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

Let C be an elliptic surface defined over a number field k, let P : C be a section, and let be a rational prime. We bound the number of points of low algebraic degree in the -division hull of P at the fibre t. Specifically, for t C(k̄) with [k(t) : k] B1 such that t is nonsingular, we obtain a bound on the number of Q t(k̄) such that [k(Q) : k] B2, and such that nQ = Pt for some n 1. This bound depends on , P, , B1, and B2, but is independent of t.

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

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