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Where can I find a reference (with carefully written proofs) for basic facts about modular curves? Namely:

  • Congruence subgroups

  • The open modular curve $Y_\Gamma$ admits the structure of a Riemann surface (dealing with the issue of orbifold points)

  • The curve $Y_\Gamma$ admits a completion at the cusps to give a closed Riemann surface $X_\Gamma$

Ideally, with no mention of the word "stack". I'm mostly interested in understanding the complex analytic side of the story.

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    $\begingroup$ Perhaps Shimura, Introduction to the arithmetic theory of automorphic functions? $\endgroup$
    – user19475
    Commented Jul 21, 2018 at 7:13
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    $\begingroup$ Another reference (maybe more accessible, depending on your taste) is Diamond-Shurman, A first course in modular forms. $\endgroup$
    – François Brunault
    Commented Jul 21, 2018 at 8:40
  • $\begingroup$ Modular Forms by Miyake is booked that I learned from and liked especially for the topics you mentioned. It’s in the same style as Shimura but a much easier read in my opinion. It doesn’t get to as many advanced topics as Shimura but treats the ones you mentioned well. $\endgroup$
    – Will Dukeminier
    Commented Jul 21, 2018 at 20:38
  • $\begingroup$ Drew Sutherland taught an undergraduate course last year at MIT. He's written up great lecture notes at math.mit.edu/classes/18.783/2017/lectures.html which contain what you're looking for. Diamond-Shurman and Miyake are also nice reads. $\endgroup$
    – skd
    Commented Jul 23, 2018 at 23:27

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