Vol. 220, No. 1, 2005

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ISSN: 0030-8730
Kepler’s small stellated dodecahedron as a Riemann surface

Matthias Weber

Vol. 220 (2005), No. 1, 167–182
Abstract

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.

Keywords
Jacobians, flat structures, small stellated dodecahedron
Mathematical Subject Classification 2000
Primary: 30F30
Milestones
Received: 3 June 1999
Revised: 27 January 2004
Accepted: 30 March 2004
Published: 1 May 2005
Authors
Matthias Weber
Department of Mathematics
Department of Mathematics
Rawles Hall
Indiana University
Bloomington, IN 47405