On Drinfeld modular forms of higher rank III: The analogue of the k/12-formula

Abstract

Continuing the work of \cite{7} and \cite{8}, we derive an analogue of the classical "$k/12$-formula" for Drinfeld modular forms of rank $r\ge 2$. Here the vanishing order ${\nu }_{\omega }(f)$ of one modular form at some point $\omega$ of the complex upper half-plane is replaced by the intersection multiplicity $ν_{\bo}(f_1,\ldots,f_{r-1})$ of $r-1$ independent Drinfeld modular forms at some point $\bo$ of the Drinfeld symmetric space $\OM^r$. We apply the formula to determine the common zeroes of $r-1$ consecutive Eisenstein series $E_{q^i-1}$, where $n-r<i<n$ for some $n\ge r$.