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Given a complex abelian variety $A = V/\Gamma$ (for $\Gamma$ being a lattice in the complex vector space $V$), one knows how to describe a holomorphic line bundle in terms of factors of automorphy: By general theory, we can describe $\mathrm{Pic}(A)$ as $H^1(\Gamma; \mathcal{O}_V^{\times})$, whose cocycles are the mentioned factors of automorphy.

As an aside I want to mention that one can actually calculate this first cohomology group in a nice way: We can look at the first Chern class $\mathrm{Pic}(A) \to H^2(A; \mathbb{Z})$. The kernel $\mathrm{Pic}^0(A)$ can be described as homomorphisms from $\Gamma$ to $U(1)$; the image $NS(A)$ consists of those $\mathbb{Z}$-valued alternating forms $E$ on $\Gamma$ (the group of which can be identified with $H^2(A;\mathbb{Z})$) that are compatible with the complex structure on $V$, i.e. $E(v,w) = E(iv, iw)$, once we extend $E$ to the $V$. These two results can even be combined to give a canonical factor of automorphy for every line bundle on $A$, thus giving a very concrete description of the cohomology group above. This is all explained in more detail in the book of Birkenhake and Lange, Chapter 2, in the context of the Appell--Humbert theorem.

On the other hand, the Picard group is also isomorphic to the divisor class group.

Question: Is there an explicit way to go from a factor of automorphy to a divisor on $A$?

My natural impulse would be do write down a theta function that is a section of the line bundle associated with the given factor of automorphy and study its divisor of zeros and poles. But I must admit that I found the literature on theta functions a bit too hard to navigate to make this work (which says more about my navigation skills than the literature).

Edit: While indeed both Birkenhake&Lange and Mumford (thanks to Donu Arapura for the hint) say some interesting (and more readable than I thought) things about theta functions, they seem to stop short of computing the divisor of zeros. But I saw some helpful things in Griffiths and Harris: If the matrix corresponding to the first Chern class of a given line bundle $\mathcal{L}$ has determinant $1$, then $\mathcal{L}$ has $1$-dimensional global sections, i.e. there is a unique divisor associated to $\mathcal{L}$. A result going back to Riemann makes this more explicit if $A$ is a Jacobian of a curve $C$: here $V$ is the dual of the space of holomorphic $1$ forms, $\Gamma$ is given by integrating against $H_1(C;\mathbb{Z})$ and one has an associated line bundle whose first Chern class corresponds essentially to integrating the wedge product of two one-forms. Then up to a translate the associated divisor is $W_{g-1}$, i.e. everything that one can write as the sum of $g-1$ points in the image of $C \to A$. If $\mathcal{L}$ is suitably chosen, then this translate is by a half of minus the image of the canonical divisor. (Chapter 2, Section 7 of Griffiths and Harris.)

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    $\begingroup$ Going to the theta function sounds like the right approach to me. It sounds like you just want an efficient reference. Have you have looked at Mumford's Abelian Varieties? He covers the basic analytic theory in chapter 1 in under 40 pages. $\endgroup$ Jul 15, 2021 at 14:20
  • $\begingroup$ If the line bundle is ample then it will correspond to natural effective divisors, i.e., the zero locus of a section, but there are no natural divisors associated to arbitrary line bundles. So it seems unlikely that there is a natural and explict way to go from all factors of automorphy to divisors. $\endgroup$
    – naf
    Jul 16, 2021 at 1:23

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