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).

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.)