Given an abelian surface $A$ as the product of two non isogenous elliptic curves $E\_1 $ and $E\_2$ over $\mathbb{C}$. We also have a smooth 2 dimensional moduli space $M$ of sheaves on $A$ with some prescribed Chern classes.


I'm interested in finding some irreducible components of $M$. I know $M$ is not empty, i.e. there is a sheaf $E$ on $A$, representing a point in $M$. The sheaf is given as $E=O\_A\oplus L\_1 \oplus L\_2\oplus (L\_1\otimes L\_2)$, where $L\_i$ is a 2-torsion line bundle coming from a 2-torsion bundle on $E\_i$. 


My first idea was to embed $A$ into $M$ via its Picard variety, since $Pic^0(A)=A$ and these line bundles have vanishing Chern classes in $NS(A)$ . If $p\in A$ and $K\_p$ is the associated line bundle, i looked at $\phi: A \rightarrow M$, $p \mapsto E\otimes K\_p$. By Atiyah's version of the Krull-Schmidt theorem we have $E\cong E\otimes K\_p$ if and only if $K\_p \in$ { $O\_A,L\_1,L\_2,L\_1\otimes L\_2$ }.


So the map $p \mapsto E\otimes K\_p$ is not injective. If two line bundles $P$ and $Q$ satisfy $P\cong L\_1\otimes Q$ or $P\cong L\_2\otimes Q$ or $P\cong (L\_1\otimes L\_2)\otimes Q$, then we get $\phi(p)=\phi(q)$ if $P$~$K\_p$ and $Q$~$K\_q$.


So my idea was to "kill" the action of $L\_1, L\_2$ and $L\_1\otimes L\_2$. So define the group $G:=${$O\_A,L\_1,L\_2,L\_1\otimes L\_2$}$\subset Pic^0(A)=A$. 


Questions:
Can we look at $A/G$? Is this an irreducible smooth projective surface? So is $A/G$ an irreducible component in $M$? Is it even an abelian surface, maybe the product product of two other elliptic curves? Or is this idea not so good at all?



Quotient of an abelian surface by a finite group, irreducible components
You are dividing a complex abelian variety by a finite subgroup, so the quotient is again an abelian variety (in particular $G$ acts freely on $A$ and $A/G$ is smooth). This is explained in any text on abelian varieties, for instance in Birkenhake-Lange. 


In addition in your case the group $G$ is equal to $G\_1\times G\_2$, were $G\_i$ is generated by $L\_i$i, and the action is the product of the action of $G\_i$ on $E\_i$, so $A/G=E\_1/G\_1\times E\_2/G\_2$.


