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Joint effective equidistribution of partial lattices in positive characteristic

Tal Horesh, Frédéric Paulin

TL;DR

This work establishes effective triple equidistribution results for primitive partial $R_\nu$-lattices in positive characteristic, simultaneously distributing their span in the Grassmannian, their shapes in Bruhat–Tits quotients, and the shapes of their orthogonal complements. The authors develop a refined LU decomposition by blocks for $\operatorname{SL}_{\mathbf d}(K_\nu)$ and adapt Gorodnik–Nevo counting in well-rounded families to control lattice counts with explicit volume-form constants and congruence variants. They introduce measured moduli spaces for Grassmannians, lattices, and correlated lattice pairs, deriving precise mass formulas and distance structures via Plücker coordinates and shape maps. The main results provide both weak-star joint equidistribution and a strengthened formulation with a moduli of paired lattices, yielding effective error terms and broad applicability in positive characteristic, extending prior real-field results to function-field settings. The findings have significant implications for arithmetic dynamics and homogeneous dynamics over function fields, offering toolkits for effective distribution in high-dimensional lattice configurations with congruence structures.

Abstract

Let $ν$ be a place of a global function field $K$ over a finite field, with associated affine function ring $R_ν$ and completion $K_ν$, and let $1 \leq \mathfrak{m}<\textbf{d}$. The aim of this paper is to prove an effective triple joint equidistribution result for primitive partial $R_ν$-lattices $Λ$ of rank $\mathfrak{m}$ in $K_ν^{\;\textbf{d}}$ as their covolume tends to infinity: of their $K_ν$-linear span $V_Λ$ in the rank-$\mathfrak{m}$ Grassmannian space of $K_ν^{\;\textbf{d}}$; of their shape in the modular quotient by $\operatorname{PGL}_\mathfrak{m}(R_ν)$ of the Bruhat-Tits buildings of $\operatorname{PGL}_\mathfrak{m}(K_ν)$; and of the shape of $Λ^\perp$ in the similar quotient for $\operatorname{PGL}_{\textbf{d}-\mathfrak{m}}(K_ν)$, where $Λ^\perp$ is the orthogonal partial $R_ν$-lattice of rank $\textbf{d}-\mathfrak{m}$ in the dual space of $K_ν^{\;\textbf{d}}$. The main tools are a new refined $\text{LU}$ decomposition by blocks of elements of $\operatorname{SL}_\textbf{d}(K_ν)$, 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 $R_ν$-lattices.

Joint effective equidistribution of partial lattices in positive characteristic

TL;DR

This work establishes effective triple equidistribution results for primitive partial -lattices in positive characteristic, simultaneously distributing their span in the Grassmannian, their shapes in Bruhat–Tits quotients, and the shapes of their orthogonal complements. The authors develop a refined LU decomposition by blocks for and adapt Gorodnik–Nevo counting in well-rounded families to control lattice counts with explicit volume-form constants and congruence variants. They introduce measured moduli spaces for Grassmannians, lattices, and correlated lattice pairs, deriving precise mass formulas and distance structures via Plücker coordinates and shape maps. The main results provide both weak-star joint equidistribution and a strengthened formulation with a moduli of paired lattices, yielding effective error terms and broad applicability in positive characteristic, extending prior real-field results to function-field settings. The findings have significant implications for arithmetic dynamics and homogeneous dynamics over function fields, offering toolkits for effective distribution in high-dimensional lattice configurations with congruence structures.

Abstract

Let be a place of a global function field over a finite field, with associated affine function ring and completion , and let . The aim of this paper is to prove an effective triple joint equidistribution result for primitive partial -lattices of rank in as their covolume tends to infinity: of their -linear span in the rank- Grassmannian space of ; of their shape in the modular quotient by of the Bruhat-Tits buildings of ; and of the shape of in the similar quotient for , where is the orthogonal partial -lattice of rank in the dual space of . The main tools are a new refined decomposition by blocks of elements of , 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 -lattices.
Paper Structure (18 sections, 26 theorems, 200 equations)

This paper contains 18 sections, 26 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Background definitions and notation
  3. On global function fields
  4. Partial lattices
  5. Orthogonal primitive partial lattices
  6. Congruence properties on primitive partial lattices
  7. Refined LU decomposition by blocks
  8. Refined LU decomposition by blocks and partial lattices
  9. Metric measured moduli spaces of partial lattices
  10. The Grassmannian spaces
  11. The spaces of unimodular full lattices
  12. The space of correlated pairs of lattices
  13. The spaces of shapes of unimodular full lattices
  14. Joint equidistribution of primitive partial lattices
  15. The correspondence between primitive partial lattices and integral group elements
  16. ...and 3 more sections

Key Result

Corollary 1.1

For the weak-star convergence of Borel measures on the locally compact space ${\widetilde{\Gamma}}_{\mathfrak m}\backslash V_0 {\cal I}_{\nu,{\mathfrak m}}\times {\widetilde{\Gamma}}_{\mathfrak n}\backslash V_0{\cal I}_{\nu,{\mathfrak n}}$, we have

Theorems & Definitions (26)

  • Corollary 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Proposition 2.8
  • ...and 16 more