2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61259We show that for all finite fields F_q, there exists a curve C over F_q of genus 3 such that the number of rational points on C is within 3 of the Serre-Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over F_q.28 pages including the appendix. Main paper by Kristin Lauter, appendix by Jean-Pierre SerreAlgebraic GeometryNumber Theory14G45 (Primary) 11G10, 14K15, 11D45 (Secondary)text