Invariants of models of genus one curves and modular forms

2019/11/04 by Manh Hung Tran, Tran, Manh Hung
Computer Science · Mathematics · #11F11 #11G05 #11Y40 #13A50 #14H50 #14Q05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1911.01350

openalex publication_date 2019/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An invariant of a model of genus one curve is a polynomial in the coefficients of the model that is stable under certain linear transformations. The classical example of an invariant is the discriminant, which characterizes the singularity of models. The ring of invariants of genus one models over a field is generated by two elements. Fisher normalized these invariants for models of degree n=2,3,4 in such a way that these invariants are moreover defined over the integers. We provide an alternative way to express these normalized invariants using modular forms. This method relies on a direct computation for the discriminants based on their own geometric properties.

Related

• Modular forms via invariant theory
• Invariant theory for the elliptic normal quintic, II. The covering map
• Capitulation discriminants of genus one curves
• Classical and minimal models of the moduli space of curves of genus two
• The structure of regular genus-one curves over imperfect fields