Variation of the canonical height of a point depending on a parameter. (English) Zbl 0618.14019

Tate, J.

Am. J. Math. 105, 287-294 (1983).

Let \(T\) be a smooth irreducible projective curve over an algebraic closure \(k\) of the field of rational numbers, \(K\) the field of rational functions on \(T\), \(E\) an elliptic curve over \(K\) and \(f: V\to T\) be the minimal model of \(E\) over \(T\). Let \(0: T\to V\) be a zero section of \(f\). For each section \(P: T\to V\) there is a divisor class \(q(P):=0^*\) (class of \((P)-(0)\)) on \(T\). Here \((P)\) is a divisor on the surface \(V\) coinciding with the image of \(P\). Let \(h_{q(P)}: T(k)\to \mathbb R\) be a real-valued Weil height function attached to \(q(P)\) [see S. Lang, “Fundamentals of Diophantine geometry.” New York etc.: Springer-Verlag (1983; Zbl 0528.14013)]. For all but finite \(t\) from \(T(k)\) the fiber \(E_ t=f^{-1}(t)\) is an elliptic curve over \(k\) and each \(P\) defines a point \(P_ t\) from \(E(k)\). Let \(\hat h_ t: E_ t(k)\to \mathbb R\) be the Tate height on \(E_ t\) attached to the origin (ibid.). For fixed \(P\) passing through the same irreducible component as 0 of each degenerate fiber the author proves that for varying \(t: \hat h_ t(P_ t)=h_{q(P)}(t)+0(1)\): (Such \(P\) constitute a subgroup of finite index in the group of sections). For additional information, including exposition of Silverman’s results [J. H. Silverman, J. Reine Angew. Math. 342, 197–211 (1983; Zbl 0505.14035)] the reader is referred to Lang’s book cited above.

Reviewer: Yu. G. Zarhin (Moskva)

MSC:

14K15	Arithmetic ground fields for abelian varieties
14J27	Elliptic surfaces, elliptic or Calabi-Yau fibrations
14H45	Special algebraic curves and curves of low genus
14H52	Elliptic curves
14G25	Global ground fields in algebraic geometry
11G05	Elliptic curves over global fields
11G50	Heights

Keywords:

variation of the canonical height; elliptic curve; real-valued Weil height function; Tate height