On the rank of the fibers of rational elliptic surfaces

Salgado, Cecília

doi:10.2140/ant.2012.6.1289

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On the rank of the fibers of rational elliptic surfaces

Cecília Salgado

Vol. 6 (2012), No. 7, 1289–1314

DOI: 10.2140/ant.2012.6.1289

Abstract

We consider an elliptic surface $π : ℰ \to ℙ^{1}$ 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 $k (ℙ^{1})$ . 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

Vol. 6, No. 7, 2012