Show that the set $\{ P \in E(\mathbb{Q})\ |\ h(P) \le M \}$ is finite, for any constant $M$.

Question

Show that the set $\{ P \in E(\mathbb{Q})\ |\ h(P) \le M \}$ is finite, for any constant $M$.

Here $h(P)$ is logarithmic height of $P$, that is, $h(P):=\log H(P)$ and $H(P)=H(x)$, for $P=(x,y) \in E(\mathbb{Q})$.

How can I show this lemma? I need it to show Mordell's Theorem.

Thanks in advance.

Answer 1

For any constant $M$ the set $\{t\in \mathbb{Q}: H(t)\leq C\}$ is finite. But given any value for $x$ there are at most two values of $y$ for which $(x,y)$ is a point of $E$. Therefore $$\{P\in E(\mathbb{Q}); h(P)\leq C\}$$ is also finite.