Namespaces
Variants
Actions

Difference between revisions of "Riemann-Hurwitz formula"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
Line 18: Line 18:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Riemann,   "Gesammelte mathematische Werke" , Dover, reprint (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Hurwitz,   "Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkte" , ''Mathematische Werke'' , '''1''' , Birkhäuser (1932) pp. 321–383</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Hurwitz,   R. Courant,   "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Nevanlinna,   "Uniformisierung" , Springer (1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S. Lang,   "Introduction to algebraic and Abelian functions" , Addison-Wesley (1972)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Riemann, "Gesammelte mathematische Werke" , Dover, reprint (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Hurwitz, "Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkte" , ''Mathematische Werke'' , '''1''' , Birkhäuser (1932) pp. 321–383</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , '''1''' , Springer (1964) {{MR|0173749}} {{ZBL|0135.12101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R. Nevanlinna, "Uniformisierung" , Springer (1967) {{MR|0228671}} {{ZBL|0152.27401}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> S. Lang, "Introduction to algebraic and Abelian functions" , Addison-Wesley (1972) {{MR|0327780}} {{ZBL|0255.14001}} </TD></TR></table>
  
  
Line 26: Line 26:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. Sect. IV.2</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.A. Griffiths,   J.E. Harris,   "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 216–219</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Hasse,   "Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei eindlichem Konstantenkörper" ''Reine Angew. Math.'' , '''172''' (1935) pp. 37–54</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H.M. Farkas,   I. Kra,   "Riemann surfaces" , Springer (1980) pp. Sect. III.6</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) pp. 216–219 {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Hasse, "Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei eindlichem Konstantenkörper" ''Reine Angew. Math.'' , '''172''' (1935) pp. 37–54</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) pp. Sect. III.6 {{MR|0583745}} {{ZBL|0475.30001}} </TD></TR></table>

Revision as of 21:56, 30 March 2012

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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann-Hurwitz_formula&oldid=23961
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article