On the connected components of Shimura varieties for CM unitary groups in odd variables

Abstract: We study the prime-to-$p$ Hecke action on the projective limit of the sets of connected components of Shimura varieties with fixed parahoric or Bruhat--Tits level at $p$. In particular, we construct infinitely many Shimura varieties for CM unitary groups in odd variables for which the considering actions are not transitive. We prove this result by giving negative examples on the question of Bruhat--Colliot-Th\'el`ene--Sansuc--Tits or its variant, which is related to the weak approximation on tori over $\mathbb{Q}$.