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Counting integral points in homogeneous spaces over function fields

Sheng Chen, Jing Liu

TL;DR

The paper studies counting integral points on non-compact symmetric homogeneous spaces X = H\\backslash G over global function fields, showing that the asymptotics are governed by Brauer--Manin obstructions encoded through local densities twisted by elements of Br_{1,P}(X,G). Building on strong approximation, equidistribution results, and the Tamagawa framework, the authors derive an explicit asymptotic formula: N({\\mathcal X}, q^n) \sim r_H q^{(1-\\eta_F)\\dim X} \sum_{\\xi \in Br_{1,P}(X,G)} (\\prod_{v \notin S} I_v(\\mathcal X, \\xi)) I_S(X, q^n, \\xi) as n \to \infty. A key feature is that, over function fields, only Br_{1,P}(X,G) contributes, reflecting the large Brauer group and the resulting Brauer--Manin obstruction structure. This advances the understanding of integral point counts in the function-field setting and provides tools for explicit computations of densities in homogeneous spaces.

Abstract

We establish the asymptotic formula for the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups over global function fields, given by the sum of the products of local densities twisted by suitable Brauer elements.

Counting integral points in homogeneous spaces over function fields

TL;DR

The paper studies counting integral points on non-compact symmetric homogeneous spaces X = H\\backslash G over global function fields, showing that the asymptotics are governed by Brauer--Manin obstructions encoded through local densities twisted by elements of Br_{1,P}(X,G). Building on strong approximation, equidistribution results, and the Tamagawa framework, the authors derive an explicit asymptotic formula: N({\\mathcal X}, q^n) \sim r_H q^{(1-\\eta_F)\\dim X} \sum_{\\xi \in Br_{1,P}(X,G)} (\\prod_{v \notin S} I_v(\\mathcal X, \\xi)) I_S(X, q^n, \\xi) as n \to \infty. A key feature is that, over function fields, only Br_{1,P}(X,G) contributes, reflecting the large Brauer group and the resulting Brauer--Manin obstruction structure. This advances the understanding of integral point counts in the function-field setting and provides tools for explicit computations of densities in homogeneous spaces.

Abstract

We establish the asymptotic formula for the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups over global function fields, given by the sum of the products of local densities twisted by suitable Brauer elements.

Paper Structure

This paper contains 5 sections, 10 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Homogeneous space and Tamagawa measure
  3. Counting integral points
  4. Orbits
  5. The asymptotic formula of $N({\mathcal{X}}, q^n)$

Key Result

Theorem 1.2

Let $G$ be a simply connected and almost $F$-simple linear algebraic group over $F$ such that $G(F_S)$ is not compact. Let $H$ be a subgroup of fixed points of some involution of $G$. Set $X=H\backslash G$ and let $\mathcal{X}$ be a finite-type separated scheme over $\mathcal{O}_S$ whose generic fib as $n\rightarrow \infty$, where $r_H$ is defined in (ri).

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2: Weil82
  • Lemma 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 3.1
  • Lemma 3.2
  • ...and 18 more