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

TL;DR

This paper investigates the prime-to-p Hecke action on the connected components of Shimura varieties for CM unitary groups in odd variables, revealing non-transitive actions and providing counterexamples related to weak approximation on tori.

Contribution

It constructs infinitely many Shimura varieties with non-transitive Hecke actions, offering new insights into their structure and related arithmetic properties.

Findings

Existence of infinitely many Shimura varieties with non-transitive prime-to-p Hecke actions

Counterexamples to Bruhat--Colliot-Thélène--Sansuc--Tits question

Connections to weak approximation on tori over bQ

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 $\mathbb{Q}$.