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)
[4]	R. Nevanlinna, "Uniformisierung" , Springer (1967)
[5]	S. Lang, "Introduction to algebraic and Abelian functions" , Addison-Wesley (1972)

Comments

The different of a mapping is the different of the extension of algebraic function fields determined by . For the latter notion cf. (the editorial comments to) Discriminant.

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2
[a2]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 216–219
[a3]	H. Hasse, "Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei eindlichem Konstantenkörper" Reine Angew. Math. , 172 (1935) pp. 37–54
[a4]	H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) pp. Sect. III.6