The Inverse Jacobian problem

To an algebraic curve C over the complex numbers one can associate a non-negative integer g, the genus, as a measure of its complexity.
One can also associate to C, via complex analysis, a g×g symmetric matrix Ω called period matrix. Because of the natural relation
between C and Ω, one can obtain information of one by studying the other. Therefore, it makes sense to consider the inverse problem:
Given a period matrix Ω, can we compute a model for the associated curve C?

In this talk, we will revise how to construct the period matrix of a curve, we will see some known results around this problem, and discuss
some application of its solution.