Variation of the canonical height for a family of polynomials (Q2872162)

Fix a number field \(k\) and a smooth projective curve \(X\) over \(k\). Let \(E\) be an elliptic surface over \(X\) and denote by \(\widehat{h}_{E_t}\) the Néron-Tate height on the fibre above \(t\in X(\overline{k})\). Given a section \(P: X\to E\), a theorem of Tate describes the behaviour of \(\widehat{h}_{E_t}(P_t)\) varying as a function of the parameter \(t\). Explicitly, Tate's theorem asserts that there exists a divisor \(D=D(E,P)\in \text{Pic}(X)\otimes \mathbb{Q}\) such that NEWLINE\[NEWLINE\widehat{h}_{E_t}(P_t)=h_D(t)+O(1)NEWLINE\]NEWLINE where \(h_D\) is the Weil height on \(X\) with respect to \(D\).NEWLINENEWLINEThe main result in the paper under review is an analogue of Tate's theorem for a one-parameter family of polynomial dynamical systems. Precisely, in this new context, if we write \(K\) for the function field \(k(X)\), then for any rational function \(f\in K(z)\) there is a canonical height \(\widehat{h}_f: \mathbb{P}^1(\overline{K})\to \mathbb{R}\) determined uniquely by the properties NEWLINE\[NEWLINE\widehat{h}_f(f(P))=\text{deg}(f)\widehat{h}_f(P)\quad\text{and}\quad \widehat{h}_f(P)=h(P)+O(1),NEWLINE\]NEWLINE and similarly to each specialization \(f_t\) at which \(f\) has good reduction. Here \(P\) stands for points in \(\mathbb{P}^1(K)\). The author proves that if \(f\in K[z]\) is a polynomial and \(P\in \mathbb{P}^1(K)\) is a rational point, then there exists a divisor \(D=D(f,P)\in \text{Pic}(X)\otimes\mathbb{Q}\) of degree \(\widehat{h}_f(P)\) such that NEWLINE\[NEWLINE\widehat{h}_{f_t}(P_t)=h_D(t)+O(1)NEWLINE\]NEWLINE as \(t\in X(\overline{k})\) varies.NEWLINENEWLINEApart from the result above, the author also compares in the paper under review, at each place of \(k\), the local canonical height with the local contribution to \(h_D\), and shows that the difference is analytic near the support of \(D\).