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rbwang1225
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##\mathbb Z_{\frac{m}n}## orbifold
Consider the identification ##z\sim ze^{2\pi i \frac{m}n}##, where ##m## and ##n## are relatively prime integers. Determine a fundamental domain for the identification.
Given two relatively prime integers ##a## and ##b##, there exist integers ##m## and ##n## s.t. ##ma+nb=1##.
By considering cases of small ##m,n##'s, I conclude that the fundamental domain is ##0\le arg(z)<gcd(\frac{2m}n,2)##, but I can't give a more rigorous proof.
I guess that needs some knowledge of basic number theory.
Any advices would be very appreciated.
Regards.
Homework Statement
Consider the identification ##z\sim ze^{2\pi i \frac{m}n}##, where ##m## and ##n## are relatively prime integers. Determine a fundamental domain for the identification.
Homework Equations
Given two relatively prime integers ##a## and ##b##, there exist integers ##m## and ##n## s.t. ##ma+nb=1##.
The Attempt at a Solution
By considering cases of small ##m,n##'s, I conclude that the fundamental domain is ##0\le arg(z)<gcd(\frac{2m}n,2)##, but I can't give a more rigorous proof.
I guess that needs some knowledge of basic number theory.
Any advices would be very appreciated.
Regards.