# Riemann-Hurwitz formula

Hurwitz formula, Hurwitz theorem

A formula that connects the genus and other invariants in a covering of Riemann surfaces (cf. Riemann surface). Let and be closed Riemann surfaces, and let be a surjective holomorphic mapping. Suppose this is an -sheeted covering, and suppose that is branched in the points with multiplicities . Suppose that and . Then the following (Riemann–Hurwitz) formula holds:

 (*)

In particular, if is the Riemann sphere, i.e. , then

Formula (*) was stated by B. Riemann [1] and proved by A. Hurwitz [2].

In the case of coverings of complete curves over a field, an analogous formula can be derived in case the covering mapping is separable (cf. Separable mapping). In that case

where is the different of . In this case one speaks of the Riemann–Hurwitz–Hasse formula. In case a branching multiplicity is divisible by the characteristic of the base field, one speaks of wild ramification, and the degree of at that point is larger than .

#### References

 [1] B. Riemann, "Gesammelte mathematische Werke" , Dover, reprint (1953) [2] A. Hurwitz, "Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkte" , Mathematische Werke , 1 , Birkhäuser (1932) pp. 321–383 [3] A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , 1 , Springer (1964) MR0173749 Zbl 0135.12101 [4] R. Nevanlinna, "Uniformisierung" , Springer (1967) MR0228671 Zbl 0152.27401 [5] S. Lang, "Introduction to algebraic and Abelian functions" , Addison-Wesley (1972) MR0327780 Zbl 0255.14001