Almost always takes an abstract object: 许下承诺 (chéngnuò, promise), 许下誓言 (shìyán, oath), 许下愿望 (yuànwàng, wish), 许下心愿 (xīnyuàn, heartfelt wish). 下 here is a directional complement marking that the act is fixed or set down.
grammar
Don't separate: 许下 is treated as a unit. The aspect particle 了 follows the whole verb (许下了), not between (✗许下了一个 — wait, that is fine; ✗许了下 — wrong).
