Invariants of models of genus one curves and modular forms

TL;DR

This paper explores invariants of genus one curve models, linking classical invariants like the discriminant to modular forms, and provides a new geometric approach to their computation.

Contribution

It offers an alternative expression for normalized invariants of genus one models using modular forms and geometric properties, extending classical invariant theory.

Findings

Invariants of genus one models are generated by two elements.

Normalized invariants can be expressed via modular forms.

A geometric method for computing discriminants is introduced.

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.