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I recently heard that the modular curves $Y(\Gamma)$ are affine, which is surprising for me (I wasn't expecting that at all).

Why is it true? Is true for other Shimura varieties?

Marsault Chabat
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  • What is the definition of the modular curve that you are using? Do you know about the j-invariant? https://en.m.wikipedia.org/wiki/J-invariant – Moishe Kohan Nov 10 '23 at 15:18
  • Hi, I think the definition doesn't matter right? If either (a quotient of) the Poincaré half plane, or a finite number of copies of that (i.e. the adelic definition) is affine then the other will be right? Even if we consider the moduli spaces of elliptic curves: it is affine iif its complex points form an affine variety right?

    I know about the $j$-invariant, but I dont know how it helps to show the affineness...

     – Marsault Chabat Nov 10 '23 at 15:27
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    No, if you do not know an actual definition, you cannot prove anything. From what you wrote, a modular curve could as well be projective, not affine! – Moishe Kohan Nov 10 '23 at 15:41
  • Ok, so I'm interested in basically every case, but we can begin with the moduli space of elliptic curve with level $N$ structure (over the cyclotomic extension). – Marsault Chabat Nov 10 '23 at 16:07
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    See this question: https://math.stackexchange.com/questions/1594240/must-a-proper-curve-minus-a-point-be-affine. Modular curves have cusps, hence they are affine. – David Loeffler Nov 10 '23 at 16:44
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    (Most Shimura varieties are neither affine nor projective, but curves are just easy.) – David Loeffler Nov 10 '23 at 16:45
  • Ooooh I see, thank you it's very enlightening!! – Marsault Chabat Nov 10 '23 at 17:16

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