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.



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


For related results, see also the paper of Baker-Hsia (Crelle, 2005).


