Question

I am looking for an article or book where the theory of heights over function fields (in any characteristic) is treated. I am especially interested in Northcott-type statements. For instance, over a function field $K$ over $\bf Q$, say, a subvariety $X$ of ${\bf P}^n_{K}$, which has a dense subset of $K$-points with bounded height, should have a model over (possibly a finite extension of) $\bf Q$. Where can I find the proof of such a statement ? When $K$ is a function field over a finite field, then one can use Hilbert schemes to get Northcott-type finiteness statements but in general, it seems that one should combine the theory of Hilbert schemes with some descent arguments. I would be grateful for any suggestions.

Answer 1

This is discussed (using somewhat older language) in Lang's Fundamentals of Diphantine Geometry. See in particular Chapter 3, Section 3, which is called "Heights in Function Fields". There is a theorem of Neron (Theorem 3.6 on page 66) which says that bounded height implies that the associated map has bounded degree. Then one should be able to use the theory Hilbert schemes (or as Lang uses in Chapter 6, the theory of Chow coordinates) to complete the argument. See also the discussion in Chapter 6, Section 5, which is entitled "Points of Bounded Height", where he uses Neron's result to analyze points of bounded height on abelian varieties; it seems that at least parts of the argument should apply generally.

Answer 2

Atsushi Moriwaki has written a beautiful paper on exactly this topic. The idea is to take an arithmetic polarization of your function field and use Arakelov intersection theory to incorporate the arithmetic of the constant field.

Arithmetic height functions over finitely generated fields. Invent. Math. 140 (2000), no. 1, 101–142.

Also, he extended his notion of polarization slightly in a follow-up article: arXiv:math/0006025 [math.NT].