# Riemann–Hurwitz formula

In mathematics, the **Riemann–Hurwitz formula**, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a *ramified covering* of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.

## Statement

For an orientable surface *S* the Euler characteristic χ(*S*) is

where *g* is the genus (the *number of handles*), since the Betti numbers are 1, 2*g*, 1, 0, 0, ... . In the case of an (*unramified*) covering map of surfaces

that is surjective and of degree *N*, we should have the formula

That is because each simplex of *S* should be covered by exactly *N* in *S*′ — at least if we use a fine enough triangulation of *S*, as we are entitled to do since the Euler characteristic is a topological invariant. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (*sheets coming together*).

Now assume that *S* and *S′* are Riemann surfaces, and that the map π is complex analytic. The map π is said to be *ramified* at a point *P* in *S*′ if there exist analytic coordinates near *P* and π(*P*) such that π takes the form π(*z*) = *z*^{n}, and *n* > 1. An equivalent way of thinking about this is that there exists a small neighborhood *U* of *P* such that π(*P*) has exactly one preimage in *U*, but the image of any other point in *U* has exactly *n* preimages in *U*. The number *n* is called the *ramification index at P* and also denoted by *e*_{P}. In calculating the Euler characteristic of *S*′ we notice the loss of *e _{P}* − 1 copies of

*P*above π(

*P*) (that is, in the inverse image of π(

*P*)). Now let us choose triangulations of

*S*and

*S′*with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then

*S′*will have the same number of

*d*-dimensional faces for

*d*different from zero, but fewer than expected vertices. Therefore we find a "corrected" formula

or as it is also commonly written

(all but finitely many *P* have *e _{P}* = 1, so this is quite safe). This formula is known as the

*Riemann–Hurwitz formula*and also as

**Hurwitz's theorem**.

## Examples

The Weierstrass -function, considered as a meromorphic function with values in the Riemann sphere, yields a map from an elliptic curve (genus 1) to the projective line (genus 0). It is a double cover (*N* = 2), with ramification at four points only, at which *e* = 2. The Riemann–Hurwitz formula then reads

with the summation taken over four values of *P*.

The formula may also be used to calculate the genus of hyperelliptic curves.

As another example, the Riemann sphere maps to itself by the function *z*^{n}, which has ramification index *n* at 0, for any integer *n* > 1. There can only be other ramification at the point at infinity. In order to balance the equation

we must have ramification index *n* at infinity, also.

## Consequences

Several results in algebraic topology and complex analysis follow.

Firstly, there are no ramified covering maps from a curve of lower genus to a curve of higher genus – and thus, since non-constant meromorphic maps of curves are ramified covering spaces, there are no non-constant meromorphic maps from a curve of lower genus to a curve of higher genus.

As another example, it shows immediately that a curve of genus 0 has no cover with *N* > 1 that is unramified everywhere: because that would give rise to an Euler characteristic > 2.

## Generalizations

For a correspondence of curves, there is a more general formula, **Zeuthen's theorem**, which gives the ramification correction to the first approximation that the Euler characteristics are in the inverse ratio to the degrees of the correspondence.

An orbifold covering of degree N between orbifold surfaces S' and S is a branched covering, so the Riemann–Hurwitz formula implies the usual formula for coverings

denoting with the orbifold Euler characteristic.

## References

- Hartshorne, Robin (1977),
*Algebraic Geometry*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052, section IV.2.