Average canonical height conjecture for higher-dimensional elliptic-curve families

Let $K=\mathbb{Q}(t_1,\ldots,t_n)$ be the function field of the parameter space, let $E/K$ be the elliptic curve in the setting of Theorem 1, and let $P\in E(K)$. For $B>1$, define

$$Ah_E^\mathbb{Q}(P)_B:=\frac{1}{\#\mathbb{Q}^n_{B}(\Delta_E,P)}\sum_{\omega \in \mathbb{Q}^n_{B}(\Delta_E,P)} \frac{\hat{h}_{E_\omega}(P_\omega)}{h(\omega)}.$$

Average canonical height conjecture. For any $P\in E(K)$,

$$\lim_{B \rightarrow \infty}Ah_E^\mathbb{Q}(P)_B=\hat{h}_E(P).$$

This conjecture seeks an averaged analogue, for parameter spaces of dimension at least two, of Silverman’s specialization theorem, because the pointwise limit can depend on the algebraic curve along which the parameters tend to infinity. The theorem preceding the conjecture establishes only a positive lower bound for the liminf when $P$ is non-torsion; the asserted limit and its identification with the canonical height remain open.