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

TL;DR

This paper establishes uniform bounds on the number of points on fibers of elliptic surfaces over number fields that are divisible by powers of a prime, depending only on specific parameters, with sharper bounds in special cases.

Contribution

It introduces uniform bounds for the divisibility of points on fibers of elliptic surfaces over number fields, depending solely on fixed parameters and providing sharper bounds in special cases.

Findings

Bound depends only on prime, surface, section, degree D, and field degree.

In some cases, bounds are sharp.

Results are uniform across fibers of bounded degree.

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 $n\geq 1$. 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.