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 h ̂ be the canonical height on E. Bounds for the difference h − ĥ are of tremendous theoretical and practical importance. It is possible to decompose h − h ̂ 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 h − ĥ, (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 (1990). We complement this by giving two methods for sharply bounding Ψυ for complex archimedean υ