A modular equality for $m$-ovoids of elliptic quadrics

An $m$-ovoid of a finite polar space $\mathcal{P}$ is a set $\mathcal{O}$ of points such that every maximal subspace of $\mathcal{P}$ contains exactly $m$ points of $\mathcal{O}$. In the case when $\mathcal{P}$ is an elliptic quadric ${\mathcal{Q}}^{-}(2r+1,q)$ of rank $r$ in ${\mathbb{F}}_q^{2r+2}$, we prove that an $m$-ovoid exists only if $m$ satisfies a certain modular equality, which depends on $q$ and $r$. This condition rules out many of the possible values of $m$. Previously, only a lower bound on $m$ was known, which we slightly improve as a byproduct of our method. We also obtain a characterization of the $m$-ovoids of ${\mathcal{Q}}^{-}(7,q)$ for $q=2$ and $(m,q)=(4,3)$.