From Farey Fractions to the Klein Quartic and beyond
From Farey Fractions to the Klein Quartic and beyond
In his 1878/79 paper "Ueber die transformation siebenter ordnung der elliptischen functionen", 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 this idea from 7 to 11 as Klein attempted to do in his follow up paper "Ueber die transformation elfter ordnung der elliptischen functionen", in 1879.
Ivrissimtzis, Ioannis
af7d236c-09e2-4b93-baa5-35fdece8a758
Singerman, David
3eeb0783-c87c-4405-81d7-e80ae4c15f8b
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
[Unknown type: UNSPECIFIED]
Abstract
In his 1878/79 paper "Ueber die transformation siebenter ordnung der elliptischen functionen", 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 this idea from 7 to 11 as Klein attempted to do in his follow up paper "Ueber die transformation elfter ordnung der elliptischen functionen", in 1879.
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From Farey fractions to the Klein quartic and beyond arXiv version
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e-pub ahead of print date: 18 September 2019
Identifiers
Local EPrints ID: 467604
URI: http://eprints.soton.ac.uk/id/eprint/467604
ISSN: 2331-8422
PURE UUID: 7018e09b-cba9-4746-adc7-c0356d76e166
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Date deposited: 15 Jul 2022 16:30
Last modified: 16 Mar 2024 21:14
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Author:
Ioannis Ivrissimtzis
Author:
James Strudwick
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