×

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
PDF BibTeX XML Cite
Full Text: EuDML EMIS