Does Lang's conjecture imply Morton-Silverman's Uniform Boundedness conjecture?

Question

I was curious to see whether the following conjecture of Morton-Silverman is (known to be) a consequence of Lang (or Lang-Vojta's) conjecture.

Conjecture. Let $D$, $N$, and $d$ be positive integers. Then, there is an integer $C=C(D,N,d)$ such that, for all number fields $K$ of degree $D$ and all endomorphisms $\varphi:\mathbb P^N_K\to \mathbb P^N_K$ of degree $d$, the set of preperiodic $K$-points of $\varphi$ is at most $C$.

This conjecture seems to be an analogue (or even generalization?) of Merel's uniform boundedness theorem for torsion on an elliptic curve over a number field.

Assuming Lang's conjecture, Caporaso-Harris-Mazur proved "Uniform Mordell". Would it suffice to apply "Uniform Mordell" to the dynatomic modular curves studied (for instance) here; https://arxiv.org/pdf/1703.04172.pdf ?

Answer 1

Uniform boundedness of torsion for elliptic curves (Mazur-Merel) and for abelian varieties (conjectured) is analogous to the dynamical conjecture on uniform boundedness of preperiodic points that Morton and I made and you have stated. An interesting and non-trivial result of Fakhruddin says that our conjecture for $\mathbb P^N$ implies uniform boundedness of torsion points on abelian varieties of any fixed dimension. But note that (AFAIK) Lang-Vojta is not known to imply uniform boundedness of torsion on abelian varieties.

The Lang-Mordell conjecture, proven by Faltings, has to do with the intersection of a finitely generated subgroup $\Gamma$ of an abelian variety $A$ with a subvariety $Y\subseteq A$. There are conjectural dynamical analogues of Lang-Mordell, too. Here's the usual statement:

Dynamical Lang-Mordell Conjecture Let $f:X\to X$ be a morphism of a variety (defined over $\mathbb C$), let $P\in X$ be a point, and let $Y\subseteq X$ be a subvariety. Then the set $$ \{n\ge0 : f^n(P)\in Y\}\qquad (*) $$ is the union of a finite set and a finite union of one-sided arithmetic progressions.

It is easy enough to formulate a uniform version of this conjecture, e.g., let $X=\mathbb P^N$, let $f$ and $P$ be defined over a number field of fixed degree, let $f$ have a fixed degree, then the number of progressions (including single-pointed ones) in $(*)$ is bounded independently of $f$ and $P$. One might then try to prove such a result from Lang-Vojta; but a first issue is that (again AFAIK), there are no general results saying that Lang-Vojta can be used to prove that the Dynamical Lang-Mordell Conjecture is true for, say, all maps of degree 2 on $\mathbb P^N$, with everything defined over $\mathbb Q$. In other words, it's not clear how to use Lang-Vojta to prove even a non-uniform Dynamical Lang-Mordell.

Finally, to answer your last question, the general conjecture certainly can't be analyzed using dynatomic modular curves, since they parameterize 1-parameter families of maps of $\mathbb P^1$. But if you wanted to prove a uniform bound for, say, all $x^2+c$, they would be a good place to start.