So there are many references where one can learn about cusps of modular curves in the analytical sense, one of them being Diamond and Shurman: A first course in modular forms. However, I'm looking to find a good source of information from the algebraic standpoint.
For example, one of the things that I'm interested in is finding the cusps of specific modular curves. Something similar has been done in the question here, where cusps of the modular group $\Gamma_0(4)$ were found. I would like to see how to get cusps on the modular curve, rather than in the modular group.
I'm also interested in Galois action on the cusps. I've heard that there are no fixed points for that action, but can't seem to find anything about it anywhere.