Variation of the canonical height on elliptic surfaces. III: Global boundedness properties
[For part II of this paper see ibid. 291-329 (1994); see the preceding review.] Let \(E \to C\) be an elliptic surface defined over a number field \(K\), let \(P : C \to E\) be a section, and for each \(t \in C (\overline K)\), let \(\widehat h (P_ t)\) be the canonical height of \(P_ t \in E_ t (\overline K)\). \textit{J. Tate} [Am. J. Math. 105, 287-294 (1983; Zbl 0618.14019)] has used a global argument to show that, up to a bounded quantity, the function \(t \mapsto \widehat h (P_ t)\) is equal to a Weil height function \(h_ C\) on \(C\). In this paper the author combines his local analyticity results [cf. part II of this paper (loc. cit.)]\ with a global boundedness theorem to precisely describe the behavior of the difference \(\widehat h (P_ t) - h_ C (t)\) as a function of \(t\).
scientific article; zbMATH DE number 645302