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 field (i.e. every algebraic extension of $K$ is separable). Recall that a non-constant map of curves $\psi:C_{1}\to C_{2}$ over $K$ is separable if the extension of function fields $K(C_{1})/\psi^{\ast}K(C_{2})$ is a separable extension of fields.

Theorem (Hurwitz Genus Formula).

Let $C_{1}$ and $C_{2}$ be two smooth curves defined over $K$ of genus $g_{1}$ and $g_{2}$, respectively. Let $\psi:C_{1}\to C_{2}$ be a non-constant and separable map. Then

 $2g_{1}-2\geq(\deg\psi)(2g_{2}-2)+\sum_{P\in C_{1}}(e_{\psi}(P)-1)$

where $e_{\psi}(P)$ is the ramification index of $\psi$ at $P$. Moreover, there is equality if and only if either $\operatorname{char}(K)=0$ or $\operatorname{char}(K)=p>0$ and $p$ does not divide $e_{\psi}(P)$ for all $P\in C_{1}$.

Example.

As an application of the Hurwitz genus formula, we show that an elliptic curve $E:y^{2}=x(x-\alpha)(x-\beta)$ defined over a field $K$ of characteristic $0$ has genus $1$. Notice that the fact that $E$ is an elliptic curve over $K$ implies that $0,\alpha$ and $\beta$ are distinct elements of $K$, otherwise $E$ would be a singular curve. We define a map:

 $\psi:E\to\mathbb{P}^{1},\quad[x,y,z]\mapsto[x,z]$

and notice that $[0,1,0]$, the “point at infinity” of $E$, maps to $[1,0]$, the point at infinity of $\mathbb{P}^{1}$. The degree of this map is $2$: generically every point in $\mathbb{P}^{1}$ has two preimages, namely $[x,y,z]$ and $[x,-y,z]$. Moreover, the genus of $\mathbb{P}^{1}$ is $0$ and the map $\psi$ is ramified exactly at $4$ points, namely $P_{1}=[0,0,1],P_{2}=[\alpha,0,1],P_{3}=[\beta,0,1]$ and the point at infinity. It is easily checked that the ramification index at each point is $e_{\psi}(P_{i})=2$. Hence, the Hurwitz formula reads:

 $2g_{1}-2=2(2\cdot 0-2)+\sum_{i=1}^{4}(e_{\psi}(P_{i})-1)=-4+4=0.$

We conclude that $g_{1}=1$, as claimed.

Title Hurwitz genus formula HurwitzGenusFormula 2013-03-22 15:57:15 2013-03-22 15:57:15 alozano (2414) alozano (2414) 6 alozano (2414) Theorem msc 14H99 RiemannRochTheorem EllipticCurve