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Manin's Conjecture for Equivariant compactifications of forms of $\mathbb{G}_a^n$

Abdulmuhsin Alfaraj

TL;DR

This work extends the $Batyrev\text{--}Manin$ framework to smooth equivariant compactifications of forms of $\mathbb{G}_a^n$ over global function fields, handling the positive-characteristic subtleties via harmonic analysis on $G(\mathbb{A}_F)$ and Denef-type local integrals. It establishes the pole structure and growth of the height zeta function, derives the asymptotics for counting points of bounded height in the averaged sense, and verifies Peyre’s leading-constant prediction in this setting by proving the Hasse principle for $G$ and computing Tamagawa measures. The paper also highlights new phenomena unique to function fields, notably purely inseparable boundary behavior in $F$-wound groups, with explicit illustrations in the $X=\mathbb{P}^{p-1}$ case. Together, these results extend the reach of Manin's conjecture to a broader arithmetic-geometry context and illuminate how inseparability influences height counting and constants in positive characteristic. The methods unify harmonic-analytic and algebro-geometric tools to address both asymptotics and constants in a function-field environment, with potential implications for further equvariant-compactification scenarios in positive characteristic.

Abstract

We prove the Batyrev-Manin conjecture for smooth equivariant compactifications of forms of $\mathbb{G}_a^n$ over a global function field $F$, assuming some conditions on the boundary divisor. To verify that the leading constant agrees with Peyre's predicition we also show that a commutative unipotent group admitting a smooth equivariant compactification satisfies the Hasse principle for algebraic groups and weak approximation. We study in detail the case of $\mathbb{P}^{p-1}$, where $p$ is the characteristic of $F$, viewed as a compactification of appropriate $F$-wound groups to illustrate new phenomena appearing in the function field setting.

Manin's Conjecture for Equivariant compactifications of forms of $\mathbb{G}_a^n$

TL;DR

This work extends the framework to smooth equivariant compactifications of forms of over global function fields, handling the positive-characteristic subtleties via harmonic analysis on and Denef-type local integrals. It establishes the pole structure and growth of the height zeta function, derives the asymptotics for counting points of bounded height in the averaged sense, and verifies Peyre’s leading-constant prediction in this setting by proving the Hasse principle for and computing Tamagawa measures. The paper also highlights new phenomena unique to function fields, notably purely inseparable boundary behavior in -wound groups, with explicit illustrations in the case. Together, these results extend the reach of Manin's conjecture to a broader arithmetic-geometry context and illuminate how inseparability influences height counting and constants in positive characteristic. The methods unify harmonic-analytic and algebro-geometric tools to address both asymptotics and constants in a function-field environment, with potential implications for further equvariant-compactification scenarios in positive characteristic.

Abstract

We prove the Batyrev-Manin conjecture for smooth equivariant compactifications of forms of over a global function field , assuming some conditions on the boundary divisor. To verify that the leading constant agrees with Peyre's predicition we also show that a commutative unipotent group admitting a smooth equivariant compactification satisfies the Hasse principle for algebraic groups and weak approximation. We study in detail the case of , where is the characteristic of , viewed as a compactification of appropriate -wound groups to illustrate new phenomena appearing in the function field setting.
Paper Structure (35 sections, 65 theorems, 206 equations)

This paper contains 35 sections, 65 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Main Theorem
  3. The leading constant
  4. Purely inseparable examples
  5. The outline of the paper
  6. Non-trivial forms of $\mathbb{G}_a^n$ in positive characterstic
  7. $F$-wound groups and forms of $\mathbb{G}_a^n$
  8. $F$-wound groups over a global function field
  9. Smooth compactifications of $F$-wound groups
  10. The Picard group of $X$
  11. The Hasse principle and weak approximation
  12. Preliminaries
  13. Proof of \ref{['thm:ShaTrivial']}
  14. An example with no smooth compactification
  15. Metrizations, heights, and measures
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $X$ be a smooth equivariant compactification of $\mathbb{G}_a^n$ over a global function field $F$. Assume that $X\setminus \mathbb{G}_a^n$ has strict normal crossings with smooth irreducible components. Then, for any anti-canonical height function $H_{\omega_X^{-1}}$, the height zeta function $Z

Theorems & Definitions (138)

  • Theorem 1.1
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Remark 1.11
  • ...and 128 more