Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 24G (Riemann Surfaces)

A covering evenly covers every base point. It is regular covering map when deck transformations act transitively on fibres. For a concrete nonregular example, restrict $z\mapsto z^3-3z$ toIt is a surjective three-sheeted unbranched cover with monodromy $S_3$, so its deck group is not transitive.

$(g,V)$ continues $(f,U)$ when they agree on a nonempty connected component reached by an overlapping chain of function elements. A complete analytic function contains every continuation. Its Riemann surface is the set of its germs; germ neighbourhoods are charts, projection records the base point, and germ evaluation gives a holomorphic map to $\mathbb{C}$.

Since $V$ is simply connected, $g$ has a primitive $G_0$. On connected $U\cap V$, $G_0-F$ has zero derivative and is constant. Adjust that constant to obtain $G=F$ on the overlap, hence the required continuation.

Differentiate a germ of a continuation of $F$ to obtain a germ of the continuation of $f$. This defines a surjective cover $R_{\tilde{F}}\to R_F$. Two primitive germs above the same derivative germ differ by an additive period. Translation by periods gives all deck transformations and acts transitively on each fibre, so the cover is regular.