A Universal Genus-Two Curve from Siegel Modular Forms

Let [...] be any point in the moduli space of genus-two curves [...] and [...] its field of moduli. We provide a universal equation of a genus-two curve [...] defined over [...], corresponding to [...], where [...] and [...] satisfy a quadratic [...] such that [...] and [...] are given in terms of r...

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Bibliographic Details
Published in:Symmetry, integrability and geometry, methods and applications integrability and geometry, methods and applications, 2017-11, Vol.13
Main Authors: Malmendier, Andreas, Shaska, Tony
Format: Article
Language:English
Subjects:
Analytic functions
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Summary:Let [...] be any point in the moduli space of genus-two curves [...] and [...] its field of moduli. We provide a universal equation of a genus-two curve [...] defined over [...], corresponding to [...], where [...] and [...] satisfy a quadratic [...] such that [...] and [...] are given in terms of ratios of Siegel modular forms. The curve [...] is defined over the field of moduli [...] if and only if the quadratic has a [...]-rational point [...]. We discover some interesting symmetries of the Weierstrass equation of [...]. This extends previous work of Mestre and others. [ProQuest: [...] denotes formulae omitted.]
ISSN:1815-0659
1815-0659
DOI:10.3842/SIGMA.2017.089