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Principal transformations between Riemann surfaces. (English) Zbl 0805.30033

Let \(X\) and \(Y\) be compact or open Riemann surfaces, \(M(X)\), \(M(Y)\) be their meromorphic function fields. A bijection \(\Phi\) of \(X\) to \(Y\) is a principal transformation provided that for every divisor \(D\) on \(X\), \(D\) is principal if and only if \(\Phi (D)\) is a principal divisor on \(Y\). We say that a principal transformation is special if it is induced by an abstract field isomorphism of meromorphic function fields of two Riemann surfaces. In this paper the author proves the following two facts: “For infinitely many pairs of compact Riemann surfaces \(X,Y\) of genus one, there exist nonspecial principal transformations” and “Let \(X\) and \(Y\) be compact Riemann surfaces of genus \(g>1\). If there exists a continuous principal transformation \(\Phi\) of \(X\) to \(Y\), then \(X\) and \(Y\) are conformally equivalent and \(\Phi\) is special”.
Reviewer: T.Kato (Yamaguchi)

MSC:

30F10 Compact Riemann surfaces and uniformization
14H05 Algebraic functions and function fields in algebraic geometry
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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