Variation of the Canonical Height On Elliptic Surfaces III: Global Boundedness Properties Original Research Article

Let E → C be an elliptic surface defined over a number field K, let P: C → E be a section, and for each t set membership, variant C(K), let image(Pt) be the canonical height of Pt set membership, variant Et (image). Tate has used a global argument to show that, up to a bounded quantity, the function t maps to image(Pt) is equal to a Weil height function hC(t) on C. In this paper we precisely describe the behavior of the difference image(Pt) − hC(t) as a function of t.