  We show the density of rational points on non-isotrivial elliptic surfaces by
studying the variation of the root numbers among the fibers of these surfaces,
conditionally to two analytic number theory conjectures (the squarefree
conjecture and Chowla's conjecture). This is a weaker statement than the one
found in a preprint of Helfgott which proves (under the same assumptions) that
the average root number is $0$ when the surface admits a place of
multiplicative reduction. However, we use a different technique. The
conjectures involved impose a restriction on the degree of the irreducible
factors of the discriminant of the surfaces.
  Moreover, we manage to drop the squarefree conjecture assumption under some
technical hypotheses, and show thus unconditionally the variation of the root
number on many elliptic surfaces, without imposing a bound for the degree of
the irreducible factors. Under the parity conjecture, this guarantees the
density of the rational points on these surfaces.
