Rational points on fibrations with few non-split fibres

TL;DR

This paper refines the fibration method for finding rational points on fibrations over the projective line, making it unconditional in certain cases and improving results under Schinzel's hypothesis by controlling the Brauer--Manin obstruction.

Contribution

It provides an unconditional version of the fibration method for low-degree non-split loci and introduces a new technique to control the Brauer--Manin obstruction in families.

Findings

Unconditional results for non-split locus degree ≤ 2.

Enhanced results under Schinzel's hypothesis.

Incorporation of Harari's technique for Brauer--Manin obstruction.

Abstract

We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the method unconditional when the degree of the non-split locus is $\leq 2$, as well as in various instances where it is $3$. We are also able to obtain improved results in the regime that is conditionally accessible under Schinzel's hypothesis, by incorporating into it, for the first time, a technique due to Harari for controlling the Brauer--Manin obstruction in families.