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arXiv · 2109.03547

Rational points on fibrations with few non-split fibres

Yonatan Harpaz·Dasheng Wei·Olivier Wittenberg
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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.

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Yonatan Harpaz, Dasheng Wei, Olivier Wittenberg. 2021-09-08. Rational points on fibrations with few non-split fibres. https://doi.org/10.1515/crelle-2022-0042

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