Serre's problem for multiple conics

TL;DR

This work proves an asymptotic formula for the frequency of globally solubility in a broad family of fibre products of conic bundles, extending Serre’s solubility-in-families program and the LRS refinements. The authors develop a robust character-sum framework that reveals a deep link between main-term contributions and the subordinate Brauer group, enabling an explicit Euler-product description of the leading constant. They verify the LRS conjecture in this setting, including the projective normalization, and further apply the machinery to the Rédei symbol, obtaining a precise distribution result via a trilinear large sieve. The results underscore the prominent role of Brauer–Manin-type obstructions in arithmetic statistics and provide a concrete, computable leading constant in a high-dimensional conic-bundle context. The techniques combine delicate combinatorics, large-sieve estimates, and a detailed local–global Brauer-analysis, with potential applicability to broader fibrations beyond diagonal conics.

Abstract

We prove the refined Loughran--Smeets conjecture of Loughran--Rome--Sofos for a wide class of varieties arising as products of conic bundles. One interesting feature of our varieties is that the subordinate Brauer group may be arbitrarily large. As an application of our methods, we answer a question of Lenstra by giving an asymptotic for the triples of integers $(a, b, c)$ for which the Rédei symbol $[a, b, c]$ takes a given value. We also make significant progress on a question of Serre on the zero loci of systems of quaternion algebras defined over $\mathbb{Q}(t_1, \dots, t_n)$.

Theorems & Definitions (64)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof