A nonabelian circle method

TL;DR

This paper introduces a nonabelian delta symbol method to count integral quaternion solutions of specific quadratic forms, providing asymptotic formulas for high dimensions and bounds for lower dimensions, advancing number theory techniques.

Contribution

The paper develops a novel nonabelian delta symbol method and applies it to count quaternion solutions, offering new asymptotic formulas and bounds in higher dimensions.

Findings

Asymptotic formula for solutions when n ≥ 9

Near-optimal bounds for n=8

Introduction of a new nonabelian delta symbol method

Abstract

We count integral quaternion zeros of $\gamma_1^2 \pm \dots \pm \gamma_n^2$, giving an asymptotic when $n\ge 9$, and a likely near-optimal bound when $n=8$. To do so, we introduce a new, nonabelian delta symbol method, which is of independent interest. Our asymptotic at height $X$ takes the form $cX^{4n-8} + O(X^{3n+\varepsilon})$ for suitable $c \in \mathbb{C}$ and any $\varepsilon>0.$ We construct special subvarieties implying that, in general, $3n+\varepsilon$ can be at best improved to $3n-2.$