**Could anyone please point out some references and different proofs for Northcott's Theorem about finiteness of the number of points of bounded height in projective varieties over number fields, and generalizations (if any)?**

I know some proofs, but I would like to inspect some references in light of the following question.

My main question, then, is: **what are the properties of height functions, that make Northcott's Theorem work?**

In other words, let $X$ be a projective variety over a number field $K/\mathbf{Q}$, and let $h : X(K)\to\mathbf{R}_{\ge 0}$ be a function.

Suppose $h$ satisfies the following property $\mathcal{P}$:

$$\#\{x\in X(K)\mid h(x)\le B\}$$ is a finite set for every $B\in\mathbf{R}_{\ge 0}$.

Can one find (necessary and) sufficient conditions on $h$ for $h$ to satisfy $\mathcal{P}$?

**Example.** If there is an ample line bundle $\mathscr{L}$ and $h = h_{\mathscr{L}}$ is the corresponding Weil height, then $h$ does satisfy $\mathcal{P}$ by Northcott's Theorem.

The idea is to extract, if possible, properties of height functions that make this implication still work. In other words, what are sufficient conditions (merely about $h$, ie. without assuming $h$ is a Weil height in the first place) to ensure $h$ has the finiteness property?

The question does not really ask for a solution to this task, but rather if this is already known to anybody.

Fundamentals of Diophantine Geometry. $\endgroup$ – Vesselin Dimitrov Jan 9 '18 at 0:35