## A Fuchsian group proof of the hyperellipticity of Riemann surfaces of genus 2.(English)Zbl 1059.30034

By the uniformization theorem, a compact Riemann surface $$S$$ of genus 2 can be viewed as $$H/K$$, where $$H$$ is the upper half plane and $$K$$ is a discrete subgroup of the real Möbius transformations ($$K$$ is called a Fuchsian group). For suitable canonical and normalized generators of K it is shown that a specific Möbius transformation induces an automorphism $$J$$ on $$S$$ such that the quotient space $$S/\langle J\rangle$$ has genus zero, a property that characterizes hyperelliptic Riemann surfaces of any genus greater than 1.

### MSC:

 30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) 30F10 Compact Riemann surfaces and uniformization
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