A quasi-linear time algorithm for computing modular polynomials in dimension 2

We propose to generalize the work of Regis Dupont for computing modular polynomials in dimension $2$ to new invariants. We describe an algorithm to compute modular polynomials for any invariants derived from theta constants and prove that this algorithm is quasi-linear.Some properties of the modular polynomials with the quotient of theta constants are analyzed.We report on experiments with our implementation.