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Hi this article http://www.ams.org/notices/200302/what-is.pdf mentioned "$p − q$ in the group of divisor classes $H^{1}(M, O^{∗})$, where $M$ is the elliptic curve and $O^{∗}$ is the sheaf of nonzero holomorphic functions on it."

In the last three years of practicing math I have just been learning divisors are formal sums of the order of a function along a cover or the 1-codimensional subvarieties. I don't know how to write the subvariety or cover in the product of the order yet but I don't see how the zero $p$ and pole $q$ are expected to fit in there.

May I have a brief explanation of the difference between the Weil and Cartier divisor and the divisor class mentioned in the article?

Ohio skateboarder . 7
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  • Plenty of textbooks deal with this in depth, and it would be a good idea to start with one of those if you're serious about the subject (if you're more interested in just decoding this article, that might be another matter). Vakil's Foundations of Algebraic Geometry touches on them in chapter 14, while Hartshorne deals with them in chapter II section 6. – KReiser Nov 06 '21 at 00:15
  • There are some other good questions on this site about this: https://math.stackexchange.com/questions/610507/correspondence-between-cartier-divisor-and-weil-divisor-hartshorne-proposition – A. Thomas Yerger Nov 06 '21 at 16:15

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