Answer to a decomposition question on tori raised by Colliot-Thélène and Sansuc

Abstract

The aim of this note is to present a simple strategy to answer negatively a decomposition question on tori posed by Colliot-Thélène and Sansuc in the article \textit{Principal Homogeneous Spaces under Flasque Tori: Applications} of 1987. We then deduce a torus $T$ over $\mathbb{Q}$ and a prime number $p$ such that $T({\mathbb{Z}}_p)T(\mathbb{Q})\ne T({\mathbb{Q}}_p)$, where $T({\mathbb{Z}}_p)$ denotes the maximal compact subgroup of $T({\mathbb{Q}}_p)$.

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