Density of rational points on isotrivial rational elliptic surfaces

Várilly-Alvarado, Anthony

doi:10.2140/ant.2011.5.659

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Density of rational points on isotrivial rational elliptic surfaces

Anthony Várilly-Alvarado

Vol. 5 (2011), No. 5, 659–690

DOI: 10.2140/ant.2011.5.659

Abstract

For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers among the fibers of these surfaces. We also prove that these surfaces satisfy a variant of weak-weak approximation. Our results are conditional on the finiteness of Tate–Shafarevich groups for elliptic curves over the field of rational numbers.

Keywords

rational elliptic surfaces, del Pezzo surfaces, root numbers

Mathematical Subject Classification 2000

Primary: 11G35

Secondary: 14G05, 11G05

Milestones

Received: 22 September 2010

Revised: 26 September 2010

Accepted: 24 October 2010

Published: 23 January 2012

Proposed: Hendrik W. Lenstra

Seconded: Andrew Granville, Bernd Sturmfels

Authors

Anthony Várilly-Alvarado
Department of Mathematics Rice University MS 136 Houston, TX 77005 United States
http://math.rice.edu/~av15/

Vol. 5, No. 5, 2011