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

$$2g_1-2\ge (\mathrm{deg}\psi )(2g_2-2)+\sum _{P\in C_1}(e_{\psi }(P)-1)$$

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

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:

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

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.