March 9th, 2013, 05:41 AM  #1 
Newbie Joined: Mar 2013 Posts: 1 Thanks: 0  Riemann surfaces
A problem in Elliptic Curves by McKean and Moll asks for a justification that the Riemann surface associated with the equation w^3 = z^2 + 1/z^2 has genus 4. I'm trying to construct the surface by using sheets and branch cuts, but without success. According to the RiemannHurwitz formula, a good possibility seems to be degree = 6, total ramification index (sum of the multiplicities each decreased by 1) = 18. This would give g = 4. Topologically the surface seems to consist of a tubular ring with 3 tubular handles attached. There appear to be 6 branch points on the sphere (CP^1). Can someone specify the slits (branch cuts) in the sheets and possibly how these are glued together? Thanks. 

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riemann, surfaces 
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