2005/05/01 by Matthias Weber · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Analytic Number Theory Research #Dodecahedron #Mathematics #Kepler #Riemann surface #Surface (topology) #Astronomy #Geometry #Physics #Stars
paper · pdf · doi:10.2140/pjm.2005.220.167
openalex publication_date 2005/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2025/11/06
We provide a new geometric computation for the Jacobian of the Riemann surface of genus 4 associated to the small stellated dodecahedron.Starting with Threlfall's description, we introduce other flat conformal geometries on this surface which are related to holomorphic 1-forms.They allow us to show that the Jacobian is isogenous to a fourfold product of a single elliptic curve whose lattice constant can be determined in two essentially different ways, yielding an unexpected relation between hypergeometric integrals.We also obtain a new platonic tessellation of the surface.