Pfaffian equations satisfied by differential modular forms

Abstract

The ring of (ordinary) isogeny covariant differential modular forms introduced in (3) was shown in (1) to be described by two basic forms introduced in (3) and (1) respectively. We prove that these two forms satisfy a simple triangular system of Pfaffian equations (in characteristic zero). The equation giving the form in (1) is integrable by quadratures which gives a closed form expression for this form; the equation giving the form in (3) is shown not to be integrable by quadratures.