From Farey fractions to the Klein quartic and beyond

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.