A Universal Genus-Two Curve from Siegel Modular Forms

Symmetry Integrability and Geometry-Methods and Applications

article

Andreas Malmendier¹  Tony Shaska² 

[1] Department of Mathematics and Statistics, Utah State University;Department of Mathematics and Statistics, Oakland University

关键词: genus-two curves;    Siegel modular forms;   

DOI  :  10.3842/SIGMA.2017.089

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown