Height difference bounds for elliptic curves over number fields

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and be the canonical height on E. Bounds for the difference are of tremendous theoretical and practical importance. It is possible to decompose as a weighted sum of continuous bounded functions Ψυ:E(Kυ)→R over the set of places υ of K. A standard method for bounding , (due to Lang, and previously employed by Silverman) is to bound each function Ψυ and sum these local ‘contributions’.In this paper, we give simple formulae for the extreme values of Ψυ for non-archimedean υ in terms of the Tamagawa index and Kodaira symbol of the curve at υ.For real archimedean υ a method for sharply bounding Ψυ was previously given by Siksek [Rocky Mountain J. Math. 25(4) (1990) 1501]. We complement this by giving two methods for sharply bounding Ψυ for complex archimedean υ.