# 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 |

#### 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 MR0463157 Zbl 0367.14001 |

[a2] | P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 216–219 MR0507725 Zbl 0408.14001 |

[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 MR0583745 Zbl 0475.30001 |

**How to Cite This Entry:**

Riemann–Hurwitz formula.

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