The Circle Method for Quadrics over Function Fields

TL;DR

This paper applies the circle method to count rational points of bounded height on quadrics over function fields, providing explicit formulas depending on various algebraic parameters.

Contribution

It introduces a novel application of the circle method to function field quadrics, deriving exact point counts with secondary terms based on algebraic properties.

Findings

Derived explicit formulas for rational points count on quadrics over function fields.

Established dependence of point counts on parity of dimension and quadratic form determinant.

Included secondary terms in the counting formulas in certain cases.

Abstract

We use the circle method to count $\mathbb{F}_q(t)$-rational points of bounded naive height on a quadric hypersurface $X\subseteq \mathbb{P}^{n-1}$ defined over $\mathbb{F}_q$, provided that $\mathrm{char}(\mathbb{F}_q)>2$ and $n\ge 3$. Viewing these points as morphisms $\mathbb{P}^1 \to X$ of fixed degree, we obtain exact formulas for their number depending on the parity of $n$ and on the determinant of the quadratic form defining $X$, including secondary terms in some cases.