〜のもとで・〜のもとに
under (guidance, conditions, law)
Noun + のもとで / のもとに
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のもとで means under someone's guidance or influence, or under certain conditions: 田中先生のもとで、研究をしている, "I'm doing research under Professor Tanaka".
It's used for teachers, supervisors and leaders (新しい社長のもとで, "under the new president"), for conditions (厳しい条件のもとで, "under strict conditions") and for protection or care (両親の愛情のもとで).
のもとに is similar but more formal, and common with principles and authority: 法のもとに、すべての人は平等だ, "all people are equal under the law".
It's written with 下 or 元 (の下で, の元で). Don't confuse it with the physical 下に, "underneath", which describes where an object sits rather than whose influence you're under.
Examples
田中先生のもとで、研究をしている。
I'm doing research under Professor Tanaka.
厳しい条件のもとで、実験が行われた。
The experiment was carried out under strict conditions.
両親の愛情のもとで育った。
I grew up surrounded by my parents' love.
法のもとに、すべての人は平等だ。
All people are equal under the law.
新しい社長のもとで、会社は大きく変わった。
Under the new president, the company changed a great deal.