Joint effective equidistribution of partial lattices in positive characteristic

Abstract

Let $\nu$ be a place of a global function field $K$ over a finite field, with associated affine function ring $R_{\nu }$ and completion $K_{\nu }$, and let $1\le \mathfrak{m}<\text{d}$. The aim of this paper is to prove an effective triple joint equidistribution result for primitive partial $R_{\nu }$-lattices $\Lambda$ of rank $\mathfrak{m}$ in $K_{\nu }^{\text{d}}$ as their covolume tends to infinity: of their $K_{\nu }$-linear span $V_{\Lambda }$ in the rank-$\mathfrak{m}$ Grassmannian space of $K_{\nu }^{\text{d}}$; of their shape in the modular quotient by ${\mathrm{PGL}}_{\mathfrak{m}}(R_{\nu })$ of the Bruhat-Tits buildings of ${\mathrm{PGL}}_{\mathfrak{m}}(K_{\nu })$; and of the shape of ${\Lambda }^{\perp }$ in the similar quotient for ${\mathrm{PGL}}_{\text{d}-\mathfrak{m}}(K_{\nu })$, where ${\Lambda }^{\perp }$ is the orthogonal partial $R_{\nu }$-lattice of rank $\text{d}-\mathfrak{m}$ in the dual space of $K_{\nu }^{\text{d}}$. The main tools are a new refined $\text{LU}$ decomposition by blocks of elements of ${\mathrm{SL}}_{\text{d}}(K_{\nu })$, techniques of Gorodnik and Nevo for counting integral points in well-rounded families of subsets of algebraic groups, and computations of volumes of various homogeneous spaces associated with partial $R_{\nu }$-lattices.