Hurwitz genus formula

The following formula, due to Hurwitz, is extremely useful when trying to compute the genus of an algebraic curve. In this entry K is a perfect fieldMathworldPlanetmath (i.e. every algebraic extensionMathworldPlanetmath of K is separable). Recall that a non-constant map of curves ψ:C1C2 over K is separable if the extension of function fields K(C1)/ψK(C2) is a separable extension of fields.

Theorem (Hurwitz Genus Formula).

Let C1 and C2 be two smooth curves defined over K of genus g1 and g2, respectively. Let ψ:C1C2 be a non-constant and separable map. Then


where eψ(P) is the ramification index of ψ at P. Moreover, there is equality if and only if either char(K)=0 or char(K)=p>0 and p does not divide eψ(P) for all PC1.


As an application of the Hurwitz genus formula, we show that an elliptic curveMathworldPlanetmath E:y2=x(x-α)(x-β) defined over a field K of characteristicPlanetmathPlanetmath 0 has genus 1. Notice that the fact that E is an elliptic curve over K implies that 0,α and β are distinct elements of K, otherwise E would be a singular curve. We define a map:


and notice that [0,1,0], the “point at infinity” of E, maps to [1,0], the point at infinity of 1. The degree of this map is 2: generically every point in 1 has two preimages, namely [x,y,z] and [x,-y,z]. Moreover, the genus of 1 is 0 and the map ψ is ramified exactly at 4 points, namely P1=[0,0,1],P2=[α,0,1],P3=[β,0,1] and the point at infinity. It is easily checked that the ramification index at each point is eψ(Pi)=2. Hence, the Hurwitz formula reads:


We conclude that g1=1, as claimed.

Title Hurwitz genus formula
Canonical name HurwitzGenusFormula
Date of creation 2013-03-22 15:57:15
Last modified on 2013-03-22 15:57:15
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Theorem
Classification msc 14H99
Related topic RiemannRochTheorem
Related topic EllipticCurve