TITLE: The genus of a curve with a group structure
QUESTION [9 upvotes]: I'm reading Milne's Elliptic Curves and came across this statement: If a nonsingular projective curve has a group structure defined by polynomial maps, then it has genus 1.
In this question a similar question was asked, and an answer given there (the one that was not accepted) someone backs this up using machinery/notation that I do not understand. Can someone give or direct me to a proof of this statement? It doesn't have to be very formal; this isn't homework. Also, in the affine case this statement clearly breaks down - the affine line has the group structure of addition, for instance. Is there some sort of correction or does the group structure bear no relation to the genus in this case?

REPLY [2 votes]: I accepted another answer but I'm going to post a more elementary (from my perspective) way to prove this. The Riemann-Hurwitz Formula gives the nice bound $|\text{Aut}(\mathcal{C})| < \infty$ for a $g(\mathcal{C}) > 1$ (we'll assume characteristic zero so all maps are tamely ramified). But each point gives the translation automorphism, and these are all clearly distinct, whence we obtain a contradiction for $g(\mathcal{C}) > 1$. So the question becomes: does the projective line have a group law? Of course not, for then a hyperelliptic curve $E$ ($g(E) > 1$) would have one too. The gist is that if we select a degree-2 map $f \colon E \to \mathbb{P}^1$, totally ramified, then it must be injective, so we can pull back the group law.