Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 089, 17 pages      arXiv:1607.08294      https://doi.org/10.3842/SIGMA.2017.089
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

A Universal Genus-Two Curve from Siegel Modular Forms

Andreas Malmendier ^a and Tony Shaska ^b
a) Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA
^b) Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA

Received July 18, 2017, in final form November 25, 2017; Published online November 30, 2017

Abstract
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.

Key words: genus-two curves; Siegel modular forms.

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