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 $ T $ and $ R $ be closed Riemann surfaces, and let $ f: T \rightarrow R $ be a surjective holomorphic mapping. Suppose this is an $ m $- sheeted covering, and suppose that $ f $ is branched in the points $ Q _ {1} \dots Q _ {s} \in T $ with multiplicities $ k _ {1} \dots k _ {s} $. Suppose that $ g = \textrm{ genus } ( T) $ and $ h = \textrm{ genus } ( R) $. Then the following (Riemann–Hurwitz) formula holds:

$$ \tag{* } 2 g - 2 = m( 2h - 2) + \sum _ {i= 1 } ^ { s } ( k _ {i} - 1). $$

In particular, if $ R $ is the Riemann sphere, i.e. $ h = \textrm{ genus } ( R) = 0 $, then

$$ g = 1 + \sum _ {i= 1 } ^ { s } \frac{k _ {i} - 1 }{2} . $$

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 $ f $ is separable (cf. Separable mapping). In that case

$$ 2g - 2 = m ( 2h- 2) + \mathop{\rm deg} ( \delta ( f ) ) , $$

where $ \delta ( f ) $ is the different of $ f $. In this case one speaks of the Riemann–Hurwitz–Hasse formula. In case a branching multiplicity $ k _ {i} $ is divisible by the characteristic of the base field, one speaks of wild ramification, and the degree of $ \delta ( f ) $ at that point is larger than $ k _ {i} - 1 $.

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 $ f $ is the different of the extension of algebraic function fields determined by $ f $. 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