From Farey fractions to the Klein quartic and beyond

Ivrissimtzis, Ioannis; Singerman, David; Strudwick, James

Abstract

In a paper published in 1878/79 Klein produced his famous 14-sided polygon representing the Klein quartic, his Riemann surface of genus 3 which has PSL(2,7) as its automorphism group. The construction and method of side pairings are fairly complicated. By considering the Farey map modulo 7 we show how to obtain a fundamental polygon for Klein’s surface using arithmetic. Now the side pairings are immediate and essentially the same as in Klein’s paper. We also extend his work from 7 to 11 as Klein also did in a follow-up paper of 1879.

Citation

Ivrissimtzis, I., Singerman, D., & Strudwick, J. (2021). From Farey fractions to the Klein quartic and beyond. Ars mathematica contemporanea (Spletna izd.), 20(1), 37-50. https://doi.org/10.26493/1855-3974.2046.cb6

• Journal Article Type: Article
• Acceptance Date: Sep 19, 2020
• Online Publication Date: Jul 14, 2021
• Publication Date: 2021
• Deposit Date: Jul 7, 2021
• Publicly Available Date: Jul 8, 2021
• Journal: Ars Mathematica Contemporanea
• Print ISSN: 1855-3966
• Electronic ISSN: 1855-3974
• Publisher: University of Primorska
• Peer Reviewed: Peer Reviewed
• Volume: 20
• Issue: 1
• Pages: 37-50
• DOI: https://doi.org/10.26493/1855-3974.2046.cb6
• Public URL: https://durham-repository.worktribe.com/output/1272634
• Related Public URLs: https://arxiv.org/abs/1909.08568