Given two smooth elliptic curves $C\_1$ and $C\_2$ over $\mathbb{C}$. Assume they are not isogenous. I'm interested in the structure of $Pic(A)$ and $Pic^{0}(A)$ for $A:=C\_1 \times C\_2$.


Reading Birkenhake/Lange - Complex Abelian Varieties, i think this has to do with correspondences of curves. Since an elliptic curve is its own Jacobian and the two curves are not isogenous, we have $Hom(C\_1,C\_2)=0$. So the space of correspondences $Corr(C\_1,C\_2)$ is trivial, i.e. every line bundle $L$ on $A$ is of the form $L=q^{\\*}M\otimes p^{\\*}N$, where q and p are the projections on the factors and $M$ and $N$ are line bundles on the factors. This implies $Pic(A)=Pic(C\_1)\times Pic(C\_2)$.


Does this impliy $Pic^{0}(A)=Pic^{0}(C\_1)\times Pic^{0}(C\_2)$? That is, is the Picard variety of $A$ isomorphic to $A$ in this case? 



Line bundles on special abelian surfaces
Yes: in fact $Pic^0(C\_1\times C\_2)=Pic^0(C\_1)\times Pic^0(C\_2)$ for any pair of curves. The fact that $C\_1$ and $C\_2$ are not isogenous in your case only affects the Neron-Severi group $Pic/Pic^0$ of $C\_1\times C\_2$, exactly for the reasons you describe.


