Variation of the canonical height in a family of rational maps (Q2439660)

Let \(d\geq 2\) be an integer and consider the rational function \(f_t(z):={z^d+t\over z}\in{\overline{\mathbb Q}}(t)(z)\). It is a degree \(d\) rational function on \(z\), defined over the function field \({\overline{\mathbb Q}}(t)\); it admits a canonical height \(\hat{h}_f: \overline{\mathbb Q}(t)\to{\mathbb R}\), satisfying the property that for each \(c(t)\in{\overline{\mathbb Q}}(t)\), \(\hat{h}_f(f(c(t))=d\cdot \hat{h}_f(c(t))\). For each specialization \(t\mapsto\lambda\in\overline{\mathbb Q}\), it makes sense to consider the specialized endomorphism \(f_\lambda\) of the line \({\mathbb P}^1_{\overline{\mathbb Q}}\) and the corresponding canonical height \(\hat{h}_{f_\lambda}\) on \({\mathbb P}^1({\overline{\mathbb Q}})\). Given a point \(c(t)\in\overline{\mathbb Q}(t)\), and algebraic specializations \(\lambda\in\overline{\mathbb Q}\) of the parameter \(t\), a natural problem is estimating the canonical height \(\hat{h}_{f_\lambda} (c(\lambda))\), as \(\lambda\) varies. The main result of the paper under review (Theorem 1.1) is the proof that for a suitable constant \(C\) depending only on \(c\) the inequality \[ \left|\hat{h}_{f_\lambda} (c(\lambda))-\hat{h}_{f}(c)\cdot h(\lambda)\right|\leq C \] holds the for all algebraic numbers \(\lambda\). This improves, for this particular class of maps, a previous result by \textit{G. S. Call} and \textit{J. H. Silverman} [Compos. Math. 89, No. 2, 163--205 (1993; Zbl 0826.14015)] who obtained the bound \(o(h(\lambda))\) instead of the present \(O(1)\). In the particular case of family of polynomial maps, the \(O(1)\) bound was already obtained by \textit{P. Ingram} [J. Reine Angew. Math. 685, 73--97 (2013; Zbl 1297.14028)]; see also [\textit{J. Tate}, Am. J. Math. 105, 287--294 (1983; Zbl 0618.14019)] which treats the case of Lattès maps depending on a parameter. As a consequence of their proof, the authors obtain a bound for the difference between the canonical height \(\hat{h}_{f_\lambda} (c(\lambda))\) and the naïve height of \(f_\lambda(c(\lambda))\); for instance, in the `constant case', when \(c\in\overline{\mathbb Q}\), they prove that \(|\hat{h}_{f_\lambda}(c) - {h(f_\lambda (c))\over d}|\) is uniformly bounded as \(\lambda\) varies. As a corollary of their main theorem, they obtain that for fixed \(c\in\overline{\mathbb Q}\), the height of the points \(\lambda\in\overline{\mathbb Q}\) such that \(c\) is pre-periodic for \(f_\lambda\), is bounded. There is an extensive literature on these kinds of questions, starting from Silverman's paper in the eighties [\textit{J. H. Silverman}, J. Reine Angew. Math. 342, 197--211 (1983; Zbl 0505.14035)]. Recently, some problems of this type have been reinterpreted in terms of unlikely intersections: see for instance the papers of \textit{D. Masser} and \textit{U. Zannier} [Math. Ann. 352, No. 2, 453--484 (2012; Zbl 1306.11047)] and \textit{M. Baker} and \textit{L. DeMarco} [Duke Math. J. 159, No. 1, 1--29 (2011; Zbl 1242.37062)], which motivated the present work.