Vol. 5, No. 5, 2011

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Density of rational points on isotrivial rational elliptic surfaces

Anthony Várilly-Alvarado

Vol. 5 (2011), No. 5, 659–690

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.

rational elliptic surfaces, del Pezzo surfaces, root numbers
Mathematical Subject Classification 2000
Primary: 11G35
Secondary: 14G05, 11G05
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
Anthony Várilly-Alvarado
Department of Mathematics
Rice University
MS 136
Houston, TX 77005
United States