Strong Lang–Silverman conjecture for exceptional points on abelian varieties

Let $g\geq 1$ be an integer. For any number field $K$, let $A/K$ be an abelian variety of dimension $g$, let $L$ be an ample symmetric line bundle on $A$, and let $P\in A(K)$. Here $\deg_L(B)$ denotes the $L$-degree of an abelian subvariety $B\subset A$, and the order of $P$ modulo $B$ means the order of its class in the quotient $A/B$. The height $\widehat{h}_{A,L}$ is the Néron–Tate height associated to $L$, and $\mathop{h_{\mathrm{F}^+}}(A/K)$ is the relative Faltings height.

Strong Lang–Silverman conjecture. There exist positive quantities $c_{33}=c_{33}(K,g)$ and $c_{34}=c_{34}(K,g)$ such that either there is an abelian subvariety $B\subset A$, with $B\neq A$, satisfying

$$\deg_L(B)\leq c_{34}\deg_L(A)$$

and such that the order of $P$ modulo $B$ is at most $c_{34}$, or $\operatorname{End}(A)\cdot P$ is Zariski dense and

\widehat{h}_{A,L}(P)\geq c_{33}\max\left\\{\mathop{h_{\mathrm{F}^+}}(A/K),1\right\\}.

This refinement is intended to account for points whose endomorphism orbit is not Zariski dense. The paper explains that an earlier proposed formulation fails and that this version changes the exceptional-point condition and includes dependence on the degree of the polarization; its resolution is not supplied here.