Variation of the Canonical Height On Elliptic-Surfaces II: : Local Analyticity 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(image), 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 on C. In this paper we study the variation of the canonical local height image(Pt). In Part III we will use our local result to show that Tate′s bounded quantity is quite well-behaved as a function of t.