On the average value of the canonical height in higher dimensional families of elliptic curves (Q2928540)

Let \(K\) be the rational function field \(\mathbb{Q}[T_1,\cdots,T_n]\) and \(\mathbf{T}=(T_1,\cdots, T_n)\). Let \(E_{\mathbf{T}}\) be an elliptic curve over \(K\) defined by \(Y^2=X^3+A(\mathbf{T})X+B(\mathbf{T})\) with \(A(\mathbf{T}), B(\mathbf{T})\in \mathbb{Z}[\mathbf{T}]\). Assume that there is no nonconstant \(g(\mathbf{T})\in \mathbb{Q}[\mathbf{T}]\) such that \(A(\mathbf{T})/g(\mathbf{Y})^4, B(\mathbf{T})/g(\mathbf{T})^6\in\mathbb{Z}[\mathbf{T}]\), and that at least one of \(A(\mathbf{T}), B(\mathbf{T})\) is not constant. Let \(\Delta_E(\mathbf{T})\) denote the discriminant of \(E\), and let \(\mathbb{Q}^n(\Delta_E)\) be the set of all \(\omega=(\omega_1,\cdots,\omega_n)\in \mathbb{Q}^n\) such that \(\Delta_E(\omega)\neq 0\). Let \(\hat{h}_{E_{\omega}}\) (resp. \(h\)) denote the canonical height on \(E_{\omega}\) (resp. the logarithmic height on \(\mathbb{P}^n_{\mathbb{Q}}\)). In this paper the author focuses on the average value of the ratio \(\hat{h}_{E_{\omega}}(P_{\omega})/h(\omega)\) for a fixed rational point \(P_{\mathbf{T}}\in E(K)\) and obtain the following theorem. Let \(\mathbb{Q}^n_B(\Delta_E)=\{\omega\in\mathbb{Q}^n;1<e^{h(\omega)}\leq B\) and \(\Delta_E(\omega)\neq 0\}, E(K)_{\text{nt}}=\{P\in E(K); P\) is not a torsion point\(\}\), \(\mathbb{Q}^n_B(\Delta_E,P)=\{\omega\in \mathbb{Q}^n_B(\Delta_E);P_{\omega}\) is defined\(\}\), and let \(\underline{Ah}_E^{\mathbb{Q}}(P)=\liminf_{B\rightarrow\infty}1/\sharp(\mathbb{Q}^n_B(\Delta_E,P))\cdot\sum_{\omega\in\mathbb{Q}^n_B(\Delta_E,P)}\hat{h}_{E_{\omega}}/h(\omega)\). Then there exists a constant \(L>0\) depending only on \(\Delta_E\) such that \(\underline{Ah}_E^{\mathbb{Q}}(P)\geq L\) holds for all \(P\in E(K)_{\text{nt}}\).