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

Abstract

An m-ovoid of a finite polar space P is a set O of points such that every maximal subspace of P contains exactly m points of O. In the case when P is an elliptic quadric Q − (2r + 1, q) of rank r in F 2r+2 q , 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 Q − (7, q) for q = 2 and (m, q) = (4, 3).