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

Ingram, Patrick

doi:10.2140/ant.2011.5.465

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

Patrick Ingram

Vol. 5 (2011), No. 4, 465–493

DOI: 10.2140/ant.2011.5.465

Abstract

Let $ℰ \to C$ be an elliptic surface defined over a number field $k$ , let $P : C \to ℰ$ be a section, and let $ℓ$ be a rational prime. We bound the number of points of low algebraic degree in the $ℓ$ -division hull of $P$ at the fibre $ℰ_{t}$ . Specifically, for $t \in C (\bar{k})$ with $[k (t) : k] \leq B_{1}$ such that $ℰ_{t}$ is nonsingular, we obtain a bound on the number of $Q \in ℰ_{t} (\bar{k})$ such that $[k (Q) : k] \leq B_{2}$ , and such that $ℓ^{n} Q = P_{t}$ for some $n \geq 1$ . This bound depends on $ℰ$ , $P$ , $ℓ$ , $B_{1}$ , 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

Vol. 5, No. 4, 2011