noun, noun which may take the genitive case particle 'no'
under the rule of; under the reign of
Formal, somewhat literary term used in historical or political contexts to indicate being under a particular ruler, government, or administration. Often appears in the pattern 〜の治下に.
この地域は長く徳川幕府の治下にあった。
This region was long under the rule of the Tokugawa shogunate.
治下の民は重税に苦しんだ。
The people under his rule suffered from heavy taxes.
支配下 is more common and neutral for 'under control/domination', while 治下 specifically implies governance or reign, often in a historical context.
Compound of 治 (chi, 'govern, rule') and 下 (ka, 'under'). The reading is on'yomi. The exact historical derivation is uncertain, but the word follows a standard Sino-Japanese pattern for 'under the rule of'.