The Circle Method for Quadrics over Function Fields

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 $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}({\mathbb{F}}_q)>2$ 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.