[noun] + の下で [noun] + の下に [noun] + の下の + [noun]
の下で means "under": something happens under a person's guidance, or within a system, a law or a set of conditions. It follows a noun, and 下 is read もと here, not した. Plenty of writers use kana instead: のもとで.
Read した, 下 is the space physically below something, as in き木のした下で, "under the tree". Read もと, it is the figurative "under" of this pattern. The kanji is the same, so the noun in front tells you which: a place or an object means した, and a person, a rule or a condition means もと.
These go beyond の下で before a verb.
の下に means the same as の下で and is more formal. It also appears in fixed phrases such as 〜~のな名のしたに下に, "in the name of".
Before a noun, the pattern becomes の下の.