NON-HYPERELLIPTIC MODULAR JACOBIANS OF DIMENSION 3

We present a method to solve in an efficient way the problem of constructing the curves given by Torelli's theorem in dimension 3 over the complex numbers: For an absolutely simple principally polarized abelian threefold A over C given by its period matrix Omega, compute a model of the curve of genus three (unique up to isomorphism) whose Jacobian, equipped with its canonical polarization, is isomorphic to A as a principally polarized abelian variety. We use this method to describe the non-hyperelliptic modular Jacobians of dimension 3. We investigate all the non-hyperelliptic new modular Jacobians Jac(C-f) of dimension 3 which are isomorphic to A(f), where f is an element of S-2(new) (X-0(N)), N <= 4000.