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

TL;DR

This paper generalizes the classical $k/12$-formula to higher rank Drinfeld modular forms, relating intersection multiplicities to zeroes of Eisenstein series, extending understanding of their structure.

Contribution

It derives an analogue of the $k/12$-formula for higher rank Drinfeld modular forms, linking intersection multiplicities to zeroes of Eisenstein series.

Findings

Derived a higher rank $k/12$-formula for Drinfeld modular forms.

Determined common zeroes of specific Eisenstein series.

Extended classical modular form results to higher-dimensional setting.

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 $r \geq 2$. 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 $\nu_{\bo}(f_1,\ldots,f_{r-1})$ 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 $n \geq r$.