From Farey fractions to the Klein quartic and beyond

TL;DR

This paper links Farey fractions to the construction of Klein's quartic and extends the approach to the case of 11, providing an arithmetic method for fundamental polygon derivation.

Contribution

It introduces an arithmetic approach to derive Klein's fundamental polygon from Farey maps and extends the method from 7 to 11.

Findings

Fundamental polygon for Klein's surface obtained via Farey map modulo 7.

Side pairings are simplified and match Klein's original construction.

Method extended from 7 to 11, aligning with Klein's subsequent work.

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.