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

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 $f:X\to {\mathbb{P}}_k^1$ from a smooth, projective $k$-variety $X$ to ${\mathbb{P}}_k^1$ 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 $\mathbb{Q}$.

17 pages, in French; final version, to appear in Mathematical Proceedings of the Cambridge Philosophical Society

References in corpus (2)

• The Hardy--Littlewood conjecture and rational points
• Norms as products of linear polynomials

Cited by in corpus (3)

• On the fibration method for zero-cycles and rational points
• Rational points on fibrations with few non-split fibres
• Towards the Brauer-Manin obstruction on varieties fibred over the projective line