Weil Height Research Articles

Variation of the Canonical Height On Elliptic Surfaces III: Global Boundedness Properties

Let E → C be an elliptic surface defined over a number field K, let P: C → E be a section, and for each t ∈ C(K), let ĥ(Pt) be the canonical height of Pt ∈ Et (K). Tate has used a global argument to show that, up to a bounded quantity, the function t ↦ ĥ(Pt) is equal to a Weil height function hC(t) on C. In this paper we precisely describe the behavior of the difference ĥ(Pt) − hC(t) as a function of t.

Variation of the Canonical Height On Elliptic-Surfaces II:: Local Analyticity Properties

Let E → C be an elliptic surface defined over a number field K, let P: C → E be a section, and for each t ∈ C(K), let ĥ(Pt) be the canonical height of Pt ∈ Et(K). Tate has used a global argument to show that, up to a bounded quantity, the function t ↦ ĥ(Pt) is equal to a Weil height function on C. In this paper we study the variation of the canonical local height λ̂(Pt). In Part III we will use our local result to show that Tate′s bounded quantity is quite well-behaved as a function of t.

The Difference Between the Weil Height and the Canonical Height on Elliptic Curves

The Difference Between the Weil Height and the Canonical Height on Elliptic Curves

The difference between the Weil height and the canonical height on elliptic curves

Estimates for the difference of the Weil height and the canonical height of points on elliptic curves are used for many purposes, both theoretical and computational. In this note we give an explicit estimate for this difference in terms of the j-invariant and discriminant of the elliptic curve. The method of proof, suggested by Serge Lang, is to use the decomposition of the canonical height into a sum of local heights. We illustrate one use for our estimate by computing generators for the Mordell-Weil group in three examples.

On the difference of the Weil height and the N�ron-Tate height

On the difference of the Weil height and the N�ron-Tate height

HEIGHT ON FAMILIES OF ABELIAN VARIETIES

Let be an abelian variety imbedded in projective space, and let be an induced invertible sheaf on . In this paper explicit bounds are determined for the difference , where is the Néron-Tate height and is the Weil height. Bibliography: 5 titles.