Document Type

Article

Journal/Book Title/Conference

Symmetry, Integrability and Geometry: Methods and Applications

Volume

13

Issue

089

Publisher

National Academy of Science of Ukraine

Publication Date

11-30-2017

First Page

1

Last Page

17

Abstract

Let p be any point in the moduli space of genus-two curves M2 and K its field of moduli. We provide a universal equation of a genus-two curve Cα,β defined over K(α, β), corresponding to p, where α and β satisfy a quadratic α2 + bβ2 = c such that b and c are given in terms of ratios of Siegel modular forms. The curve Cα,β is defined over the field of moduli K if and only if the quadratic has a K-rational point (α, β). We discover some interesting symmetries of the Weierstrass equation of Cα,β . This extends previous work of Mestre and others.

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