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Let $C$ be a smooth projective curve over a field $k$ and $S \subset C$ a finite number of points. A modulus is simply a divisor supported on $S$. What is the divisor class group with modulus?

I have seen the definition $$ \mathrm{coker}(k(C)^\times \to \bigoplus_{x \in X-S} \mathbb{Z} \oplus \bigoplus_{x \in S} k(C)^\times / 1+\mathfrak{m}_x) $$ but it does not speak to me... What is the meaning of this group, the intuition behind the definition? How is it related to class field theory?

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    $\begingroup$ See Serre Algebraic Groups and Class Fields. $\endgroup$
    – abx
    Commented Jan 20, 2014 at 10:30
  • $\begingroup$ Could you give me a more precise reference, e.g. chapter, section? $\endgroup$
    – sruobera
    Commented Jan 20, 2014 at 10:47
  • $\begingroup$ I would say the whole book is about the subject; in particular chapters 5 and 6. $\endgroup$
    – abx
    Commented Jan 20, 2014 at 10:52
  • $\begingroup$ Ok, I will have a look at Serre's book. But maybe you can give me some guidelines? $\endgroup$
    – sruobera
    Commented Jan 20, 2014 at 18:01
  • $\begingroup$ Serre does it much better than I could. And I don't know what you are interested in. You better look at the book. $\endgroup$
    – abx
    Commented Jan 20, 2014 at 18:18

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