Non-hyperelliptic modular Jacobians of dimension 3

Oyono, Roger

doi:10.1090/S0025-5718-08-02174-1

Non-hyperelliptic modular Jacobians of dimension 3
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Math. Comp. 78 (2009), 1173-1191 Request permission

Abstract:

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 $\mathbb {C}$ 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 $\textrm {Jac}(C_f)$ of dimension $3$ which are isomorphic to $A_f$, where $f\in S_2^\textrm {new}(X_0 (N)),$ $N\leq 4000.$

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