Dichotomous point counts over finite fields
arXiv:2202.10427 · doi:10.1016/j.jnt.2023.03.001
Abstract
We establish a near dichotomy between randomness and structure for the point counts of arbitrary projective cubic threefolds over finite fields. Certain "special" subvarieties, not unlike those in the Manin conjectures, dominate. We also prove new general results for projective hypersurfaces. Our work continues a line of inquiry initiated by Hooley.
26 pages; reorganized paper; further improved exposition; minor corrections; accepted version