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A Universal Genus-Two Curve from Siegel Modular Forms
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\mathfrak{p} $ be any point in the moduli space of genus-two curves $\mathcal{M}_2$ and $K$ its field of moduli. We provide a universal equation of a genus-two curve $\mathcal C_{\alpha, \beta}$ defined over $K(\alpha, \beta)$, corresponding to $\mathfrak{p}$, where $\alpha $ and $\beta$ satisfy a quadratic $\alpha^2+ b \beta^2= c$ such that $b$ and $c$ are given in terms of ratios of Siegel modular forms. The curve $\mathcal C_{\alpha, \beta}$ is defined over the field of moduli $K$ if and only if the quadratic has a $K$-rational point $(\alpha, \beta)$. We discover some interesting symmetries of the Weierstrass equation of $\mathcal C_{\alpha, \beta}$. This extends previous work of Mestre and others.
Keywords: genus-two curves; Siegel modular forms. 
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     author = {Andreas Malmendier and Tony Shaska},
     title = {A {Universal} {Genus-Two} {Curve} from {Siegel} {Modular} {Forms}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a88/}
}
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Andreas Malmendier; Tony Shaska. A Universal Genus-Two Curve from Siegel Modular Forms. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a88/

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