On Lifting the Hyperelliptic Involution

Abstract

Let ${W_p}$ stand for a compact Riemann surface of genus p. (1) Let ${W_q}$ be hyperelliptic, and let n be a positive integer. Then there exists an unramified covering of n sheets, ${W_p} \to {W_q}$, where ${W_p}$ is hyperelliptic. (2) Let ${W_{2n + 1}} \to {W_2}$ be an unramified Galois covering with a dihedral group as Galois group, and let n be odd. Then ${W_{2n + 1}}$ is elliptic hyperelliptic (bi-elliptic). (3) Let ${W_4} \to {W_2}$ be an unramified non-Galois covering of three sheets. Then ${W_4}$ is hyperelliptic.