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