Principes locaux-globaux pour certaines fibrations en torseurs sous un tore

TL;DR

This paper investigates the Brauer-Manin obstruction to the Hasse principle and weak approximation for certain fibrations in torsors under a torus over number fields, extending recent results under Schinzel's hypothesis.

Contribution

It generalizes recent work by analyzing fibrations in torsors under tori, providing new results on the Hasse principle and weak approximation under Schinzel's hypothesis.

Findings

Unconditional results over $\\mathbb{Q}$ when non-split fibres are defined over $\mathbb{Q}$.

Extension of Wei's work to more general fibrations in torsors under tori.

Application of Schinzel's hypothesis to study the Brauer-Manin obstruction.

Abstract

Let $k$ be a number field and let $T$ be a $k$-torus. Consider a fibration in torsors under $T$, i.e. a morphism $f: X \to \mathbb{P}^1_k$ from a smooth, projective $k$-variety $X$ to $\mathbb{P}^1_k$ such that the generic fibre $X_\eta \to \eta$ is a smooth compactification of a principal homogeneous space under $T \times_k \eta$. We study the Brauer-Manin obstruction to the Hasse principle and weak approximation for $X$, under Schinzel's hypothesis, thereby generalizing recent work of Wei. Our results are unconditional if $k = \mathbb{Q}$ and the non-split fibres of $f$ are defined over $\mathbb{Q}$.