A07
- DOI: https://doi.org/10.1090/S0002-9939-08-09262-9
- MathSciNet review: 2390505