The difference between the ordinary height and the canonical height on elliptic curves (Q2469219)

Let \(E\) be an elliptic curve over a number field \(K\). Write \(h\) for the Weil logarithmic height on points of \(E\) over \(K\), and \(\widehat{h}\) for the Néron-Tate canonical logarithmic height. It is well known that the two heights are comparable, meaning that there exist constants \(c_1,c_2\) depending only on the model of \(E\) and on \(K\) such that \[ c_1 \leq h(P) -\widehat{h}(P) \leq c_2, \] for all \(P \in E(K)\). Effective bounds on the constants \(c_1,c_2\) have been investigated by different authors. One possible approach is to decompose the difference \(h-\widehat{h}\) as \[ h(P)-\widehat{h}(P) = \frac{1}{[K:\mathbb Q]} \sum_{v \in M_K} \Psi_v(P), \] where \(M_K\) is the set of all places of \(K\) and \(\Psi_v : E(K_v) \rightarrow \mathbb R\) are bounded continuous functions for all \(v \in M_K\), where \(K_v\) stands for the completion of \(K\) at \(v\). To obtain explicit bounds on \(h(P)-\widehat{h}(P)\) it is now sufficient to produce estimates on \(\Psi_v(P)\). In particular, this approach has recently been used by \textit{J. E. Cremona, M. Prickett} and \textit{S. Siksek} [J. Number Theory 116, No. 1, 42--68 (2006; Zbl 1162.11032)]. In the paper under review, the author carries the method of Cremona et al. further, replacing certain duplication maps on elliptic curves by general multiplication and producing new sharper bounds on this height difference.