I know this is a naive question, but what is the relation between the Tate curve and cusps on a modular curve?
Naive googling seems to suggest that level structures on the Tate curve (up to isomorphism) are in correspondence with the cusps of a modular curve. However, more googling says that the fibers over the cusps are n-gons (https://mathoverflow.net/questions/51147/what-objects-do-the-cusps-of-modular-curve-classify).
Suppose we are working over a fixed field. Then, the special fiber of the Tate curve is a nodal curve or 1-gon. Then, where do the n-gons come from for n bigger than 1? If I take formal completions at the cusps of a modular curve, how are the fibers related to the Tate curve?