Belyi Maps from Zeroes of Hypergeometric Polynomials

Abstract

The evaluation of low-degree hypergeometric polynomials to zero defines algebraic hypersurfaces in the affine space of the free parameters and the argument of the hypergeometric function. This article investigates the algebraic surfaces defined by the hypergeometric equation F12(−N,b;c;z)=0 with N=3 or N=4. As a captivating application, these surfaces parametrize certain families of genus 0 Belyi maps. Thereby, this article contributes to the systematic enumeration of Belyi maps.

Keywords

• Gauss hypergeometric function
• Belyi map (of genus 0)
• elliptic surfaces