Secondary terms in asymptotics for the number of zeros of quadratic forms over number fields

Abstract: Let $Q$ be a nondegenerate quadratic form on a vector space $V$ of even dimension $n$ over a number field $F$. Via the circle method or automorphic methods one can give good estimates for smoothed sums over the number of zeros of the quadratic form whose coordinates are of size at most $X$ (properly interpreted). For example, when $F=\mathbb{Q}$ and $\dim V>4$ Heath-Brown has given an asymptotic of the form \begin{align} \label{HB:esti} c_1X^{n-2}+O_{Q,\varepsilon,f}(X^{n/2+\varepsilon}) \end{align} for any $\varepsilon>0$. Here $c_1 \in \mathbb{C}$ and $f \in \mathcal{S}(V(\mathbb{R}))$ is a smoothing function. We refine Heath-Brown's work to give an asymptotic of the form $$ c_1X^{n-2}+c_2X^{n/2}+O_{Q,\varepsilon,f}(X^{n/2+\varepsilon-1}) $$ over any number field. Here $c_2 \in \mathbb{C}$. Interestingly the secondary term $c_2$ is the sum of a rapidly decreasing function on $V(\mathbb{R})$ over the zeros of $Q^{\vee}$, the form whose matrix is inverse to the matrix of $Q$. We also prove analogous results in the boundary case $n=4$, generalizing and refining Heath-Brown's work in the case $F=\mathbb{Q}$.