Given an explicit description (as an intersection) of an abelian surface $A$ is there an algorithm for computing the period lattice of the surface? For the specific examples that I am interested in, the ideal of $A$ has been obtained by Weil restriction from the affine model of an elliptic curve.

It seems to me that it's best to go back up to the quadratic extension and view it as a product of two elliptic curves $E \times E^\sigma$. If you can compute the Weierstrass equations for these elliptic curves, you can compute their $j$ invariants, and then you just need to find $\tau_1,\tau_2$ such that $j(\tau_1)=j(E)$, $j(\tau_2)=j(E^\sigma)$. Then the lattice is $\mathbb C^2 /((1,0),(\tau_1,0),(0,1),(0,\tau_2))$. So if you have a Weierstrass equation for the curve you're Weil restricting, you're done, modulo the analytic process of inverting $j$, which should be computable to an arbitrary degree of accuracy, and exactly computable if your $j$ is a special value. A question I'm not sure about is, if you have the equation for the Weilrestricted surface in some other form, how you can get it into productofWeierstrass form. 

