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I just read the definition of a divisor of an elliptic curve [1] and I'm not able to understand the syntax. So my questions are targeting the following steps:

1.) What are formal symbols, specially in that case?

2.) How are those coeefficient determined? Is there any "ahh, I see" example?

3.) About what can I think, if I'm confronted with divisors? Are those a kind of an ideal? I know, that they perform an additive group, but I'm not sure, >>how<< to think.

[1] http://people.cs.nctu.edu.tw/~rjchen/ECC2009/22_Divisor.pdf p. 3, def. 1.1

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    $\begingroup$ This appears to be a "power point" presentation from a conference ECC 2008 on elliptic curve cryptography. As such the "definitions" section is more of a summary of notation that is presumed to be already known to its specialized audience. See this thesis Mathematical Foundations of Elliptic Curve Cryptography for a more self-contained presentation of Divisors (Sec. 1.3). $\endgroup$ – hardmath Jun 27 '17 at 12:02
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    $\begingroup$ The set of (Weil) divisors on a smooth projective curve $\mathcal C/K$ is just the free abelian group generated by the points of $\mathcal C$ in an algebraic clsure of $K$. A formal symbol is just one of these points, considered as one of the elements of the basis of this free abelian group. $\endgroup$ – Bernard Jun 27 '17 at 12:10

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