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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.

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    $\begingroup$ The Northcott property (which really extends to algebraic points of a bounded degree, rather than just the $K$-rational points), is a pretty trivial generalization of the observation that there are finitely many integer polynomials with a bound on the degree and the coefficients. See the first chapter of Bombieri and Gubler's book. Your example is not quite right; you need the line bundle to be ample (and, properly speaking, metrized - if you want to have an actual function rather than an $O(1)$ class of functions). $\endgroup$
    – Vesselin Dimitrov
    Commented Jan 9, 2018 at 0:31
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    $\begingroup$ Regarding non-trivial generalizations, Moriwaki's height functions also have the finiteness property under the suitable conditions. It is also worth noting that height functions such as $\sum_v \log^+{|\cdot|}$ arise abstractly from any any field with a set of valuations that satisfy a product formula $\sum_v \log{|\xi|_v} = 0$, and the Northcott finiteness property is then sometimes imposed as an axiom. This was done by Lang in his treatment of Roth's theorem in Fundamentals of Diophantine Geometry. $\endgroup$
    – Vesselin Dimitrov
    Commented Jan 9, 2018 at 0:35

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If you don't require any further properties, then I don't think you get a particularly interesting class of functions. For example, $X(K)$ is countable, list them as $P_1,P_2,\ldots$. Define $h(P_n)=n$. Then you get the "Northcott Property." One of the important things about height functions is that they are, in some sense, geometric. In other words, if one has a map between varieties, then they have some functorial properties relative to the map. Roughly speaking, "nice" height functions have two properties: (1) They measure complexity, and there are finitely many objects of bounded complexity. (2) I tend to say they transform geometry into number theory in a functorial manner, but in a more abstract sense, they transform geometry into complexity relations.

BTW, if you don't impose further conditions, you can even get a "super-Northcott Property," since $X(\mathbb Q)$ is countable, listing the points gives a "height" function such that there are finitely many algebraic points of bounded height!

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