# Is the canonical height of a totally p-adic point on an abelian variety bounded away from zero?

Inspired by the result of Schinzel and Smyth that a totally real number other than $0$ and $\pm 1$ has height at least $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.240659\ldots$, Bombieri and Zannier discovered that totally $p$-adic algebraic numbers which are not roots of $1$ likewise have height bounded away from $0$. More precisely, the proved that the lim inf of the height over the totally p-adic algebraic numbers is between $\frac{1}{2}\frac{\log{p}}{p+1}$ and $\frac{\log{p}}{p-1}$; cf. Ch. 4.6 of the Bombieri-Gubler book (Heights in diophantine geometry). Both results (real and $p$-adic) can be explained as consequences of a Galois equidistribution property for algebraic numbers of small height.

Similarly, for the totally real points of an abelian variety, the Archimedean Galois equidistribution of Szpiro-Ullmo-Zhang implies (S. W. Zhang, Equidistribution of small points on abelian varieties, Corollary 2) that the height of a totally real (non-torsion) point on an abelian variety is bounded away from $0$.

Now, there are various non-Archimedean equidistribution results in the literature in the spirit of the Szpiro-Ullmo-Zhang theorem. Does any of them imply a lower bound on the height of a totally $p$-adic non-torsion point on an abelian variety? Is there anything otherwise published on this problem?

Added a little later. By W. Gubler's "tropical equidistribution theorem" (which is really a statement about the equidistribution of $p$-adic valuations mod 1), we do know that the answer is positive for abelian varieties which do not have potentially good reduction at some place above $p$ (that is to say, they acquire some $\mathbb{G}_m$-part at some place above $p$).

But what about the proper case?

An example. The level $N$ division field $K_N := \mathbb{Q}(A[N])$ of the abelian variety contains the level $N$ cyclotomic field $C_N = \mathbb{Q}(\mu_N)$, so a prime which splits completely in $K_N$ is congruent to $1 \mod{N}$, hence $> N$. Hence, for a given $p$, the abelian variety contains only finitely many totally p-adic torsion points.

In my paper Mesures et équidistribution sur les espaces de Berkovich, Crelle, 2006, I had proved an equidistribution theorem in the good reduction case. The limit measure is a Dirac mass at a single point of the Berkovich space (the one whose reduction is the generic point of the special fiber). Since the Galois orbit of a totally $p$-adic point is contained in a compact subset disjoint from that point, this implies that there exist a dense Zariski open subset $U$ and a strictly positive real number $c$ such that the Néron-Tate height of any totally $p$-adic point of $U$ is at least $c$.
• 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. – ACL Feb 28 '13 at 7:53
• 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. – ACL Feb 28 '13 at 7:55
• 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 $r$. – ACL Feb 28 '13 at 13:17