Horst Zimmer, Universität des Saarlandes (Koautor: Susanne Schmitt)
There are essentially two height functions on an elliptic curve over a global field, the naive or Weil height and the canonical or Néron-Tate height. The first is good for calculations but bad for the theory and the second is, on the contrary, bad for calculations but good for the theory. It is therefore worthwile to estimate the difference between the two height functions. This was done at first by Dem'janenko and the author, later by Silverman and Siksek. The point is that the Weil height can be modified and, in the number field case, be replaced by another modified height.
Estimates of the differences between various heights can be obtained in a simple manner. In particular, it makes sense to compare the estimates given by the author with those obtained by Silverman and Siksek.