I am looking for an algorithm to obtain the period matrix tau= (tau1 & tau12\ tau12 & tau2) from the Igusa invariants of a genus two curve. More precisely:
I consider a family of genus two curves given in hyperlliptic form y^2 = f(x,z1, z2, z3). Here, f is a polynomial of degree six in x and z1, z2, z3 are moduli parametrizing the family. Now I can compute the Igusa invariants I2(z1, z2, z3), ... I10(z1, z2, z3). But what I am interested in is the dependence tau(z1, z2, z3). I know that the absolute Igusa invariants x1, x2, x3 can be written in terms of Siegel modular forms of genus two, but it seems difficult to invert these relations.
Thanks a lot for your help in advance.