Timeline for Is the canonical height of a totally p-adic point on an abelian variety bounded away from zero?
Current License: CC BY-SA 3.0
11 events
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| Mar 2, 2013 at 6:57 | comment | added | Vesselin Dimitrov | Thank you very much indeed for all those explanations, and especially for the references! | |
| Feb 28, 2013 at 13:21 | comment | added | ACL | (followed) The most beautiful result in this direction is due to Thuillier (unpublished thesis, tel.archives-ouvertes.fr/tel-00010990, Théorème 4.3.8) for points on a curve: if the measures associated to the points converge to $\nu$ and the measure associated to the metrized line bundle is $\nu$, then the heights of the points are at least the difference of the height of the curve and the energy $E(\mu-\nu)$ of the difference of the probability measures (this energy is strictly negative if $\mu\neq\nu$). | |
| Feb 28, 2013 at 13:17 | comment | added | ACL | 2) If the height of the variety is zero, you may have a strictly positive lower bound for points which additional constraints forbid the equidistribution property. This would be the case for totally $p$-adic points in $\mathbf G_m$, for example. Now, a trick of Autissier allowed him to reverse the argument and give a semi-explicit lower bound in such situations. For example, he proves that if you impose that all points avoir a a ball of radius $r$ in an Abelian variety, then their heights must be at least some function of $r$. | |
| Feb 28, 2013 at 13:14 | comment | added | ACL | 1) If you can prove that the height of the variety is strictly positive, this gives a lower bound which is valid for every (generic) point. This is basically the equivalence of Bogomolov's original conjecture with the formulation of Philippon and Zhang (the height of a subvariety in an Abelian variety is strictly positive unless this is a translate of an Abelian subvariety by a torsion point). | |
| Feb 28, 2013 at 13:13 | comment | added | ACL | [SUZ] and all subsequent equidistribution statements in Arakelov geometry say: if a sequence of points in a variety is small (meaning the height converge to the height of the variety) and generic (no strict subvariety contains a subsequence), then the Galois orbits converge to some canonical measure. This is used in two ways. | |
| Feb 28, 2013 at 11:18 | comment | added | Vesselin Dimitrov | Thank you very much for those explanations! Thus, the [SUZ] equidist. thm. implies that the lim inf (under Zariski topology) of the height of a totally real point is bounded away from zero; but to get a lower bound over all non-torsion totally real points, you need Ullmo's idea. And to get an effective lower bound on the height of a non-torsion point, you need the David-Philippon analytic geometry approach. However, the bound on the lim inf from [SUZ] is already effective - do I understand this right? Thus, I wondered whether your p-adic equid. thm. would likewise give effective $U$ and $c$. | |
| Feb 28, 2013 at 7:55 | comment | added | ACL | The approach of David-Philippon to the Bogomolov conjecture furnishes (in most cases) a geometric lower bound for the height, but since it relies crucially on complex analytic geometry (using volume estimates of tubes), I do not think that this approach will lead to a strictly positive lower bound for the height of generic totally $p$-adic points. At least, this is not straightforward. | |
| Feb 28, 2013 at 7:53 | comment | added | ACL | The proof of the Bogomolov conjecture, which concerns points on subvarieties of Abelian varieties (or tori), amounts to showing that the height of the subvariety is strictly positive, since then the height of points will be generically larger than the height of the subvariety. But in your case, the ambient variety has height zero, so having a lower bound for totally $p$-adic points is a different question. And, in both cases, the equidistribution approach to the Bogomolov conjecture (as proved by Ullmo and Zhang) does not shed much light on the actual lower bound for the height. | |
| Feb 28, 2013 at 0:07 | vote | accept | Vesselin Dimitrov | ||
| Feb 27, 2013 at 23:47 | comment | added | Vesselin Dimitrov | Thanks a lot, this was very helpful! I will look into those two papers. Is there any lower bound on the number $c$ that you get from your equidistribution theorem? E.g., when you fix $A$, how does the optimal $c$ vary with $p$? Is it always $> \epsilon/p$ as in the $\mathbb{G}_m$ case, with $\epsilon > 0$ a constant independent of $p$? | |
| Feb 27, 2013 at 23:32 | history | answered | ACL | CC BY-SA 3.0 |