Pure Maths Colloquium - From Farey fractions to the Klein quartic - Dr Ioannis Ivrissimtzis (Durham) Seminar

Event details

Abstract: The Farey map is a triangulation of the extended complex upper-half plane with vertices the rationals and infinity and edges the semi-circles with endpoints a/b and c/d, where ad-bc is +1 or -1. In this talk we discuss the construction of the Klein quartic, a compact Riemann surface of genus 3, as the quotient of the Farey map by the principal congruence subgroup mod 7 of the modular group. Our construction is very similar to Klein's original construction in his seminal 'On the order-seven transformation of elliptic functions', as they are both based on 7 pair identifications of the 14 edges of a polygon. The main difference is that in our case, each edge of the fundamental polygon is the union of three Farey edges, while in Klein's construction each edge is the union of one Farey edge and a semi-circle with endpoints a/b and c/d, where now ad-bc is +2 or -2. The higher uniformity of our construction allows for an easier extension to higher order curves and we use it to obtain drawings of fundamental polygons for the surface with order-eleven symmetry introduced by Klein in his follow-up paper. (joint work with David Singerman and James Strudwick)