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Mathematics of Computation

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Non-hyperelliptic modular Jacobians of dimension 3

Author: Roger Oyono
Journal: Math. Comp. 78 (2009), 1173-1191
MSC (2000): Primary 14C34, 14G35; Secondary 11G10, 11F11, 14H42
Published electronically: September 3, 2008
MathSciNet review: 2476578
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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 $ {\rm Jac}(C_f)$ of dimension $ 3$ which are isomorphic to $ A_f$, where $ f\in S_2^{\rm new}(X_0 (N)),$ $ N\leq 4000.$

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Additional Information

Roger Oyono
Affiliation: Équipe GAATI, Université de Polynésie Française, BP 6570, 98702 Faa’a, Tahiti, Polynésie Française

Keywords: Modular curves, modular Jacobians, non-hyperelliptic curves of genus~3, Torelli's theorem, theta functions
Received by editor(s): February 5, 2007
Received by editor(s) in revised form: March 5, 2008
Published electronically: September 3, 2008
Additional Notes: The research of this paper was done while the author was a Ph.D. student at the Institut für Experimentelle Mathematik (IEM) of the university of Essen under the supervision of Gerhard Frey
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.