Invariants of models of genus one curves via modular forms and determinantal representations

Preprint, 2019

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 will 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. In the case of the discriminant of ternary cubics over the complex numbers, we perform another approach using determinantal representations with a connection to theta functions. Both of these two approaches link a genus one model to a Weierstrass form.