From Farey fractions to the Klein quartic and beyond

Ioannis Ivrissimtzis, David Singerman, James Strudwick

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.


Keywords


Riemann surfaces, Klein quartic, regular maps, Farey tessellation, modular group, principal congruence subgroups

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DOI: https://doi.org/10.26493/1855-3974.2046.cb6

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications