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.


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