Specializations of elliptic surfaces, and divisibility in the Mordell-Weil group

Abstract

Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic) degree at most $D$ over $k$, such that ${\ell }^{n}Q=P_t$, for some $n\ge 1$. The bound obtained depends only on $\ell$, the surface and section in question, $D$, and the degree $[k(t):k]$; that is, it is uniform across all fibres of bounded degree. In special cases, we obtain more specific, in some instances sharp, bounds.

Introduction re-written, and minor additions. The results of the paper are largely unchanged, with one small observation added

References in corpus (1)

• Torsion points on elliptic curves over function fields and a theorem of Igusa