under the rule of; under the government of
Used to describe a territory, people, or period being governed by a particular authority. Often appears in the pattern 〜の統治下.
The island was under foreign rule for many years.
In the region under governance, new laws were enacted.
支配下 is similar but can imply control or domination more broadly, not necessarily official governance.
Compound of 統治 (rule, governance) and 下 (under).