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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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


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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Additional Information
  • Roger Oyono
  • Affiliation: Équipe GAATI, Université de Polynésie Française, BP 6570, 98702 Faa’a, Tahiti, Polynésie Française
  • Email:
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 78 (2009), 1173-1191
  • MSC (2000): Primary 14C34, 14G35; Secondary 11G10, 11F11, 14H42
  • DOI:
  • MathSciNet review: 2476578