Vol. 5, No. 4, 2011

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

Patrick Ingram

Vol. 5 (2011), No. 4, 465–493
Abstract

Let $\mathsc{ℰ}\to C$ be an elliptic surface defined over a number field $k$, let $P:C\to \mathsc{ℰ}$ be a section, and let $\ell$ be a rational prime. We bound the number of points of low algebraic degree in the $\ell$-division hull of $P$ at the fibre ${\mathsc{ℰ}}_{t}$. Specifically, for $t\in C\left(\stackrel{̄}{k}\right)$ with $\left[k\left(t\right):k\right]\le {B}_{1}$ such that ${\mathsc{ℰ}}_{t}$ is nonsingular, we obtain a bound on the number of $Q\in {\mathsc{ℰ}}_{t}\left(\stackrel{̄}{k}\right)$ such that $\left[k\left(Q\right):k\right]\le {B}_{2}$, and such that ${\ell }^{n}Q={P}_{t}$ for some $n\ge 1$. This bound depends on $\mathsc{ℰ}$, $P$, $\ell$, ${B}_{1}$, and ${B}_{2}$, but is independent of $t$.

Keywords
elliptic surface, specialization theorem
Mathematical Subject Classification 2000
Primary: 11G05
Secondary: 14J27, 14G05
Milestones
Received: 1 October 2009
Revised: 10 March 2010
Accepted: 21 August 2010
Published: 21 December 2011

Proposed: Barry Mazur
Seconded: Andrew Granville
Authors
 Patrick Ingram Department of Pure Mathematics University of Waterloo Waterloo, ON N2L 3G1 Canada Department of Mathematics Colorado State University Fort Collins, CO 80523-1874 United States