Serre's problem on the density of isotropic fibres in conic bundles

TL;DR

This paper proves the true order of magnitude for the number of rational points on fibers of certain conic bundles over $Q$, confirming Serre's conjecture for bundles of rank up to 3.

Contribution

It establishes the asymptotic behavior of rational points on non-singular conic bundles of rank at most 3, solving a problem posed by Serre in 1990.

Findings

Confirmed the conjectured order of magnitude for $N( pi,B)$ for rank ≤ 3.

Applied advanced sieve methods and divisor sum estimates.

Extended understanding of rational points distribution on conic bundles.

Abstract

Let $\pi:X\to \mathbb{P}^1_{\mathbb{Q}}$ be a non-singular conic bundle over $\mathbb{Q}$ having $n$ non-split fibres and denote by $N(\pi,B)$ the cardinality of the fibres of Weil height at most $B$ that possess a rational point. Serre showed in $1990$ that a direct application of the large sieve yields $$N(\pi,B)\ll B^2(\log B)^{-n/2}$$ and raised the problem of proving that this is the true order of magnitude of $N(\pi,B)$ under the necessary assumption that there exists at least one smooth fibre with a rational point. We solve this problem for all non-singular conic bundles of rank at most $3$. Our method comprises the use of Hooley neutralisers, estimating divisor sums over values of binary forms, and an application of the Rosser-Iwaniec sieve.