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arXiv · 1305.0756

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

Arne Smeets
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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 $X_η\to η$ is a smooth compactification of a principal homogeneous space under $T \times_k η$. 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 $\mathbb{Q}$.

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Arne Smeets. 2015-01-11. Principes locaux-globaux pour certaines fibrations en torseurs sous un tore. https://doi.org/10.1017/s0305004114000577

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