Improved computation of fundamental domains for arithmetic Fuchsian groups

Rickards, James

doi:10.1090/mcom/3777

Improved computation of fundamental domains for arithmetic Fuchsian groups
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by James Rickards;

Math. Comp. 91 (2022), 2929-2954

DOI: https://doi.org/10.1090/mcom/3777

Published electronically: August 3, 2022

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Abstract:

A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the improvements by comparing running times versus the live Magma implementation.

References

Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Mikhail Belolipetsky, Tsachik Gelander, Alexander Lubotzky, and Aner Shalev, Counting arithmetic lattices and surfaces, Ann. of Math. (2) 172 (2010), no. 3, 2197–2221. MR 2726109, DOI 10.4007/annals.2010.172.2197
Michelle Chu and Han Li, Small generators of cocompact arithmetic Fuchsian groups, Proc. Amer. Math. Soc. 144 (2016), no. 12, 5121–5127. MR 3556258, DOI 10.1090/proc/13177
Lassina Dembélé and John Voight, Explicit methods for Hilbert modular forms, Elliptic curves, Hilbert modular forms and Galois deformations, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Basel, 2013, pp. 135–198. MR 3184337, DOI 10.1007/978-3-0348-0618-3_{4}
U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comp. 44 (1985), no. 170, 463–471. MR 777278, DOI 10.1090/S0025-5718-1985-0777278-8
Konstantin Golubev and Amitay Kamber, Cutoff on hyperbolic surfaces, Geom. Dedicata 203 (2019), 225–255. MR 4027593, DOI 10.1007/s10711-019-00433-5
Aurel Page, Computing arithmetic Kleinian groups, Math. Comp. 84 (2015), no. 295, 2361–2390. MR 3356030, DOI 10.1090/S0025-5718-2015-02939-1
The PARI Group, Univ. Bordeaux, Pari/gp version 2.14.0, 2022, available from http://pari.math.u-bordeaux.fr/.
James Rickards, Intersections of Closed Geodesics on Shimura Curves, ProQuest LLC, Ann Arbor, MI, 2021. Thesis (Ph.D.)–McGill University (Canada). MR 4336941
James Rickards, Fundamental domains for Shimura curves, https://github.com/JamesRickards-Canada/Fundamental-Domains-for-Shimura-curves, 2022.
Raphael S. Steiner, Small diameters and generators for arithmetic lattices in $\text {SL}_2(\mathbb {R})$ and certain Ramanujan graphs, https://arxiv.org/abs/2207.12684, 2022.
John Voight, Computing fundamental domains for Fuchsian groups, J. Théor. Nombres Bordeaux 21 (2009), no. 2, 469–491 (English, with English and French summaries). MR 2541438
John Voight, Quaternion algebras, Graduate Texts in Mathematics, vol. 288, Springer, Cham, [2021] ©2021. MR 4279905, DOI 10.1007/978-3-030-56694-4