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 Ω, 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 ∈ S new 2 (X 0 (N)), N < 4000.

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