Consider:
- Silverman, Ex V.5.4: Elliptic curves $E/\mathbb{F}_q$ and $E'/\mathbb{F}_q$ are isogenous if and only if $\#E(\mathbb{F}_q) = \# E'(\mathbb{F}_q)$.
- Silverman, Proposition 4.12: any finite subgroup $\Phi \subset E$ induces an isogeny $E \rightarrow E'$, where $E'$ has group structure $E / \Phi$.
- Silverman Theorem 2.3: Any non-constant morphism of curves is surjective.
Why do these three facts not contradict each other? In other words, if by quotienting with some nontrivial subgroup $\Phi \subset E(\mathbb{F}_q)$ I can induce a surjective isogeny $E \rightarrow E'$, why doesn't $E'$ have strictly fewer rational points than $E$? This is blowing my mind. Thank you.
Best Answer
Surjective means here surjective from $E(\overline k)$ to $E'(\overline k)$ where $\overline k$ is the algebraic closure of $k=\Bbb F_q$. The image of $E(k)$ may be a proper subgroup of $E'(k)$. There may be points $Q\in E'(k)$ which are images of elements of $E(\overline k)$ all lying outside $E(k)$.