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

Abstract

Let $\pi :X\to {\mathbb{P}}_{\mathbb{Q}}^1$ 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.

References in corpus (2)

• Uniformly counting rational points on conics
• What is the probability that a random integral quadratic form in $n$ variables has an integral zero?

Cited by in corpus (2)

• Rational points on del Pezzo surfaces of degree four
• Counting rational points on quartic del Pezzo surfaces with a rational conic