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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.

Maximilian

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    $\begingroup$ I don't know if I understand your question properly. I think you know the Thomae Theorem that expresses the 4th powers of theta constants in terms of the tableaux invariants of the 6 points on the line, don't you? $\endgroup$
    – IMeasy
    Jan 31, 2014 at 12:44
  • $\begingroup$ No to be honest I don't know it. Can you recommend me a good reference? So to perhaps reformulate my question: In the case of an elliptic curve, I can compute the J-function from the Weierstrass normal form and use the Fourier expansion in exp(2 pi i tau) of the J-function to compute the modulus tau in terms of the parameters parametrizing my curve. I am looking for an analog on in case of genus two curves. $\endgroup$
    – Maximilian
    Feb 1, 2014 at 13:15

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It's par 5-6-7 of chapter eight of Asterisque 165 by Dolgachev and Ortland. About your question on EC, it seems reasonable, you should check especially par 6. Let us know what you find out!

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