On the variation of the root number in families of elliptic curves

Abstract

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.