Rational points on fibrations with few non-split fibres

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 $\le 2$, 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.

arXiv comment46 pages; v2: exposition improved

References in corpus (7)

• On the fibration method for zero-cycles and rational points
• The Hardy--Littlewood conjecture and rational points
• Higher dimensional analogues of Châtelet surfaces
• The Hasse norm principle for $A_n$-extensions
• On the square-free representation function of a norm form and nilsequences
• Principes locaux-globaux pour certaines fibrations en torseurs sous un tore
• Strong approximation for a family of norm varieties

Cited by in corpus (3)

• Supersolvable descent for rational points
• Retour sur l'arithmétique des intersections de deux quadriques, avec un appendice par A. Kuznestov
• Polynomials represented by norm forms via the beta sieve