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- Mar 22, 2013

- 573

Now, define an operation \(\displaystyle \boxplus\) over the points of the 2-sphere \(\displaystyle \Gamma\) by constructing \(\displaystyle A \boxplus B\) by joining them and finding the intersection of \(\displaystyle AB\) with the equatorial plane of \(\displaystyle \Gamma\); call it \(\displaystyle C\) and construct \(\displaystyle \zeta^{-1}(C)\) to map it on \(\displaystyle \Gamma\) again through usual methods.

Note that this creates a group \(\displaystyle \Gamma\) where the elements are the points on the surface of the sphere. One can easily check that it is also abelian.

Now, as the point at infinity creates a projective version of this arithmetic, it is a projective variety, as well as an abelian one. So if one applies the Mordell-Weil theorem, we get an evidence of finiteness of the generating set.

I am trying to get a structural information of this peculiarly constructed group. I don't have any specific questions at this moment, just a confirmation of my work up to the last statements. It is also very much appreciated if one can derive a strong result concerning this.

Balarka

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