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March 9th, 2013, 05:41 AM   #1
Joined: Mar 2013

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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 Riemann-Hurwitz 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?

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riemann, surfaces

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