We prove the following special case of Mazur's conjecture on the topology of
rational points. Let E be an elliptic curve over Q with
j-invariant 1728. For a class of elliptic pencils which are quadratic
twists of E by quartic polynomials, the rational points on the projective
line with positive rank fibres are dense in the real topology. This extends
results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic
polynomials.
For the proof, we investigate a highly singular rational curve on the Kummer
surface K associated to a product of two elliptic curves over Q,
which previously appeared in publications by Mestre, Kuwata-Wang and Satg\'e.
We produce this curve in a simpler manner by finding algebraic equations which
give a direct proof of rationality. We find that the same equations give rise
to rational curves on a class of more general surfaces extending the Kummer
construction. This leads to further applications apart from Mazur's conjecture,
for example the existence of rational points on simultaneous twists of
superelliptic curves.
Finally, we give a proof of Mazur's conjecture for the Kummer surface K
without any restrictions on the j-invariants of the two elliptic curves.Comment: 14 pages, same content as published version except for added remark
acknowledging overlap with prior work by Ula