Explicit period lattices for abelian surfaces

2012-09-27T23:27:36Z

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.

Answer by Will Sawin for Explicit period lattices for abelian surfaces

2012-09-28T02:41:09Z

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 Weil-restricted surface in some other form, how you can get it into product-of-Weierstrass form.

Explicit period lattices for abelian surfaces - MathOverflow

Explicit period lattices for abelian surfaces

Answer by Will Sawin for Explicit period lattices for abelian surfaces