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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?

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    $\begingroup$ Every cusp has a width, which is a positive integer. Cusps of width $n$ will correspond to level structures on $Tate(q^n)$; geometrically these correspond to $n$-gons. $\endgroup$ – David Loeffler Dec 19 '15 at 9:50
  • $\begingroup$ Thanks for commenting! Sorry for being clueless here, but how do you get an $n$-gon from $Tate(q^n)$? (Is it true that $Tate(q^n)$ somehow gives the curve over the completion at the cusps?) $\endgroup$ – Munchlax Dec 19 '15 at 17:02
  • $\begingroup$ I think section 2.5.3 of Hida's book looks relevant. There, he finds an etale map $E'\rightarrow Tate(q)$ over $Spec(\mathbb{Z}[[q^{1/n}]])\rightarrow Spec(\mathbb{Z}[[q]])$ that restricts to $Tate(q^{1/n})\rightarrow Tate(q)$ over the generic fiber and the quotient map from the $n$-gon to 1-gon over the special fiber. Do you know why we'd like the map to be etale, or if there are other references? $\endgroup$ – Munchlax Dec 20 '15 at 4:42

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