Elliptic curves have a group structure over the rational points. Why is this impossible for curves having genus greater than 1? I read somewhere that this impossibility is implied by Faltings theorem, which states that the number of rationals is finite. I don't see the implication. The group over the rational points could be finite.



Group over algebraic curves having genus greater than 1
Let $G/k$ be a proper smooth connected non-trivial group variety. For $1 \neq x \in G$, the translation by $x$ has no fixed point, so by the Lefschetz fixed point formula for $\ell$-adic cohomology (using $G$ connected), the Euler characteristic of $G$ is $0$. If $G/k$ is $1$-dimensional, this implies that $0 = \chi(G) = 2-2g$, so $g=1$.


