[noun] + の下で(のもとで) [noun] + の下に(のもとに)
Note: 下 is read もと in this pattern. The noun typically represents a person (leader/supervisor), system, law, rule, condition, or principle.
The grammar pattern の下で (のもとで) is used to express that something takes place under a certain condition, authority, supervision, or influence. It connects a noun representing a leader, system, condition, or principle to an action or situation that occurs within that framework.
While the kanji した下 normally reads した or した, in this expression it is read もと. This is a fixed reading specific to this pattern.
The most common use of の下で is to describe doing something under the guidance, leadership, or supervision of a person or authority.
の下で can also describe actions or states that exist within the framework of a particular system, law, or set of rules.
It can also describe something happening under certain conditions or circumstances.
A closely related form is の下に (のもとに), which is often interchangeable with の下で. However, の下に tends to emphasize the state of being under something (a more static relationship), while の下で emphasizes the action that takes place within that context. For example, せんせい先生のしたに下にあつまる集まる (gather under the teacher) focuses on the gathering toward the teacher, whereas せんせい先生のもと下でまなぶ学ぶ (study under the teacher) focuses on the activity of studying.