Pfaffian equations satisfied by differential modular forms

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

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Published 1 January 2004