Finite degree holomorphic covers of compact Riemann surfaces
Let \(X_1, X_2\) be compact Riemann surfaces and \(K>1\). Here the authors prove the existence of an arithmetic Riemann surface \(Y\) (i.e. a Riemann surface defined over an algebraic field) and finite degree \(K\)-quasiregular maps \(r_i: Y \to X_i\), \(i=1,2\), which are unbranched outside the preimages of \(\{0,1,\infty \}\) by a Belyi map. They also prove the existence of branched ramified covering \(Z_i \to X_i\), \(i=1,2\), such that there is a \(K\)-quasiconformal homeomorphism \(Z_1 \to Z_2\). The latter result is an asymptotic form of a conjecture of Ehrenpreis which states that \(X_1, X_2\) have finite degree unbranched coverings which are arbitrarly near in the common moduli space. The authors discuss the full conjecture and give more results on arithmetic curves.
scientific article; zbMATH DE number 5080002
scientific article