Riemann-Hurwitz formula

In mathematics , the classic Riemann-Hurwitz formula (also known as the Hurwitz theorem) makes a statement about the holomorphic mappings between compact Riemann surfaces and relates the order of branches and the number of leaves to the topological gender (number of "holes") of the two surfaces .

The formula is named after Bernhard Riemann and Adolf Hurwitz .

statement

Let and be compact Riemannian surfaces of topological gender or and a -leaved branched holomorphic overlay . denote the total branching order of . Then the following relationship applies between these quantities: ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle g}$ ${\ displaystyle g ^ {\ prime}}$ ${\ displaystyle f: X \ rightarrow Y}$ ${\ displaystyle n}$ ${\ displaystyle b}$ ${\ displaystyle f}$

{\ displaystyle 2g-2 = b + n \ cdot (2g ^ {\ prime} -2)}

.

The total branch order is defined as the sum of all branch orders :

{\ displaystyle b: = \ sum _ {P \ in X} \ left (e (P) -1 \ right)}

where denotes the multiplicity of the mapping in the point . The compactness of guarantees that there are only finitely many branch points and that the sum is therefore finite. ${\ displaystyle e (P)}$ ${\ displaystyle f}$ ${\ displaystyle P}$ ${\ displaystyle X}$

Application example

The Riemann-Hurwitz formula is particularly useful for calculating the topological gender of a Riemann surface. For example, be the Riemann area of the algebraic function . This defines a -leaf branched overlay on the Riemann number sphere (gender ). It can also be stated that there are exactly branch points, all with branch order . Used in the formula is obtained for the sex of : . ${\ displaystyle X}$ ${\ displaystyle {\ sqrt [{n}] {1-z ^ {n}}}}$ ${\ displaystyle n}$ ${\ displaystyle 0}$ ${\ displaystyle n}$ ${\ displaystyle n-1}$ ${\ displaystyle X}$ ${\ displaystyle g = {\ frac {(n-1) (n-2)} {2}}}$

Generalizations

Algebraic Curves

For non-singular projective algebraic curves over an algebraically closed field, the Riemann-Hurwitz formula also applies, in the following formulation:

{\ displaystyle 2g-2 = n \ cdot (2g'-2) + \ mathrm {deg} \, R}

where denotes the branch divisor. ${\ displaystyle \ textstyle R: = \ sum _ {P \ in X} \ mathrm {len} \ left (\ Omega _ {X / Y} \ right) _ {P} \ cdot P}$

Explanation of Notation: The (arithmetic) Gender a non-singular projective curve is defined as the dimension of the first cohomology group of the sheaf of Zariski regular functions . In the event that the curves are considered over the basic field of the complex numbers , this definition of gender agrees with the topological gender and it is then only a reformulation of the classical statement with the help of algebra. ${\ displaystyle g}$ ${\ displaystyle X}$ ${\ displaystyle g: = \ dim H ^ {1} (X, {\ mathcal {O}} _ {X})}$

Since a non-constant morphism between such algebraic curves is automatically surjective, it induces a monomorphism of the associated function fields. This can be viewed as a body expansion. The degree of body expansion is finite and represents the algebraic equivalent of the number of leaves. ${\ displaystyle f: X \ rightarrow Y}$ ${\ displaystyle f ^ {*}: k (Y) \ rightarrow k (X)}$ ${\ displaystyle k (Y): k (X)}$ ${\ displaystyle n}$

${\ displaystyle \ Omega _ {X / Y}}$ denotes the sheaf of relative differentials. If the branch points are tame , i.e. H. if the basic body has characteristics or if the characteristic of the basic body does not divide the multiplicities for any point , then the following applies , thus then corresponds to the total branching order. ${\ displaystyle 0}$ ${\ displaystyle e (P)}$ ${\ displaystyle P \ in X}$ ${\ displaystyle \ mathrm {len} \ left (\ Omega _ {X / Y} \ right) _ {P} = e (P) -1}$ ${\ displaystyle \ textstyle \ mathrm {deg} R = \ sum _ {P \ in X} \ left (e (P) -1 \ right)}$

Number theory

The formula can be applied in a modified form to extensions of algebraically non-closed fields and is used in number theory .

literature

Otto Forster : Riemann surfaces (= Heidelberg pocket books 184). Springer, Berlin et al. 1977, ISBN 3-540-08034-1 (English: Lectures on Riemann Surfaces (= Graduate Texts in Mathematics 81). Corrected 2nd printing. Ibid 1991, ISBN 3-540-90617-7 ).
Robin Hartshorne : Algebraic Geometry (= Graduate Texts in Mathematics 52). Springer, New York et al. 1977, ISBN 0-387-90244-9 .
Klaus Lamotke : Riemann surfaces. Springer, Berlin et al. 2005, ISBN 3-540-57053-5 .