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.

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

:2 − 2"g"

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

:π: "S′" → "S"

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

:χ("S"′) = "N"χ("S").

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").

Near a point "P" of "S" where "e" sheets come together, "e" = "e_P" being called the "ramification index", we notice the loss of "e − 1" copies of "P" "above" "P" (in π⁻¹("P")). Therefore we expect a 'corrected' formula

:χ("S"′) = "N"χ("S") − Σ ("e_P − 1")

the sum being taken over all "P" in "S" (almost all "P" have "e_P" = 1, so this is quite safe). This is the Riemann-Hurwitz formula, but for a special though important case (namely where there is just one point in which the sheets above "P" come together, or equivalently the local monodromy is a cyclic permutation). In the most general case the final sum must be replaced by the sum of terms

:"e_P − c_P"

where "c_P" is the number of points of "S"′ above "P", or equivalently the number of cycles of the local monodromy acting on the sheets.

To give an example, any elliptic curve (genus 1) maps to the projective line (genus 0) as a double cover ("N" = 2), with ramification at four points only, at which "e" = 2. We can check that this then reads

:0 = 2·2 − Σ 1

with the summation taken over four values of "P". This covering comes from the Weierstrass pe-function as meromorphic function, with values considered as lying in the Riemann sphere.The formula may also be used to check the value of the genus formula of the 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

:2 = "n"·2 − ("n" − 1) − ("e"_∞ − 1)

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

The formula may be used to prove theorems. For 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.

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.

References

* | year=1977, section IV.2.