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) over cap be the canonical height on E. Bounds for the difference h - (h) over cap are of tremendous theoretical and practical importance. It is possible to decompose h - L as a weighted sum of continuous bounded functions Psi(upsilon) : E(K-upsilon) -> R over the set of places upsilon of K. A standard method for bounding h - (h) over cap, (due to Lang, and previously employed by Silverman) is to bound each function Psi(upsilon) and sum these local 'contributions'. In this paper, we give simple formulae for the extreme values of Psi(upsilon), for non-archimedean upsilon in terms of the Tamagawa index and Kodaira symbol of the curve at upsilon. For real archimedean v a method for sharply bounding Psi(upsilon) was previously given by Siksek [Rocky Mountain J. Math. 25(4) (1990) 1501]. We complement this by giving two methods for sharply bounding Psi(upsilon) for complex archimedean upsilon. (c) 2005 Elsevier Inc. All rights reserved.