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.
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$\begingroup$ I think such questions have been studied by people working in arithmetic dynamics, though I could not find a precise reference. Lucien Szpiro might be a good person to ask. $\endgroup$– nafNov 11, 2012 at 11:20
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$\begingroup$ @Ulrich: I also thought about that but couldn't find anything. Thank you for the suggestion. It is a good idea to ask Szpiro. $\endgroup$– Damian RösslerNov 11, 2012 at 12:48
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1$\begingroup$ I thought this statement (for an arbitrary constant field) could be found in Buium's book "Differential algebra and diophantine geometry" but I just looked and can't find the statement. $\endgroup$– Felipe VolochJan 21, 2013 at 19:11
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1$\begingroup$ Hi Damian. Things are slightly more complicated. What you get via Hilbert schemes is a variety defined over the constant field which dominates (generically finitely to one) your given variety. But in characteristic $p$, Moret-Bailly constructed (in Séminaire sur les pinceaux de courbes de genre au moins 2) an Abelian surface which is isogeneous to a constant one, but not constant... The paper of Chatzidakis and Hrushovski does something in this direction, but at in a birational setting. $\endgroup$– ACLJan 21, 2013 at 22:27
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$\begingroup$ @Felipe Voloch. Thank you for pointing the book of Buium. I also think that that is the place to look (but I couldn't find it either...). $\endgroup$– Damian RösslerJan 23, 2013 at 13:49
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2 Answers
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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.
answered Jan 21, 2013 at 20:40
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$\begingroup$ Thank you for providing me with such precise references. I am sure that the result of Néron that you quote will turn out to be useful. I am aware of the Hilbert scheme arguments used to prove the Lang-Néron theorem but they don't seem to suffice to prove the descent results I have in mind, because I cannot make use of the theory of the $K|k$-trace. But maybe I am missing an obvious point ? $\endgroup$ Jan 23, 2013 at 13:48
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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].
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$\begingroup$ Thank you very much for pointing out this reference. I agree that Moriwaki addresses the problem I mention but he doesn't give any descent arguments, which is what I am looking for (Moriwaki shows that over $\bf Q$, I consider the "wrong" height and he is probably right about that but for the time being I want to use the purely geometric height). $\endgroup$ Jan 23, 2013 at 13:40