Namespaces
Variants
Actions

Difference between revisions of "Riemann-Hurwitz formula"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
r0819201.png
 +
$#A+1 = 22 n = 0
 +
$#C+1 = 22 : ~/encyclopedia/old_files/data/R081/R.0801920 Riemann\ANDHurwitz formula,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''Hurwitz formula, Hurwitz theorem''
 
''Hurwitz formula, Hurwitz theorem''
  
A formula that connects the genus and other invariants in a covering of Riemann surfaces (cf. [[Riemann surface|Riemann surface]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r0819201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r0819202.png" /> be closed Riemann surfaces, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r0819203.png" /> be a surjective holomorphic mapping. Suppose this is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r0819204.png" />-sheeted covering, and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r0819205.png" /> is branched in the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r0819206.png" /> with multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r0819207.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r0819208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r0819209.png" />. Then the following (Riemann–Hurwitz) formula holds:
+
A formula that connects the genus and other invariants in a covering of Riemann surfaces (cf. [[Riemann surface|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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
2 g - 2  = m( 2h - 2) + \sum _ {i= 1 } ^ { s }  ( k _ {i} - 1).
 +
$$
  
In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192011.png" /> is the Riemann sphere, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192012.png" />, then
+
In particular, if $  R $
 +
is the Riemann sphere, i.e. $  h = \textrm{ genus } ( R) = 0 $,  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192013.png" /></td> </tr></table>
+
$$
 +
= 1 + \sum _ {i= 1 } ^ { s } 
 +
\frac{k _ {i} - 1 }{2}
 +
.
 +
$$
  
 
Formula (*) was stated by B. Riemann [[#References|[1]]] and proved by A. Hurwitz [[#References|[2]]].
 
Formula (*) was stated by B. Riemann [[#References|[1]]] and proved by A. Hurwitz [[#References|[2]]].
  
In the case of coverings of complete curves over a field, an analogous formula can be derived in case the covering mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192014.png" /> is separable (cf. [[Separable mapping|Separable mapping]]). In that case
+
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|Separable mapping]]). In that case
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192015.png" /></td> </tr></table>
+
$$
 +
2g - 2  = m ( 2h- 2) +  \mathop{\rm deg} ( \delta ( f  ) ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192016.png" /> is the different of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192017.png" />. In this case one speaks of the Riemann–Hurwitz–Hasse formula. In case a branching multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192018.png" /> is divisible by the characteristic of the base field, one speaks of wild ramification, and the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192019.png" /> at that point is larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192020.png" />.
+
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====
 
====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) {{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>
 
<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>
 
 
  
 
====Comments====
 
====Comments====
The different of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192021.png" /> is the different of the extension of algebraic function fields determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081920/r08192022.png" />. For the latter notion cf. (the editorial comments to) [[Discriminant|Discriminant]].
+
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|Discriminant]].
  
 
====References====
 
====References====
 
<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>
 
<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>

Latest revision as of 08:11, 6 June 2020


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
How to Cite This Entry:
Riemann-Hurwitz formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann-Hurwitz_formula&oldid=48541
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article