Optimal bounds for the difference between Weil height and Néron-Tate height on elliptic curves over \(\overline{\mathbb Q}\) (Q2851134)

Let \(E\) be an elliptic curve defined over a number field \(k\). In order to study the distribution of \(k\)-rational points on \(E\), it is common to define a height function on the algebraic points of \(E\). If \(E\) is given in Weierstrass form in the projective plane, an easy height to define and calculate is the Weil height: NEWLINE\[NEWLINE h(x:y:z) =[k(P):\mathbb{Q}]^{-1} \sum_v \log\max\{|x|_v,|z|_v\} NEWLINE\]NEWLINE where \(k(P)\) denotes the field of definition of the point \(P=[x:z]\) over \(\mathbb{Q}\), and the sum is over all places \(v\) of \(k(P)\). However, there is a height that enjoys more pleasant properties with respect to the group law on \(E\), called the Néron-Tate height function: NEWLINE\[NEWLINE\hat{h}(P) =\lim_{n\to\infty}n^{-2}h(nP)NEWLINE\]NEWLINENEWLINENEWLINESadly, \(\hat{h}\) is not easy to calculate, unlike \(h\). The purpose of this paper is to describe an algorithm to compute the maximum difference between \(h\) and \(\hat{h}\), in order to marry the theoretical power of \(\hat{h}\) with the computational ease of \(h\).NEWLINENEWLINEThe algorithm itself builds on work of \textit{J. E. Cremona, M. Prickett} and \textit{S. Siksek} [J. Number Theory 116, No. 1, 42--68 (2006; Zbl 1162.11032)], among others, and uses a careful study of local height functions to control \(h-\hat{h}\). Several examples are given in the last section that illustrate the algorithm.