Height difference bounds for elliptic curves over number fields

Abstract

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 - h ^ are of tremendous theoretical and practical importance. It is possible to decompose h - h ^ as a weighted sum of continuous bounded functions Ψ υ : E ( K υ ) → R over the set of places υ of K. A standard method for bounding h - h ^ , (due to Lang, and previously employed by Silverman) is to bound each function Ψ υ and sum these local ‘contributions’. In this paper, we give simple formulae for the extreme values of Ψ υ 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 Ψ υ was previously given by Siksek [Rocky Mountain J. Math. 25(4) (1990) 1501]. We complement this by giving two methods for sharply bounding Ψ υ for complex archimedean υ .