Theorem 1 (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 1   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.