group homomorphisms of elliptic curves

Question

Let $q ‎\geq‎ 37$ be a prime power, $E_1$ and $E_2$ be two elliptic curves defined over $\Bbb{F}_q$, and $g:E_1 \rightarrow E_2$ be a rational map which is a group homomorphisms. show that $g$ is injective iff it is surjective.

Answer 1

It is enough to show that if one $|E_i|$ divides the other (which happens if the map is either injective or surjective), then $|E_1|=|E_2|$.

But note that by Hasse $\sqrt{\frac{|E_1|}{|E_2|}} \geq \frac{\sqrt{q}-1}{\sqrt{q}+1} > \frac{5}{7}$. Now, $\frac{7}{5}=1.4 <\sqrt{2}$, therefore, if $i \neq j$, $\frac{|E_i|}{|E_j|} > \frac{1}{2}$. The conclusion follows.