On the isomorphism theorem of the meromorphic function fields (Q800524)

This paper is concerned with compact Riemann surfaces whose fields of meromorphic functions are isomorphic (but which are not necessarily related by a conformal or anticonformal bijection). For the case of genus 1, cf. \textit{M. Heins}, Complex function theory (1968; Zbl 0155.11501), p. 391. For subsequent work, cf. \textit{M. Nakai} and \textit{L. Sario}, Hokkaido Math. J. 10, Special Issue, 531-545 (1981; Zbl 0492.30033). In this paper the set \(I_ g\cap H_ g\) is studied. Here \(I_ g\) is the set of compact Riemann surfaces R of genus g (\(\geq 2)\) having the property that a Riemann surface S whose field of meromorphic functions is isomorphic to the corresponding field for R is conformally equivalent (in the above sense) to R; \(H_ g\) is the set of hyperelliptic Riemann surfaces of genus g. A theorem concerning the zero Lebesgue measure of the set of parameters in the normalized representations \(y^ 2=x(x-1)\prod^{2g- 1}_{1}(x-a_ k)\) associated with the members of \(I_ g\cap H_ g\) is treated. The study appeals to a paper of the author ''A counterexample on the isomorphism theorem of meromorphic functions'', given in the references with merely the indication ''to appear''.