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

TL;DR

This paper investigates how root numbers vary across fibers of non-isotrivial elliptic surfaces, linking this variation to the density of rational points, under certain conjectural assumptions in number theory.

Contribution

It introduces a new technique to study root number variation on elliptic surfaces, relaxing some conjectural restrictions and providing unconditional results in specific cases.

Findings

Conditional density of rational points on elliptic surfaces.

Variation of root numbers can be shown unconditionally under certain hypotheses.

The method differs from Helfgott's approach, avoiding some restrictions.

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.