The Riemann-Hurwitz formula

Abstract

Let ϕ : S → T be a surjective holomorphic map between compact Riemann surfaces. There is a formula relating the various invariants involved: the genus of S, the genus of T, the degree of ϕ and the amount of ramification. Riemann used this formula in case T has genus zero. Contemporaries referred to this general formula as ”Riemann’s theorem”. Proofs were given by Zeuthen and Hurwitz. We discuss this formula in its historical context, and in modern generalizations.