Canonical Heights and the Arithmetic Complexity of Morphisms on Projective Space

The theory of canonical heights on abelian varieties originated with the work of Néron [10] and Tate (first described in print by Manin [8]) in 1965. Tate’s simple and elegant limit construction uses a Cauchy sequence telescoping sum argument. Néron’s construction, which is via more delicate local tools, has proven to be fundamental for understanding the deeper properties of the canonical height. Canonical heights appear prominently in the conjecture of Birch and Swinnerton-Dyer, so early efforts to check the conjecture numerically required the computation of ĥ(P ) to at least a few decimal places. In the mid-1970’s, John Coates used Tate’s limit definition/construction