Algebra & Number Theory

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

Patrick Ingram

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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 tC(k̄) with [k(t):k]B1 such that t is nonsingular, we obtain a bound on the number of Qt(k̄) such that [k(Q):k]B2, and such that nQ=Pt for some n1. This bound depends on , P, , B1, and B2, but is independent of t.

Article information

Algebra Number Theory, Volume 5, Number 4 (2011), 465-493.

Received: 1 October 2009
Revised: 10 March 2010
Accepted: 21 August 2010
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 14J27: Elliptic surfaces 14G05: Rational points

elliptic surface specialization theorem


Ingram, Patrick. Specializations of elliptic surfaces, and divisibility in the Mordell–Weil group. Algebra Number Theory 5 (2011), no. 4, 465--493. doi:10.2140/ant.2011.5.465.

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