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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the arithmetic dimension of triangle groups
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by Steve Nugent and John Voight PDF
Math. Comp. 86 (2017), 1979-2004 Request permission

Abstract:

Let $\Delta =\Delta (a,b,c)$ be a hyperbolic triangle group, a Fuchsian group obtained from reflections in the sides of a triangle with angles $\pi /a,\pi /b,$ $\pi /c$ drawn on the hyperbolic plane. We define the arithmetic dimension of $\Delta$ to be the number of split real places of the quaternion algebra generated by $\Delta$ over its (totally real) invariant trace field. Takeuchi has determined explicitly all triples $(a,b,c)$ with arithmetic dimension $1$, corresponding to the arithmetic triangle groups. We show more generally that the number of triples with fixed arithmetic dimension is finite, and we present an efficient algorithm to completely enumerate the list of triples of bounded arithmetic dimension.
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Additional Information
  • Steve Nugent
  • Affiliation: Lady Margaret Hall College, Norham Gardens, Oxford OX2 6QA, United Kingdom
  • Email: steve.nugent@me.com
  • John Voight
  • Affiliation: 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): November 7, 2015
  • Received by editor(s) in revised form: January 26, 2016
  • Published electronically: November 8, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 86 (2017), 1979-2004
  • MSC (2010): Primary 11F03, 20H10; Secondary 11R52, 14G35
  • DOI: https://doi.org/10.1090/mcom/3147
  • MathSciNet review: 3626545