On Drinfeld modular forms of higher rank VII: Expansions at the boundary

We study expansions of Drinfeld modular forms of rank $r\ge 2$ along the boundary of moduli varieties. Product formulas for the discriminant forms ${\mathrm{\Delta }}_{\mathfrak{n}}$ are developed, which are analogous with Jacobi’s formula for the classical elliptic discriminant. The vanishing orders are described through values at $s=1-r$ of partial zeta functions of the underlying Drinfeld coefficient ring $A$. We show linear independence properties for Eisenstein series, which allow to split spaces of modular forms into the subspaces of cusp forms and of Eisenstein series, and give various characterizations of the boundary condition for modular forms.