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

##### 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
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