Canonical heights, for example the N\’eron-Tate height on an abelian variety, are one of the key tools for studying both abelian varieties and algebraic dynamical systems. Given a family \(X\to T\) of dynamical systems with a section \(P:T\to X\), the content of several theorems and conjectures is essentially that the canonical height of \(P_t\) on each fiber should be given by a height \(h_P\) on \(T\). I’ll show how, using adelic line bundles, this problem can be translated into arithmetic intersection theory, and prove the corresponding statement for a one-parameter family of abelian varieties.

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