Ivrissimtzis, Ioannis and Singerman, David and Strudwick, James (2021) 'From Farey fractions to the Klein quartic and beyond.', Ars Mathematica Contemporanea, 20 (1). pp. 37-50.
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.
|Full text:||(AM) Accepted Manuscript|
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|Publisher Web site:||https://doi.org/10.26493/1855-3974.2046.cb6|
|Publisher statement:||This work is licensed under https://creativecommons.org/licenses/by/4.0/|
|Date accepted:||19 September 2020|
|Date deposited:||08 July 2021|
|Date of first online publication:||14 July 2021|
|Date first made open access:||08 July 2021|
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