From Farey fractions to the Klein quartic and beyond
From Farey fractions to the Klein quartic and beyond
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.
37-50
Ivrissimtzis, Ioannis
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Singerman, David
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Strudwick, James
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Ivrissimtzis, Ioannis
af7d236c-09e2-4b93-baa5-35fdece8a758
Singerman, David
3eeb0783-c87c-4405-81d7-e80ae4c15f8b
Strudwick, James
2ea49295-91bf-4647-ad4f-a6aa9a391a1c
Ivrissimtzis, Ioannis, Singerman, David and Strudwick, James
(2021)
From Farey fractions to the Klein quartic and beyond.
Ars Mathematica Contemporanea, 20 (1), .
(doi:10.26493/1855-3974.2046.cb6).
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.
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Accepted/In Press date: 21 September 2020
e-pub ahead of print date: 14 July 2021
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Local EPrints ID: 475538
URI: http://eprints.soton.ac.uk/id/eprint/475538
ISSN: 1855-3966
PURE UUID: 21febadb-6229-4ee3-810a-69117052cdc9
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Date deposited: 21 Mar 2023 17:41
Last modified: 17 Mar 2024 01:05
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Author:
Ioannis Ivrissimtzis
Author:
James Strudwick
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