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Recently, I read 'the red book of varieties and schemes' written by Professor Daivd Mumford. At page 65, he wrote that ' A much deeper connetion is given by class field theory, between the tower of number fields and the tower of coverings of an algebraic curve defined over a finite field.'. In this paragraph, he wanted to give an example of the analogy vetween arithmetic and geometric questions.
I know the main theorems of CFT, but I'm not familiar with the proof. Thus, I'm interesting with this result and wish it can give me a interesting proof. However, when I searched this result, I can only find 'geometric CFT', which is over function fields.
So I'm wondering if there is some reference about this result? Or the 'geometric CFT' is linked with this result?
Thanks in advance.

Zhibin
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  • This post, and others, have many references for books on class field theory, with proofs. Do you know the notes of Milne? – Dietrich Burde Oct 02 '22 at 14:47
  • There's a deep connection much more broadly between number fields and global function fields (that is, function fields of algebraic curves over finite fields). This is known as the "function field analogy" (you can find lots of literature on it under this name). Under this correspondence, class field theory for number fields does correspond to geometric class field theory, is probably what Mumford was referring to. – Daniel Hast Oct 02 '22 at 15:00
  • @DanielHast Thank you for your comments. I’ve searched it, but it seems that it is some kind philosophy but not a precise theorem? For example, if I have the tower of number fields, how can I get the tower of coverings of an algebraic curve? Maybe I missed something, I’ll search more. – Zhibin Oct 03 '22 at 10:14
  • @DietrichBurde Thank you for your comments. I’ve read this. It is a good reference but maybe not the one I’m looking for. Thanks again. – Zhibin Oct 03 '22 at 10:15
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    Right, it's not a precise theorem, and I don't think Mumford was referring to a precise theorem either. – Daniel Hast Oct 03 '22 at 12:12

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