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

TL;DR

This paper proves the Batyrev-Manin conjecture for certain smooth equivariant compactifications of algebraic groups over function fields, confirming Peyre's prediction and exploring new phenomena in the function field context.

Contribution

It establishes the conjecture for equivariant compactifications of forms of ^n over function fields and verifies Peyre's constant, also analyzing specific cases like ^{p-1} in characteristic p.

Findings

Confirmed Batyrev-Manin conjecture in the function field setting.

Verified Peyre's constant matches the leading term.

Identified new phenomena in the case of ^{p-1} in characteristic p.

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.