Vol. 220, No. 1, 2005

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 331: 1
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
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