Let $(G, X)$ be a Shimura datum. Take a prime number $p$ and a Bruhat–Tits subgroup $K_p$ of $G(\mathbb{Q}_p)$. Consider the projective limit of the sets of connected components of Shimura varieties for $(G,X)$ whose level at $p$ are given by $K_p$. It is equipped with the prime-to-$p$ Hecke action. Then we discuss the question whether the above action is transitive, which is motivated by the theory of mod $p$ reductions of Shimura varieties. In this talk, we give infinitely many projective systems of the Shimura varieties for CM unitary groups in odd variables for which the considered question is negative. To achieve this goal, we study a question related to the weak approximation on certain tori over $\mathbb{Q}$.

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