Vol. 6, No. 7, 2012

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On the rank of the fibers of rational elliptic surfaces

Cecília Salgado

Vol. 6 (2012), No. 7, 1289–1314
Abstract

We consider an elliptic surface π : 1 defined over a number field k and study the problem of comparing the rank of the special fibers over k with that of the generic fiber over k(1). We prove, for a large class of rational elliptic surfaces, the existence of infinitely many fibers with rank at least equal to the generic rank plus two.

Keywords
elliptic surface, rational surface, Mordell–Weil group, elliptic curve
Mathematical Subject Classification 2010
Primary: 14J27
Secondary: 11G05, 14D99
Milestones
Received: 18 November 2010
Revised: 19 December 2011
Accepted: 24 January 2012
Published: 4 December 2012
Authors
Cecília Salgado
Mathematisch Instituut
Universiteit Leiden
Niels Bohrweg, 1
2333 CA Leiden
The Netherlands
Instituto de Matemática
Universidade Federal do Rio de Janeiro
21941-909 - Rio de Janeiro, RJ
Brazil